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Learner Teacher Parent [Statistics and probability16 units · 157 skills](https://www.khanacademy.org/math/statistics-probability) [Unit 1Analyzing categorical data](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data) [Unit 2Displaying and comparing quantitative data](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data) [Unit 3Summarizing quantitative data](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data) [Unit 4Modeling data distributions](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data) [Unit 5Exploring bivariate numerical data](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data) [Unit 6Study design](https://www.khanacademy.org/math/statistics-probability/designing-studies) [Unit 7Probability](https://www.khanacademy.org/math/statistics-probability/probability-library) [Unit 8Counting, permutations, and combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations) [Unit 9Random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library) [Unit 10Sampling distributions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library) [Unit 11Confidence intervals](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample) [Unit 12Significance tests (hypothesis testing)](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample) [Unit 13Two-sample inference for the difference between groups](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples) [Unit 14Inference for categorical data (chi-square tests)](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests) [Unit 15Advanced regression (inference and transforming)](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming) [Unit 16Analysis of variance (ANOVA)](https://www.khanacademy.org/math/statistics-probability/analysis-of-variance-anova-library) Course challenge Test your knowledge of the skills in this course.[Start Course challenge](https://www.khanacademy.org/math/statistics-probability/test/x83aef204:course-challenge) 1. [Math](https://www.khanacademy.org/math) # Statistics and probability 15,200 possible mastery points Mastered Proficient Familiar Attempted Not started Quiz Unit test 1. [Unit 1Unit 1: Analyzing categorical data](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data) 1. [Individuals, variables, and categorical & quantitative data](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/e/individuals-variables-categorical-quantitative-data) 2. [Read picture graphs (multi-step problems)](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/e/reading_pictographs_2) 3. [Create bar graphs](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/e/creating_bar_charts_1) 4. [Read bar graphs (2-step problems)](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/e/reading_bar_charts_2) 5. [Analyzing categorical data: Quiz 1](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/quiz/analyzing-categorical-data-quiz-1) 6. [Read two-way frequency tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/reading-two-way-frequency-tables) 7. [Create two-way frequency tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/two-way-frequency-tables) 8. [Create two-way relative frequency tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/two-way-relative-frequency-tables) 9. [Analyze two-way frequency tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/reading-two-way-relative-frequency-tables) 10. [Interpret two-way tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/frequencies-of-bivariate-data) 11. [Trends in categorical data](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/trends-in-categorical-data) 12. [Analyzing categorical data: Quiz 2](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/quiz/analyzing-categorical-data-quiz-2) 13. [Identify marginal and conditional distributions](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/distributions-in-two-way-tables/e/identifying-marginal-conditional-distributions) 14. [Marginal distributions](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/distributions-in-two-way-tables/e/calculating-marginal-distributions) 15. [Conditional distributions](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/distributions-in-two-way-tables/e/calculating-conditional-distributions) 16. [Analyzing categorical data: Quiz 3](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/distributions-in-two-way-tables/quiz/analyzing-categorical-data-quiz-3) 17. [Analyzing categorical data: Unit test](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/distributions-in-two-way-tables/test/analyzing-categorical-data-unit-test) 2. [Unit 2Unit 2: Displaying and comparing quantitative data](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data) 1. [Creating frequency tables](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/creating-frequency-tables) 2. [Creating dot plots](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/creating-dot-plots) 3. [Reading dot plots & frequency tables](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/analyzing-with-dot-plots) 4. [Create histograms](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/creating-histograms) 5. [Read histograms](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/reading-histograms) 6. [Reading stem and leaf plots](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/reading_stem_and_leaf_plots) 7. [Displaying and comparing quantitative data: Quiz 1](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/quiz/displaying-describing-data-quiz-1) 8. [Shape of distributions](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/shape-of-distributions) 9. [Clusters, gaps, & peaks in data distributions](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/clusters--gaps--peaks--and-outliers) 10. [Comparing distributions](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/comparing-populations) 11. [Comparing data displays](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/comparing-data-displays) 12. [Comparing data distributions](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/interpreting-and-comparing-data-distributions) 13. [Comparing center and spread](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/comparing-distribution) 14. [Displaying and comparing quantitative data: Quiz 2](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/quiz/displaying-describing-data-quiz-2) 15. [Displaying and comparing quantitative data: Unit test](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/more-on-data-displays/test/displaying-describing-data-unit-test) 3. [Unit 3Unit 3: Summarizing quantitative data](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data) 1. [Mean, median, and mode](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/e/mean_median_and_mode) 2. [Calculating the mean](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/e/calculating-the-mean) 3. [Calculating the median](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/e/calculating-the-median) 4. [Summarizing quantitative data: Quiz 1](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/quiz/summarizing-quantitative-data-quiz-1) 5. [Calculating the mean: data displays](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/e/calculating-the-mean-from-various-data-displays) 6. [Calculating the median: data displays](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/e/calculating-the-median-from-data-displays) 7. [Effects of shifting, adding, & removing a data point](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/e/effects-of-shifting-adding-removing-data-point) 8. [Missing value given the mean](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/e/find-a-missing-value-given-the-mean) 9. [Median & range puzzlers](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/e/median-and-range-puzzlers) 10. [Summarizing quantitative data: Quiz 2](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/quiz/summarizing-quantitative-data-quiz-2) 11. [Interquartile range (IQR)](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/interquartile-range-iqr/e/calculating-the-interquartile-range--iqr-) 12. [Standard deviation of a population](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/e/standard_deviation_of_a_population) 13. [Variance](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/e/variance) 14. [Sample and population standard deviation](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/e/standard_deviation) 15. [Summarizing quantitative data: Quiz 3](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/quiz/summarizing-quantitative-data-quiz-3) 16. [Creating box plots](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/e/box-plots) 17. [Reading box plots](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/e/analyzing-data-with-box-plots) 18. [Interpreting quartiles](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/e/interpreting-quartiles-on-box-plots) 19. [Identifying outliers](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/e/identifying-outliers) 20. [Mean absolute deviation (MAD)](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/other-measures-of-spread/e/calculating-the-mean-absolute-deviation--mad-) 21. [Summarizing quantitative data: Quiz 4](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/other-measures-of-spread/quiz/summarizing-quantitative-data-quiz-4) 22. [Summarizing quantitative data: Unit test](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/other-measures-of-spread/test/summarizing-quantitative-data-unit-test) 4. [Unit 4Unit 4: Modeling data distributions](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data) 1. [Calculate percentiles](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/percentiles/e/calculating-percentiles) 2. [Calculating z-scores](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/z-scores/e/z_scores_1) 3. [Transforming data](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/effects-of-linear-transformations/e/transforming-data) 4. [Properties of density curves](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/density-curve/e/density-curves) 5. [Area under density curves](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/density-curve/e/area-under-density-curves) 6. [Modeling data distributions: Quiz 1](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/density-curve/quiz/modeling-distributions-of-data-quiz-1) 7. [Empirical rule](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distributions-library/e/empirical_rule) 8. [Normal distribution: Area above or below a point](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/e/z_scores_2) 9. [Normal distribution: Area between two points](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/e/z_scores_3) 10. [Normal calculations in reverse](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/e/normal-calculations-3) 11. [Modeling data distributions: Quiz 2](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/quiz/modeling-distributions-of-data-quiz-2) 12. [Modeling data distributions: Unit test](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/more-on-normal-distributions/test/modeling-distributions-of-data-unit-test) 5. [Unit 5Unit 5: Exploring bivariate numerical data](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data) 1. [Constructing scatter plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/e/constructing-scatter-plots) 2. [Making appropriate scatter plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/e/making-good-scatter-plots) 3. [Positive and negative linear associations from scatter plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/e/positive-and-negative-linear-correlations-from-scatter-plots) 4. [Describing trends in scatterplots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/e/interpreting-scatter-plots) 5. [Correlation coefficient intuition](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/scatterplots-and-correlation/e/correlation-coefficient-intuition) 6. [Exploring bivariate numerical data: Quiz 1](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/scatterplots-and-correlation/quiz/describing-relationships-quantitative-data-quiz-1) 7. [Eyeballing the line of best fit](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/e/plotting_the_line_of_best_fit) 8. [Estimating equations of lines of best fit, and using them to make predictions](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/e/equations-of-lines-of-best-fit-to-make-predictions) 9. [Estimating slope of line of best fit](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/e/linear-models-of-bivariate-data) 10. [Exploring bivariate numerical data: Quiz 2](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/quiz/describing-relationships-quantitative-data-quiz-2) 11. [Calculating and interpreting residuals](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/e/calculating-interpreting-residuals) 12. [Calculating the equation of the least-squares line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/e/calculating-equation-least-squares) 13. [Interpreting slope and y-intercept for linear models](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/e/interpreting-slope-and-y-intercept-of-lines-of-best-fit) 14. [Exploring bivariate numerical data: Quiz 3](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/quiz/describing-relationships-quantitative-data-quiz-3) 15. [Residual plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/e/residual-plots) 16. [Effects of influential points](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/e/influential-points) 17. [Exploring bivariate numerical data: Quiz 4](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/quiz/describing-relationships-quantitative-data-quiz-4) 18. [Exploring bivariate numerical data: Unit test](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/test/describing-relationships-quantitative-data-unit-test) 6. [Unit 6Unit 6: Study design](https://www.khanacademy.org/math/statistics-probability/designing-studies) 1. [Statistical questions](https://www.khanacademy.org/math/statistics-probability/designing-studies/statistics-overview/e/statistical-questions) 2. [Valid claims](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/e/valid-claims) 3. [Making inferences from random samples](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/e/making-inferences-from-random-samples) 4. [Identify the population and sample](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/e/identifying-population-sample) 5. [Study design: Quiz 1](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/quiz/designing-studies-quiz-1) 6. [Using probability to make fair decisions](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/e/using-probability-to-make-fair-decisions) 7. [Simple random samples](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/e/simple-random-samples) 8. [Sampling methods](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/e/sampling-methods) 9. [Study design: Quiz 2](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/quiz/designing-studies-quiz-2) 10. [Types of statistical studies](https://www.khanacademy.org/math/statistics-probability/designing-studies/types-studies-experimental-observational/e/types-of-statistical-studies) 11. [Experiment designs](https://www.khanacademy.org/math/statistics-probability/designing-studies/experiments-stats-library/e/experiment-designs) 12. [Study design: Quiz 3](https://www.khanacademy.org/math/statistics-probability/designing-studies/experiments-stats-library/quiz/designing-studies-quiz-3) 13. [Study design: Unit test](https://www.khanacademy.org/math/statistics-probability/designing-studies/experiments-stats-library/test/designing-studies-unit-test) 7. [Unit 7Unit 7: Probability](https://www.khanacademy.org/math/statistics-probability/probability-library) 1. [Simple probability](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/e/probability_1) 2. [Comparing probabilities](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/e/understanding-probability) 3. [Subsets of sample spaces](https://www.khanacademy.org/math/statistics-probability/probability-library/probability-sample-spaces/e/describing-subsets-of-sample-spaces) 4. [Basic set notation](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-ops/e/basic_set_notation) 5. [Probability: Quiz 1](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-ops/quiz/probability-library-quiz-1) 6. [Experimental probability](https://www.khanacademy.org/math/statistics-probability/probability-library/experimental-probability-lib/e/finding-probability) 7. [Making predictions with probability](https://www.khanacademy.org/math/statistics-probability/probability-library/experimental-probability-lib/e/using-probability-to-make-predictions) 8. [Interpret results of simulations](https://www.khanacademy.org/math/statistics-probability/probability-library/randomness-probability-and-simulation/e/interpreting-results-simulations) 9. [Adding probabilities](https://www.khanacademy.org/math/statistics-probability/probability-library/addition-rule-lib/e/adding-probability) 10. [Two-way tables, Venn diagrams, and probability](https://www.khanacademy.org/math/statistics-probability/probability-library/addition-rule-lib/e/two-way-tables-venn-diagrams-probability) 11. [Probability: Quiz 2](https://www.khanacademy.org/math/statistics-probability/probability-library/addition-rule-lib/quiz/probability-library-quiz-2) 12. [Sample spaces for compound events](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/e/sample-spaces-for-compound-events) 13. [Independent probability](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/e/independent_probability) 14. [Probabilities of compound events](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/e/compound-events) 15. [Probability of "at least one" success](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/e/probability-at-least-one-success) 16. [Dependent probability](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-dependent/e/dependent_probability) 17. [Calculate conditional probability](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/e/calculating-conditional-probability) 18. [Dependent and independent events](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/e/identifying-dependent-and-independent-events) 19. [Probability: Quiz 3](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/quiz/probability-library-quiz-3) 20. [Probability: Unit test](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/test/probability-library-unit-test) 8. [Unit 8Unit 8: Counting, permutations, and combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations) 1. [The counting principle](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/counting-principle-factorial/e/fundamental-counting-principle) 2. [Permutations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/permutation-lib/e/permutations_1) 3. [Combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinations-lib/e/combinations_1) 4. [Permutations & combinations with overcounting](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinations-lib/e/permutations_and_combinations_2) 5. [Probability with permutations and combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/e/probability_with_perm_comb) 6. [Counting, permutations, and combinations: Unit test](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/test/counting-permutations-and-combinations-unit-test) 9. [Unit 9Unit 9: Random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library) 1. [Constructing probability distributions](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/constructing-probability-distributions) 2. [Probability models](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/probability-models) 3. [Probability with discrete random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/probability-discrete-random-variables) 4. [Expected value](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/expected_value) 5. [Mean (expected value) of a discrete random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/mean-expected-value-discrete-random-variable) 6. [Standard deviation of a discrete random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/standard-deviation-discrete-random-variable) 7. [Probability in density curves](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-continuous/e/probability-density-curves) 8. [Probability in normal density curves](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-continuous/e/probability-normal-density-curves) 9. [Transforming random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/transforming-random-variable/e/transforming-random-variables) 10. [Combining random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/e/combining-random-variables) 11. [Combining normal random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/e/combine-normal-random-variables) 12. [Identifying binomial variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/e/identifying-binomial-variables) 13. [Binomial probability formula](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/e/binomial-probability) 14. [Calculating binomial probability](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/e/calculating-binomial-probability) 15. [Mean and standard deviation of a binomial random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-mean-standard-dev-formulas/e/mean-standard-deviation-binomial-variable) 16. [Binomial vs. geometric random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/e/binomial-vs-geometric-variables) 17. [Geometric probability](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/e/geometric-probability) 18. [Cumulative geometric probability](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/e/cumulative-geometric-probability) 19. [Expected value with empirical probabilities](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/e/expected-value-with-empirical-probabilities) 20. [Expected value with calculated probabilities](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/e/expected-value-with-calculated-probabilities) 21. [Making decisions with expected values](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/e/making-decisions-with-expected-values) 22. [Random variables: Unit test](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/poisson-distribution/test/random-variables-stats-library-unit-test) 10. [Unit 10Unit 10: Sampling distributions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library) 1. [Biased and unbiased estimators](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/what-is-a-sampling-distribution/e/biased-unbiased-estimators) 2. [The normal condition for sample proportions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/e/normal-condition-sample-proportions) 3. [Mean and standard deviation of sample proportions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/e/sampling-distribution-sample-proportion-mean-standard-deviation) 4. [Finding probabilities with sample proportions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/e/finding-probabilities-sample-proportions) 5. [Mean and standard deviation of sample means](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/e/mean-standard-deviation-sample-means) 6. [Sample means and the central limit theorem](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/e/sample-means-central-limit-theorem) 7. [Finding probabilities with sample means](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/e/finding-probabilities-sample-means) 8. [Sampling distributions: Unit test](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/test/sampling-distributions-library-unit-test) 11. [Unit 11Unit 11: Confidence intervals](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample) 1. [Conditions for a z interval for a proportion](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/e/conditions-one-sample-z-interval-proportion) 2. [Finding the critical value z\* for a desired confidence level](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/e/finding-critical-value-z-for-confidence-level) 3. [Calculating a z interval for a proportion](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/e/calculating-one-sample-z-interval-proportion) 4. [Sample size and margin of error in a z interval for p](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/e/sample-size-margin-error-one-sample-z-interval) 5. [Conditions for a t interval for a mean](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/e/conditions-one-sample-t-interval-mean) 6. [Finding the critical value t\* for a desired confidence level](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/e/finding-critical-value-t) 7. [Calculating a t interval for a mean](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/e/calculating-one-sample-t-interval-mean) 8. [Sample size and margin of error in a confidence interval for a mean](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/e/sample-size-margin-error-one-sample-t-interval-mean) 9. [Confidence intervals: Unit test](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/old-confidence-interval-videos/test/confidence-intervals-one-sample-unit-test) 12. [Unit 12Unit 12: Significance tests (hypothesis testing)](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample) 1. [Simple hypothesis testing](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/e/hypothesis-testing-with-simulations) 2. [Writing null and alternative hypotheses](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/e/writing-null-and-alternative-hypotheses-informal) 3. [Estimating P-values from simulations](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/e/estimating-p-values-and-making-conclusions) 4. [Type I vs Type II error](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/error-probabilities-and-power/e/type-i-error-type-ii-error-power) 5. [Error probabilities and power](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/error-probabilities-and-power/e/error-probabilities-power) 6. [Writing hypotheses for a test about a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/e/writing-hypotheses-one-sample-z-test-proportion) 7. [Conditions for a z test about a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/e/checking-conditions-one-sample-z-test-proportion) 8. [Calculating the test statistic in a z test for a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/e/test-stat-one-sample-z-test-p) 9. [Calculating the P-value in a z test for a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/e/p-value-one-sample-z-test-p) 10. [Making conclusions in a z test for a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/e/conclusions-one-sample-z-test-proportion) 11. [Writing hypotheses for a test about a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/e/writing-hypotheses-one-sample-t-test-mean) 12. [Conditions for a t test about a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/e/checking-conditions-one-sample-t-test-mean) 13. [Calculating the test statistic in a t test for a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/e/test-statistic-t-test) 14. [Calculating the P-value in a t test for a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/e/p-value-t-test-mean) 15. [Making conclusions in a t test for a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/e/conclusions-one-sample-t-test-mean) 16. [Significance tests (hypothesis testing): Unit test](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/more-significance-testing-videos/test/significance-tests-one-sample-unit-test) 13. [Unit 13Unit 13: Two-sample inference for the difference between groups](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples) This unit's exercises do not count toward course mastery. 14. [Unit 14Unit 14: Inference for categorical data (chi-square tests)](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests) 1. [Expected counts in a goodness-of-fit test](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/e/expected-counts-chi-square-goodness-of-fit-test) 2. [Conditions for a goodness-of-fit test](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/e/checking-conditions-chi-square-goodness-of-fit-test) 3. [Test statistic and P-value in a goodness-of-fit test](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/e/test-statistic-p-value-chi-square-goodness-of-fit-test) 4. [Conclusions in a goodness-of-fit test](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/e/conclusions-chi-square-goodness-of-fit-test) 5. [Expected counts in chi-squared tests with two-way tables](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/e/expected-counts-chi-square-test-two-way-tables) 6. [Test statistic and P-value in chi-square tests with two-way tables](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/e/test-statistics-p-values-chi-square-tests-two-way-tables) 7. [Making conclusions in chi-square tests for two-way tables](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/e/conclusions-chi-square-tests-for-two-way-tables) 8. [Inference for categorical data (chi-square tests): Unit test](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/test/inference-categorical-data-chi-square-tests-unit-test) 15. [Unit 15Unit 15: Advanced regression (inference and transforming)](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming) This unit's exercises do not count toward course mastery. 16. [Unit 16Unit 16: Analysis of variance (ANOVA)](https://www.khanacademy.org/math/statistics-probability/analysis-of-variance-anova-library) This unit does not include exercises. 17. Course challenge Test your knowledge of the skills in this course.[Start Course challenge](https://www.khanacademy.org/math/statistics-probability/test/x83aef204:course-challenge) [Unit 1: Analyzing categorical data](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data) Unit mastery: 0% ### Analyzing one categorical variable 1. [Identifying individuals, variables and categorical variables in a data set](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/v/identifying-individuals-variables-and-categorical-variables-in-a-data-set) 2. [Individuals, variables, and categorical & quantitative data](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/e/individuals-variables-categorical-quantitative-data) 3. [Reading pictographs](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/v/reading-pictographs) 4. [Read picture graphs (multi-step problems)](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/e/reading_pictographs_2) 5. [Reading bar graphs](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/v/reading-bar-graphs) 6. [Reading bar graphs: Harry Potter](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/v/reading-bar-charts-1) 7. [Creating a bar graph](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/v/creating-bar-charts-1) 8. [Create bar graphs](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/e/creating_bar_charts_1) 9. [Reading bar charts: comparing two sets of data](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/v/reading-bar-charts-2) 10. [Read bar graphs (2-step problems)](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/e/reading_bar_charts_2) 11. [Reading bar charts: putting it together with central tendency](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/v/reading-bar-charts-3) 12. [Reading pie graphs (circle graphs)](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/v/reading-pie-graphs-circle-graphs) 13. [Picture graphs (pictographs) review](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/a/picture-graphs-review) 14. [Bar graphs review](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/one-categorical-variable/a/bar-graphs-review) ### Two-way tables 1. [Two-way frequency tables and Venn diagrams](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/v/two-way-frequency-tables-and-venn-diagrams) 2. [Two-way frequency tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/a/two-way-frequency-tables) 3. [Read two-way frequency tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/reading-two-way-frequency-tables) 4. [Create two-way frequency tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/two-way-frequency-tables) 5. [Two-way relative frequency tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/v/two-way-relative-frequency-tables) 6. [Create two-way relative frequency tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/two-way-relative-frequency-tables) 7. [Analyze two-way frequency tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/reading-two-way-relative-frequency-tables) 8. [Interpreting two-way tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/v/interpreting-two-way-tables) 9. [Interpret two-way tables](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/frequencies-of-bivariate-data) 10. [Categorical data example](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/v/video-games-and-violence-bivariate-data) 11. [Analyzing trends in categorical data](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/v/analyzing-trends-categorical-data) 12. [Trends in categorical data](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/e/trends-in-categorical-data) 13. [Two-way relative frequency tables and associations](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/a/two-way-relative-frequency-tables-super-powers) 14. [Two-way tables review](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/a/two-way-tables-review) ### Distributions in two-way tables 1. [Marginal and conditional distributions](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/distributions-in-two-way-tables/v/marginal-distribution-and-conditional-distribution) 2. [Identify marginal and conditional distributions](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/distributions-in-two-way-tables/e/identifying-marginal-conditional-distributions) 3. [Marginal distributions](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/distributions-in-two-way-tables/e/calculating-marginal-distributions) 4. [Conditional distributions](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/distributions-in-two-way-tables/e/calculating-conditional-distributions) 5. [Conditional distributions and relationships](https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/distributions-in-two-way-tables/a/conditional-distributions-relationships-ap) [Unit 2: Displaying and comparing quantitative data](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data) Unit mastery: 0% ### Displaying quantitative data with graphs 1. [Representing data](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/v/ways-to-represent-data) 2. [Frequency tables & dot plots](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/v/frequency-tables-and-dot-plots) 3. [Creating frequency tables](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/creating-frequency-tables) 4. [Creating dot plots](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/creating-dot-plots) 5. [Reading dot plots & frequency tables](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/analyzing-with-dot-plots) 6. [Dot plots and frequency tables review](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/a/dot-plots-and-frequency-tables-review) 7. [Creating a histogram](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/v/histograms-intro) 8. [Histograms](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/v/histograms) 9. [Interpreting a histogram](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/v/interpreting-histograms) 10. [Create histograms](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/creating-histograms) 11. [Read histograms](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/reading-histograms) 12. [Histograms review](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/a/histograms-review) 13. [Stem-and-leaf plots](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/v/u08-l1-t2-we3-stem-and-leaf-plots) 14. [Reading stem and leaf plots](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/v/reading-stem-and-leaf-plots) 15. [Reading stem and leaf plots](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/e/reading_stem_and_leaf_plots) 16. [Stem and leaf plots review](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/quantitative-data-graphs/a/stem-and-leaf-plots-review) ### Describing and comparing distributions 1. [Shapes of distributions](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/v/shapes-of-distributions) 2. [Shape of distributions](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/shape-of-distributions) 3. [Clusters, gaps, peaks & outliers](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/v/examples-analyzing-clusters-gaps-peaks-and-outliers-for-distributions) 4. [Clusters, gaps, & peaks in data distributions](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/clusters--gaps--peaks--and-outliers) 5. [Comparing distributions with dot plots (example problem)](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/v/comparing-swim-times-at-the-olympics) 6. [Comparing distributions](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/comparing-populations) 7. [Comparing dot plots, histograms, and box plots](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/v/comparing-dot-plots-histograms-and-box-plots) 8. [Comparing data displays](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/comparing-data-displays) 9. [Example: Comparing distributions](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/v/comparing-distributions) 10. [Comparing data distributions](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/interpreting-and-comparing-data-distributions) 11. [Comparing center and spread](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/comparing-features-distributions/e/comparing-distribution) ### More on data displays 1. [Reading line graphs](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/more-on-data-displays/v/u08-l1-t2-we2-reading-line-graphs) 2. [Misleading line graphs](https://www.khanacademy.org/math/statistics-probability/displaying-describing-data/more-on-data-displays/v/misleading-line-graphs) [Unit 3: Summarizing quantitative data](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data) Unit mastery: 0% ### Measuring center in quantitative data 1. [Statistics intro: Mean, median, & mode](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/v/statistics-intro-mean-median-and-mode) 2. [Mean, median, & mode example](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/v/mean-median-and-mode) 3. [Mean, median, and mode](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/e/mean_median_and_mode) 4. [Calculating the mean](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/a/calculating-the-mean) 5. [Calculating the mean](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/e/calculating-the-mean) 6. [Calculating the median](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/e/calculating-the-median) 7. [Choosing the "best" measure of center](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/a/choosing-the-best-measure-of-center) 8. [Mean, median, and mode review](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/a/mean-median-and-mode-review) ### More on mean and median 1. [Calculating the mean: data displays](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/e/calculating-the-mean-from-various-data-displays) 2. [Calculating the median: data displays](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/e/calculating-the-median-from-data-displays) 3. [Comparing means of distributions](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/v/comparing-distribution-means) 4. [Means and medians of different distributions](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/v/comparing-means-and-medians) 5. [Impact on median & mean: removing an outlier](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/v/impact-on-median-and-mean-when-removing-lowest-value-example) 6. [Impact on median & mean: increasing an outlier](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/v/impact-on-median-and-mean-when-increasing-highest-value) 7. [Effects of shifting, adding, & removing a data point](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/e/effects-of-shifting-adding-removing-data-point) 8. [Mean as the balancing point](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/a/mean-as-the-balancing-point) 9. [Missing value given the mean](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/v/using-mean-to-find-missing-value) 10. [Missing value given the mean](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/e/find-a-missing-value-given-the-mean) 11. [Median & range puzzlers](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/v/median-and-range-puzzle) 12. [Median & range puzzlers](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-mean-median/e/median-and-range-puzzlers) ### Interquartile range (IQR) 1. [Interquartile range (IQR)](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/interquartile-range-iqr/v/calculating-interquartile-range-iqr) 2. [Interquartile range (IQR)](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/interquartile-range-iqr/e/calculating-the-interquartile-range--iqr-) 3. [Comparing range and interquartile range (IQR)](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/interquartile-range-iqr/a/range-and-interquartile-range-worksheet) 4. [Interquartile range review](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/interquartile-range-iqr/a/interquartile-range-review) ### Variance and standard deviation of a population 1. [Measures of spread: range, variance & standard deviation](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/v/range-variance-and-standard-deviation-as-measures-of-dispersion) 2. [Variance of a population](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/v/variance-of-a-population) 3. [Population standard deviation](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/v/population-standard-deviation) 4. [The idea of spread and standard deviation](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/a/introduction-to-standard-deviation) 5. [Calculating standard deviation step by step](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/a/calculating-standard-deviation-step-by-step) 6. [Standard deviation of a population](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/e/standard_deviation_of_a_population) 7. [Mean and standard deviation versus median and IQR](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/v/mean-and-standard-deviation-versus-median-and-iqr) 8. [Concept check: Standard deviation](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/a/concept-check-standard-deviation) 9. [Statistics: Alternate variance formulas](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/v/statistics-alternate-variance-formulas) ### Variance and standard deviation of a sample 1. [Sample variance](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/v/sample-variance) 2. [Sample standard deviation and bias](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/v/sample-standard-deviation-and-bias) 3. [Variance](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/e/variance) 4. [Sample and population standard deviation](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/e/standard_deviation) 5. [Population and sample standard deviation review](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/a/population-and-sample-standard-deviation-review) ### More on standard deviation 1. [Why we divide by n - 1 in variance](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-on-standard-deviation/v/another-simulation-giving-evidence-that-n-1-gives-us-an-unbiased-estimate-of-variance) 2. [Simulation showing bias in sample variance](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-on-standard-deviation/v/simulation-showing-bias-in-sample-variance) 3. [Simulation providing evidence that (n-1) gives us unbiased estimate](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-on-standard-deviation/v/simulation-providing-evidence-that-n-1-gives-us-unbiased-estimate) 4. [Unbiased estimate of population variance](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-on-standard-deviation/a/unbiased-estimate-of-population-variance) 5. [Review and intuition why we divide by n-1 for the unbiased sample variance](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/more-on-standard-deviation/v/review-and-intuition-why-we-divide-by-n-1-for-the-unbiased-sample-variance) ### Box and whisker plots 1. [Worked example: Creating a box plot (odd number of data points)](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/v/box-and-whisker-plot-exercise-example) 2. [Worked example: Creating a box plot (even number of data points)](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/v/another-example-constructing-box-plot) 3. [Constructing a box plot](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/v/constructing-a-box-and-whisker-plot) 4. [Creating box plots](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/e/box-plots) 5. [Reading box plots](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/v/reading-box-and-whisker-plots) 6. [Reading box plots](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/e/analyzing-data-with-box-plots) 7. [Interpreting box plots](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/v/interpreting-box-plots) 8. [Interpreting quartiles](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/e/interpreting-quartiles-on-box-plots) 9. [Box plot review](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/a/box-plot-review) 10. [Judging outliers in a dataset](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/v/judging-outliers-in-a-dataset) 11. [Identifying outliers](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/e/identifying-outliers) 12. [Identifying outliers with the 1.5xIQR rule](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/a/identifying-outliers-iqr-rule) ### Other measures of spread 1. [Range and mid-range](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/other-measures-of-spread/v/range-and-mid-range) 2. [Mean absolute deviation (MAD)](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/other-measures-of-spread/v/mean-absolute-deviation) 3. [Mean absolute deviation example](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/other-measures-of-spread/v/mean-absolute-deviation-example) 4. [Mean absolute deviation (MAD)](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/other-measures-of-spread/e/calculating-the-mean-absolute-deviation--mad-) 5. [Mean absolute deviation (MAD) review](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/other-measures-of-spread/a/mean-absolute-deviation-mad-review) [Unit 4: Modeling data distributions](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data) Unit mastery: 0% ### Percentiles 1. [Calculating percentile](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/percentiles/v/calculating-percentile) 2. [Calculate percentiles](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/percentiles/e/calculating-percentiles) 3. [Analyzing a cumulative relative frequency graph](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/percentiles/v/analyzing-a-cumulative-relative-frequency-graph) 4. [Cumulative relative frequency graph problem](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/percentiles/a/cumulative-relative-frequency-graph-problem) ### Z-scores 1. [Normal distribution problem: z-scores (from ck12.org)](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/z-scores/v/ck12-org-normal-distribution-problems-z-score) 2. [Comparing with z-scores](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/z-scores/v/comparing-with-z-scores) 3. [Calculating z-scores](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/z-scores/e/z_scores_1) 4. [Z-scores review](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/z-scores/a/z-scores-review) ### Effects of linear transformations 1. [How parameters change as data is shifted and scaled](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/effects-of-linear-transformations/v/how-parameters-change-as-data-is-shifted-and-scaled) 2. [Transforming data](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/effects-of-linear-transformations/e/transforming-data) 3. [Transforming data problem](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/effects-of-linear-transformations/a/transforming-data-problem) ### Density curves 1. [Density Curves](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/density-curve/v/density-curves) 2. [Median, mean and skew from density curves](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/density-curve/v/median-mean-and-skew-from-density-curves) 3. [Density curve worked example](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/density-curve/v/density-curve-worked-example) 4. [Properties of density curves](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/density-curve/e/density-curves) 5. [Worked example finding area under density curves](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/density-curve/v/worked-example-finding-area-under-density-curves) 6. [Area under density curves](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/density-curve/e/area-under-density-curves) ### Normal distributions and the empirical rule 1. [Qualitative sense of normal distributions](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distributions-library/v/ck12-org-normal-distribution-problems-qualitative-sense-of-normal-distributions) 2. [Normal distribution problems: Empirical rule](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distributions-library/v/ck12-org-normal-distribution-problems-empirical-rule) 3. [Standard normal distribution and the empirical rule (from ck12.org)](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distributions-library/v/ck12-org-exercise-standard-normal-distribution-and-the-empirical-rule) 4. [More empirical rule and z-score practice (from ck12.org)](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distributions-library/v/ck12-org-more-empirical-rule-and-z-score-practice) 5. [Empirical rule](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distributions-library/e/empirical_rule) 6. [Normal distributions review](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distributions-library/a/normal-distributions-review) ### Normal distribution calculations 1. [Standard normal table for proportion below](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/v/z-table-for-proportion-below) 2. [Standard normal table for proportion above](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/v/z-table-for-proportion-above) 3. [Standard normal table for proportion between values](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/v/z-table-for-proportion-between-values) 4. [Normal distribution: Area above or below a point](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/e/z_scores_2) 5. [Normal distribution: Area between two points](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/e/z_scores_3) 6. [Finding z-score for a percentile](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/v/finding-z-score-for-a-percentile) 7. [Threshold for low percentile](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/v/threshold-for-low-percentile) 8. [Normal calculations in reverse](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/e/normal-calculations-3) 9. [Basic normal calculations](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distribution-calculation/a/basic-normal-calculations) ### More on normal distributions 1. [Deep definition of the normal distribution](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/more-on-normal-distributions/v/introduction-to-the-normal-distribution) 2. [Normal distribution excel exercise](https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/more-on-normal-distributions/v/normal-distribution-excel-exercise) [Unit 5: Exploring bivariate numerical data](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data) Unit mastery: 0% ### Introduction to scatterplots 1. [Constructing a scatter plot](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/v/constructing-scatter-plot) 2. [Constructing scatter plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/e/constructing-scatter-plots) 3. [Making appropriate scatter plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/e/making-good-scatter-plots) 4. [Example of direction in scatterplots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/v/scatter-plot-interpreting) 5. [Scatter plot: smokers](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/v/trends-in-smoking-scatter-plot) 6. [Bivariate relationship linearity, strength and direction](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/v/bivariate-relationship-linearity-strength-and-direction) 7. [Positive and negative linear associations from scatter plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/e/positive-and-negative-linear-correlations-from-scatter-plots) 8. [Describing trends in scatterplots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/e/interpreting-scatter-plots) 9. [Positive and negative associations in scatterplots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/a/constructing-and-interpreting-a-scatterplot) 10. [Outliers in scatter plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/a/outliers-in-scatter-plots) 11. [Clusters in scatter plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/a/clusters-in-scatter-plots) 12. [Describing scatterplots (form, direction, strength, outliers)](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/a/describing-scatterplots-form-direction-strength-outliers) 13. [Scatterplots and correlation review](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-scatterplots/a/scatterplots-and-correlation-review) ### Correlation coefficients 1. [Example: Correlation coefficient intuition](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/scatterplots-and-correlation/v/correlation-coefficient-intuition-examples) 2. [Correlation coefficient intuition](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/scatterplots-and-correlation/e/correlation-coefficient-intuition) 3. [Calculating correlation coefficient r](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/scatterplots-and-correlation/v/calculating-correlation-coefficient-r) 4. [Correlation coefficient review](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/scatterplots-and-correlation/a/correlation-coefficient-review) ### Introduction to trend lines 1. [Fitting a line to data](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/v/fitting-a-line-to-data) 2. [Estimating the line of best fit exercise](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/v/estimating-the-line-of-best-fit-exercise) 3. [Eyeballing the line of best fit](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/e/plotting_the_line_of_best_fit) 4. [Estimating with linear regression (linear models)](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/v/example-estimating-from-regression-line) 5. [Estimating equations of lines of best fit, and using them to make predictions](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/e/equations-of-lines-of-best-fit-to-make-predictions) 6. [Line of best fit: smoking in 1945](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/v/smoking-1945-extrapolation) 7. [Estimating slope of line of best fit](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/e/linear-models-of-bivariate-data) 8. [Equations of trend lines: Phone data](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/a/equations-of-trend-lines-phone-data) 9. [Linear regression review](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/a/linear-regression-review) ### Least-squares regression equations 1. [Introduction to residuals and least squares regression](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/v/introduction-to-residuals-and-least-squares-regression) 2. [Introduction to residuals](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/a/introduction-to-residuals) 3. [Calculating residual example](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/v/calculating-residual-example) 4. [Calculating and interpreting residuals](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/e/calculating-interpreting-residuals) 5. [Calculating the equation of a regression line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/v/calculating-the-equation-of-a-regression-line) 6. [Calculating the equation of the least-squares line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/e/calculating-equation-least-squares) 7. [Interpreting slope of regression line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/v/interpreting-slope-of-regression-line) 8. [Interpreting y-intercept in regression model](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/v/interpreting-y-intercept-in-regression-model) 9. [Interpreting a trend line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/v/interpreting-trend-line) 10. [Interpreting slope and y-intercept for linear models](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/regression-library/e/interpreting-slope-and-y-intercept-of-lines-of-best-fit) ### Assessing the fit in least-squares regression 1. [Residual plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/v/residual-plots) 2. [Residual plots](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/e/residual-plots) 3. [R-squared intuition](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/a/r-squared-intuition) 4. [R-squared or coefficient of determination](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/v/r-squared-or-coefficient-of-determination) 5. [Standard deviation of residuals or Root-mean-square error (RMSD)](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/v/standard-deviation-of-residuals-or-root-mean-square-error-rmsd) 6. [Interpreting computer regression data](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/v/interpreting-computer-regression-data) 7. [Interpreting computer output for regression](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/a/interpreting-computer-output-regression) 8. [Impact of removing outliers on regression lines](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/v/impact-of-removing-outliers-on-regression-lines) 9. [Effects of influential points](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/assessing-the-fit-in-least-squares-regression/e/influential-points) ### More on regression 1. [Squared error of regression line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/v/squared-error-of-regression-line) 2. [Proof (part 1) minimizing squared error to regression line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/v/proof-part-1-minimizing-squared-error-to-regression-line) 3. [Proof (part 2) minimizing squared error to regression line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/v/proof-part-2-minimizing-squared-error-to-line) 4. [Proof (part 3) minimizing squared error to regression line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/v/proof-part-3-minimizing-squared-error-to-regression-line) 5. [Proof (part 4) minimizing squared error to regression line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/v/proof-part-4-minimizing-squared-error-to-regression-line) 6. [Regression line example](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/v/regression-line-example) 7. [Second regression example](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/v/second-regression-example) 8. [Calculating R-squared](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/v/calculating-r-squared) 9. [Covariance and the regression line](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/v/covariance-and-the-regression-line) [Unit 6: Study design](https://www.khanacademy.org/math/statistics-probability/designing-studies) Unit mastery: 0% ### Statistical questions 1. [Statistical questions](https://www.khanacademy.org/math/statistics-probability/designing-studies/statistics-overview/v/understanding-statistical-questions) 2. [Statistical and non statistical questions](https://www.khanacademy.org/math/statistics-probability/designing-studies/statistics-overview/v/statistical-questions) 3. [Statistical questions](https://www.khanacademy.org/math/statistics-probability/designing-studies/statistics-overview/e/statistical-questions) ### Sampling and observational studies 1. [Reasonable samples](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/v/reasonable-samples) 2. [Valid claims](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/e/valid-claims) 3. [Making inferences from random samples](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/e/making-inferences-from-random-samples) 4. [Identifying a sample and population](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/v/identifying-a-sample-and-population) 5. [Identify the population and sample](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/e/identifying-population-sample) 6. [Examples of bias in surveys](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/v/examples-of-bias-in-surveys) 7. [Example of undercoverage introducing bias](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/v/example-of-under-coverage-introducing-bias) 8. [Correlation and causality](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/v/correlation-and-causality) 9. [Identifying bias in samples and surveys](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/a/identifying-bias-in-samples-and-surveys) 10. [Simulation and randomness: Random digit tables](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-and-surveys/a/simulations-and-randomness-random-digit-tables) ### Sampling methods 1. [Picking fairly](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/v/probability-decisions) 2. [Using probability to make fair decisions](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/e/using-probability-to-make-fair-decisions) 3. [Techniques for generating a simple random sample](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/v/techniques-for-generating-a-simple-random-sample) 4. [Simple random samples](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/e/simple-random-samples) 5. [Techniques for random sampling and avoiding bias](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/v/techniques-for-random-sampling-and-avoiding-bias) 6. [Sampling methods](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/e/sampling-methods) 7. [Sampling methods review](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/a/sampling-methods-review) 8. [Samples and surveys](https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/a/worksheet-samples-and-surveys) ### Types of studies (experimental vs. observational) 1. [Types of statistical studies](https://www.khanacademy.org/math/statistics-probability/designing-studies/types-studies-experimental-observational/v/types-of-statistical-studies) 2. [Worked example identifying experiment](https://www.khanacademy.org/math/statistics-probability/designing-studies/types-studies-experimental-observational/v/worked-example-identifying-experiment) 3. [Worked example identifying observational study](https://www.khanacademy.org/math/statistics-probability/designing-studies/types-studies-experimental-observational/v/worked-example-identifying-observational-study) 4. [Worked example identifying sample study](https://www.khanacademy.org/math/statistics-probability/designing-studies/types-studies-experimental-observational/v/worked-example-identifying-sample-study) 5. [Observational studies and experiments](https://www.khanacademy.org/math/statistics-probability/designing-studies/types-studies-experimental-observational/a/observational-studies-and-experiments) 6. [Types of statistical studies](https://www.khanacademy.org/math/statistics-probability/designing-studies/types-studies-experimental-observational/e/types-of-statistical-studies) 7. [Appropriate statistical study example](https://www.khanacademy.org/math/statistics-probability/designing-studies/types-studies-experimental-observational/v/analyzing-statistical-study) ### Experiments 1. [Introduction to experiment design](https://www.khanacademy.org/math/statistics-probability/designing-studies/experiments-stats-library/v/introduction-to-experiment-design) 2. [Matched pairs experiment design](https://www.khanacademy.org/math/statistics-probability/designing-studies/experiments-stats-library/v/matched-pairs-experiment-design) 3. [The language of experiments](https://www.khanacademy.org/math/statistics-probability/designing-studies/experiments-stats-library/a/the-language-of-experiments) 4. [Principles of experiment design](https://www.khanacademy.org/math/statistics-probability/designing-studies/experiments-stats-library/a/principles-of-experiment-design) 5. [Experiment designs](https://www.khanacademy.org/math/statistics-probability/designing-studies/experiments-stats-library/e/experiment-designs) 6. [Random sampling vs. random assignment (scope of inference)](https://www.khanacademy.org/math/statistics-probability/designing-studies/experiments-stats-library/a/scope-of-inference-random-sampling-assignment) [Unit 7: Probability](https://www.khanacademy.org/math/statistics-probability/probability-library) Unit mastery: 0% ### Basic theoretical probability 1. [Intro to theoretical probability](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/v/basic-probability) 2. [Probability: the basics](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/a/probability-the-basics) 3. [Simple probability: yellow marble](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/v/simple-probability) 4. [Simple probability: non-blue marble](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/v/probability-1-module-examples) 5. [Simple probability](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/e/probability_1) 6. [Intuitive sense of probabilities](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/v/intuitive-sense-of-probabilities) 7. [Comparing probabilities](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/e/understanding-probability) 8. [The Monty Hall problem](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/v/monty-hall-problem) ### Probability using sample spaces 1. [Example: All the ways you can flip a coin](https://www.khanacademy.org/math/statistics-probability/probability-library/probability-sample-spaces/v/coin-flipping-example) 2. [Die rolling probability](https://www.khanacademy.org/math/statistics-probability/probability-library/probability-sample-spaces/v/events-and-outcomes-2) 3. [Subsets of sample spaces](https://www.khanacademy.org/math/statistics-probability/probability-library/probability-sample-spaces/v/describing-subsets-of-sample-spaces-exercise) 4. [Subsets of sample spaces](https://www.khanacademy.org/math/statistics-probability/probability-library/probability-sample-spaces/e/describing-subsets-of-sample-spaces) ### Basic set operations 1. [Intersection and union of sets](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-ops/v/intersection-and-union-of-sets) 2. [Relative complement or difference between sets](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-ops/v/relative-complement-or-difference-between-sets) 3. [Universal set and absolute complement](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-ops/v/universal-set-and-absolute-complement) 4. [Subset, strict subset, and superset](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-ops/v/subset-strict-subset-and-superset) 5. [Bringing the set operations together](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-ops/v/bringing-the-set-operations-together) 6. [Basic set notation](https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-ops/e/basic_set_notation) ### Experimental probability 1. [Experimental probability](https://www.khanacademy.org/math/statistics-probability/probability-library/experimental-probability-lib/v/experimental-probability) 2. [Experimental probability](https://www.khanacademy.org/math/statistics-probability/probability-library/experimental-probability-lib/e/finding-probability) 3. [Theoretical and experimental probabilities](https://www.khanacademy.org/math/statistics-probability/probability-library/experimental-probability-lib/v/comparing-theoretical-to-experimental-probabilites) 4. [Making predictions with probability](https://www.khanacademy.org/math/statistics-probability/probability-library/experimental-probability-lib/v/making-predictions-with-probability) 5. [Making predictions with probability](https://www.khanacademy.org/math/statistics-probability/probability-library/experimental-probability-lib/e/using-probability-to-make-predictions) 6. [Simulation and randomness: Random digit tables](https://www.khanacademy.org/math/statistics-probability/probability-library/experimental-probability-lib/a/simulations-and-randomness-random-digit-tables) ### Randomness, probability, and simulation 1. [Experimental versus theoretical probability simulation](https://www.khanacademy.org/math/statistics-probability/probability-library/randomness-probability-and-simulation/v/experimental-versus-theoretical-probability-simulation) 2. [Theoretical and experimental probability: Coin flips and die rolls](https://www.khanacademy.org/math/statistics-probability/probability-library/randomness-probability-and-simulation/a/theoretical-and-experimental-probability-coin-flips-and-die-rolls) 3. [Random number list to run experiment](https://www.khanacademy.org/math/statistics-probability/probability-library/randomness-probability-and-simulation/v/random-number-list-to-run-experiment) 4. [Random numbers for experimental probability](https://www.khanacademy.org/math/statistics-probability/probability-library/randomness-probability-and-simulation/v/random-numbers-for-experimental-probability) 5. [Interpret results of simulations](https://www.khanacademy.org/math/statistics-probability/probability-library/randomness-probability-and-simulation/e/interpreting-results-simulations) 6. [Statistical significance of experiment](https://www.khanacademy.org/math/statistics-probability/probability-library/randomness-probability-and-simulation/v/statistical-significance-experiment) ### Addition rule 1. [Probability with Venn diagrams](https://www.khanacademy.org/math/statistics-probability/probability-library/addition-rule-lib/v/probability-with-playing-cards-and-venn-diagrams) 2. [Addition rule for probability](https://www.khanacademy.org/math/statistics-probability/probability-library/addition-rule-lib/v/addition-rule-for-probability) 3. [Addition rule for probability (basic)](https://www.khanacademy.org/math/statistics-probability/probability-library/addition-rule-lib/a/addition-rule-for-probability-basic) 4. [Adding probabilities](https://www.khanacademy.org/math/statistics-probability/probability-library/addition-rule-lib/e/adding-probability) 5. [Two-way tables, Venn diagrams, and probability](https://www.khanacademy.org/math/statistics-probability/probability-library/addition-rule-lib/e/two-way-tables-venn-diagrams-probability) ### Multiplication rule for independent events 1. [Sample spaces for compound events](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/v/compound-sample-spaces) 2. [Sample spaces for compound events](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/e/sample-spaces-for-compound-events) 3. [Compound probability of independent events](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/v/compound-probability-of-independent-events) 4. [Probability of a compound event](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/v/probability-from-compound-sample-space) 5. ["At least one" probability with coin flipping](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/v/getting-at-least-one-heads) 6. [Free-throw probability](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/v/lebron-asks-about-the-chances-of-making-10-free-throws) 7. [Three-pointer vs free-throw probability](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/v/lebron-asks-about-free-throws-versus-three-pointers) 8. [Probability without equally likely events](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/v/frequency-probability-and-unfair-coins) 9. [Independent events example: test taking](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/v/independent-events-2) 10. [Die rolling probability with independent events](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/v/independent-events-3) 11. [Independent probability](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/e/independent_probability) 12. [Probabilities of compound events](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/e/compound-events) 13. [Probabilities involving "at least one" success](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/a/probabilities-involving-at-least-one-success) 14. [Probability of "at least one" success](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/e/probability-at-least-one-success) ### Multiplication rule for dependent events 1. [Dependent probability introduction](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-dependent/v/introduction-to-dependent-probability) 2. [Dependent probability: coins](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-dependent/v/dependent-probability-example-1) 3. [Dependent probability example](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-dependent/v/dependent-probability-example-2) 4. [Independent & dependent probability](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-dependent/v/independent-events-1) 5. [Dependent probability](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-dependent/e/dependent_probability) 6. [The general multiplication rule](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-dependent/a/general-multiplication-rule) 7. [Dependent probability](https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-dependent/v/analyzing-dependent-probability) ### Conditional probability and independence 1. [Calculating conditional probability](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/v/calculating-conditional-probability) 2. [Conditional probability explained visually](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/v/conditional-probability2) 3. [Conditional probability using two-way tables](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/a/conditional-probability-using-two-way-tables) 4. [Calculate conditional probability](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/e/calculating-conditional-probability) 5. [Conditional probability tree diagram example](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/v/conditional-probability-tree-diagram-example) 6. [Tree diagrams and conditional probability](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/a/tree-diagrams-conditional-probability) 7. [Conditional probability and independence](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/v/testing-independence-from-experimental-data) 8. [Conditional probability and independence](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/a/check-independence-conditional-probability) 9. [Analyzing event probability for independence](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/v/independent-dependent-probability) 10. [Dependent and independent events](https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/e/identifying-dependent-and-independent-events) [Unit 8: Counting, permutations, and combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations) Unit mastery: 0% ### Counting principle and factorial 1. [Count outcomes using tree diagram](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/counting-principle-factorial/v/tree-diagram-to-count-outcomes) 2. [Counting outcomes: flower pots](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/counting-principle-factorial/v/counting-pot-and-flower-scenarios) 3. [The counting principle](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/counting-principle-factorial/e/fundamental-counting-principle) ### Permutations 1. [Permutation formula](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/permutation-lib/v/permutation-formula) 2. [Zero factorial or 0\!](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/permutation-lib/v/zero-factorial-or-0) 3. [Factorial and counting seat arrangements](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/permutation-lib/v/factorial-and-counting-seat-arrangements) 4. [Possible three letter words](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/permutation-lib/v/possible-three-letter-words) 5. [Ways to arrange colors](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/permutation-lib/v/permutations-and-combinations-1) 6. [Ways to pick officers](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/permutation-lib/v/permutations-and-combinations-2) 7. [Permutations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/permutation-lib/e/permutations_1) ### Combinations 1. [Intro to combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinations-lib/v/introduction-to-combinations) 2. [Combination formula](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinations-lib/v/combination-formula) 3. [Handshaking combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinations-lib/v/handshaking-combinations) 4. [Combination example: 9 card hands](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinations-lib/v/permutations-and-combinations-3) 5. [Combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinations-lib/e/combinations_1) 6. [Permutations & combinations with overcounting](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinations-lib/e/permutations_and_combinations_2) ### Combinatorics and probability 1. [Probability using combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/probability-using-combinations) 2. [Probability & combinations (2 of 2)](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/probability-and-combinations-part-2) 3. [Example: Different ways to pick officers](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/probability-of-dependent-events) 4. [Example: Combinatorics and probability](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/probability-of-dependent-events-2) 5. [Getting exactly two heads (combinatorics)](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/getting-exactly-two-heads-combinatorics) 6. [Exactly three heads in five flips](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/exactly-three-heads-in-five-flips) 7. [Generalizing with binomial coefficients (bit advanced)](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/generalizing-with-binomial-coefficients-bit-advanced) 8. [Example: Lottery probability](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/permutations-and-combinations-4) 9. [Probability with permutations and combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/e/probability_with_perm_comb) 10. [Conditional probability and combinations](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/conditional-probability-and-combinations) 11. [Mega millions jackpot probability](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/mega-millions-jackpot-probability) 12. [Birthday probability problem](https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/combinatorics-probability/v/birthday-probability-problem) [Unit 9: Random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library) Unit mastery: 0% ### Discrete random variables 1. [Random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/v/random-variables) 2. [Discrete and continuous random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/v/discrete-and-continuous-random-variables) 3. [Constructing a probability distribution for random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/v/discrete-probability-distribution) 4. [Constructing probability distributions](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/constructing-probability-distributions) 5. [Probability models example: frozen yogurt](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/v/constructing-probability-model-from-observations) 6. [Probability models](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/probability-models) 7. [Valid discrete probability distribution examples](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/v/valid-discrete-probability-distribution-examples) 8. [Probability with discrete random variable example](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/v/example-analyzing-discrete-probability-distribution) 9. [Probability with discrete random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/probability-discrete-random-variables) 10. [Mean (expected value) of a discrete random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/v/expected-value-of-a-discrete-random-variable) 11. [Expected value](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/expected_value) 12. [Mean (expected value) of a discrete random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/mean-expected-value-discrete-random-variable) 13. [Expected value (basic)](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/a/expected-value-basic) 14. [Variance and standard deviation of a discrete random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/v/variance-and-standard-deviation-of-a-discrete-random-variable) 15. [Standard deviation of a discrete random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/standard-deviation-discrete-random-variable) ### Continuous random variables 1. [Probability density functions](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-continuous/v/probability-density-functions) 2. [Probabilities from density curves](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-continuous/v/probabilities-from-density-curves) 3. [Probability in density curves](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-continuous/e/probability-density-curves) 4. [Probability in normal density curves](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-continuous/e/probability-normal-density-curves) ### Transforming random variables 1. [Impact of transforming (scaling and shifting) random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/transforming-random-variable/v/impact-of-scaling-and-shifting-random-variables) 2. [Example: Transforming a discrete random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/transforming-random-variable/v/example-transforming-a-discrete-random-variable) 3. [Transforming random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/transforming-random-variable/e/transforming-random-variables) ### Combining random variables 1. [Mean of sum and difference of random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/v/mean-of-sum-and-difference-of-random-variables) 2. [Variance of sum and difference of random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/v/variance-of-sum-and-difference-of-random-variables) 3. [Intuition for why independence matters for variance of sum](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/v/intuition-for-why-independence-matters-for-variance-of-sum) 4. [Deriving the variance of the difference of random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/v/variance-of-differences-of-random-variables) 5. [Combining random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/a/combining-random-variables-article) 6. [Combining random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/e/combining-random-variables) 7. [Example: Analyzing distribution of sum of two normally distributed random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/v/analyzing-distribution-of-sum-of-two-normally-distributed-random-variables) 8. [Example: Analyzing the difference in distributions](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/v/analyzing-the-difference-in-distributions) 9. [Combining normal random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/a/combining-normal-random-variables) 10. [Combining normal random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/combine-random-variables/e/combine-normal-random-variables) ### Binomial random variables 1. [Binomial variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/binomial-variables) 2. [Recognizing binomial variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/recognizing-binomial-variables) 3. [10% Rule of assuming "independence" between trials](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/10-percent-rule-independence) 4. [Identifying binomial variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/e/identifying-binomial-variables) 5. [Binomial distribution](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/binomial-distribution) 6. [Visualizing a binomial distribution](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/visualizing-a-binomial-distribution) 7. [Binomial probability example](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/probability-of-making-2-shots-in-6-attempts) 8. [Generalizing k scores in n attempts](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/generalizing-k-scores-in-n-attempts) 9. [Free throw binomial probability distribution](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/free-throw-binomial-probability-distribution) 10. [Graphing basketball binomial distribution](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/graphing-basketball-binomial-distribution) 11. [Binompdf and binomcdf functions](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/ti-84-binompdf-and-binomcdf-functions) 12. [Binomial probability (basic)](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/a/binomial-probability-basic) 13. [Binomial probability formula](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/e/binomial-probability) 14. [Calculating binomial probability](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/e/calculating-binomial-probability) ### Binomial mean and standard deviation formulas 1. [Mean and variance of Bernoulli distribution example](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-mean-standard-dev-formulas/v/mean-and-variance-of-bernoulli-distribution-example) 2. [Bernoulli distribution mean and variance formulas](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-mean-standard-dev-formulas/v/bernoulli-distribution-mean-and-variance-formulas) 3. [Expected value of a binomial variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-mean-standard-dev-formulas/v/expected-value-of-binomial-variable) 4. [Variance of a binomial variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-mean-standard-dev-formulas/v/variance-of-binomial-variable) 5. [Finding the mean and standard deviation of a binomial random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-mean-standard-dev-formulas/v/finding-the-mean-and-standard-deviation-of-a-binomial-random-variable) 6. [Mean and standard deviation of a binomial random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-mean-standard-dev-formulas/e/mean-standard-deviation-binomial-variable) ### Geometric random variables 1. [Geometric random variables introduction](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/v/geometric-random-variables-introduction) 2. [Binomial vs. geometric random variables](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/e/binomial-vs-geometric-variables) 3. [Probability for a geometric random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/v/probability-for-a-geometric-random-variable) 4. [Geometric probability](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/e/geometric-probability) 5. [Cumulative geometric probability (greater than a value)](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/v/cumulative-geometric-probability-greater-than-a-value) 6. [Cumulative geometric probability (less than a value)](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/v/cumulative-geometric-probability-less-than-a-value) 7. [TI-84 geometpdf and geometcdf functions](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/v/ti-84-geometpdf-and-geometcdf-functions) 8. [Cumulative geometric probability](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/e/cumulative-geometric-probability) 9. [Proof of expected value of geometric random variable](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-geometric/v/proof-of-expected-value-of-geometric-random-variable) ### More on expected value 1. [Term life insurance and death probability](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/v/term-life-insurance-and-death-probability) 2. [Getting data from expected value](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/v/empirical-data-expected-value) 3. [Expected profit from lottery ticket](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/v/expected-value-profit-lottery-ticket) 4. [Expected value while fishing](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/v/expected-value-wager) 5. [Comparing insurance with expected value](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/v/expected-value-insurance) 6. [Expected value with empirical probabilities](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/e/expected-value-with-empirical-probabilities) 7. [Expected value with calculated probabilities](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/e/expected-value-with-calculated-probabilities) 8. [Making decisions with expected values](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/e/making-decisions-with-expected-values) 9. [Law of large numbers](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/v/law-of-large-numbers) ### Poisson distribution 1. [Poisson process 1](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/poisson-distribution/v/poisson-process-1) 2. [Poisson process 2](https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/poisson-distribution/v/poisson-process-2) [Unit 10: Sampling distributions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library) Unit mastery: 0% ### What is a sampling distribution? 1. [Introduction to sampling distributions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/what-is-a-sampling-distribution/v/introduction-to-sampling-distributions) 2. [Sample statistic bias worked example](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/what-is-a-sampling-distribution/v/sample-statistic-bias-worked-example) 3. [Biased and unbiased estimators](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/what-is-a-sampling-distribution/e/biased-unbiased-estimators) ### Sampling distribution of a sample proportion 1. [Sampling distribution of sample proportion part 1](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/v/sampling-distribution-of-sample-proportion-part-1) 2. [Sampling distribution of sample proportion part 2](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/v/sampling-distribution-of-sample-proportion-part-2) 3. [Normal conditions for sampling distributions of sample proportions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/v/normal-conditions-for-sampling-distributions-of-sample-proportions) 4. [The normal condition for sample proportions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/e/normal-condition-sample-proportions) 5. [Mean and standard deviation of sample proportions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/e/sampling-distribution-sample-proportion-mean-standard-deviation) 6. [Probability of sample proportions example](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/v/probability-of-sample-proportions-example) 7. [Finding probabilities with sample proportions](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/e/finding-probabilities-sample-proportions) 8. [Sampling distribution of a sample proportion example](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/a/sampling-distribution-sample-proportion-example) ### Sampling distribution of a sample mean 1. [Inferring population mean from sample mean](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/statistics-sample-vs-population-mean) 2. [Central limit theorem](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/central-limit-theorem) 3. [Sampling distribution of the sample mean](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-of-the-sample-mean) 4. [Sampling distribution of the sample mean (part 2)](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-of-the-sample-mean-2) 5. [Standard error of the mean](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/standard-error-of-the-mean) 6. [Example: Probability of sample mean exceeding a value](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-example-problem) 7. [Mean and standard deviation of sample means](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/e/mean-standard-deviation-sample-means) 8. [Sample means and the central limit theorem](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/e/sample-means-central-limit-theorem) 9. [Finding probabilities with sample means](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/e/finding-probabilities-sample-means) 10. [Sampling distribution of a sample mean example](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/a/sampling-distribution-sample-mean-example) [Unit 11: Confidence intervals](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample) Unit mastery: 0% ### Introduction to confidence intervals 1. [Confidence intervals and margin of error](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/introduction-to-confidence-intervals/v/confidence-intervals-and-margin-of-error) 2. [Confidence interval simulation](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/introduction-to-confidence-intervals/v/confidence-interval-simulation) 3. [Interpreting confidence level example](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/introduction-to-confidence-intervals/v/interpreting-confidence-intervals-example) 4. [Interpreting confidence levels and confidence intervals](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/introduction-to-confidence-intervals/a/interpreting-confidence-levels-and-confidence-intervals) ### Estimating a population proportion 1. [Confidence interval example](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/v/confidence-interval-example) 2. [Margin of error 1](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/v/margin-of-error-1) 3. [Margin of error 2](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/v/margin-of-error-2) 4. [Conditions for valid confidence intervals for a proportion](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/v/conditions-for-valid-confidence-intervals) 5. [Conditions for confidence interval for a proportion worked examples](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/v/conditions-for-confidence-intervals-worked-examples) 6. [Reference: Conditions for inference on a proportion](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/a/conditions-inference-one-proportion) 7. [Conditions for a z interval for a proportion](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/e/conditions-one-sample-z-interval-proportion) 8. [Critical value (z\*) for a given confidence level](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/v/critical-value-for-a-given-confidence-level) 9. [Finding the critical value z\* for a desired confidence level](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/e/finding-critical-value-z-for-confidence-level) 10. [Example constructing and interpreting a confidence interval for p](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/v/example-calculating-confidence-interval) 11. [Calculating a z interval for a proportion](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/e/calculating-one-sample-z-interval-proportion) 12. [Interpreting a z interval for a proportion](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/a/interpreting-z-interval-proportion) 13. [Determining sample size based on confidence and margin of error](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/v/determining-sample-size-based-on-confidence-and-margin-of-error) 14. [Sample size and margin of error in a z interval for p](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/e/sample-size-margin-error-one-sample-z-interval) ### Estimating a population mean 1. [Introduction to t statistics](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/v/introduction-to-t-statistics) 2. [Simulation showing value of t statistic](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/v/simulation-showing-value-of-t-statistic) 3. [Conditions for valid t intervals](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/v/conditions-for-valid-t-intervals) 4. [Reference: Conditions for inference on a mean](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/a/reference-conditions-inference-one-mean) 5. [Conditions for a t interval for a mean](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/e/conditions-one-sample-t-interval-mean) 6. [Example finding critical t value](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/v/example-finding-critical-t-value) 7. [Finding the critical value t\* for a desired confidence level](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/e/finding-critical-value-t) 8. [Example constructing a t interval for a mean](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/v/calculating-a-one-sample-t-interval-for-a-mean) 9. [Calculating a t interval for a mean](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/e/calculating-one-sample-t-interval-mean) 10. [Confidence interval for a mean with paired data](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/v/confidence-interval-for-a-mean-difference) 11. [Making a t interval for paired data](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/a/one-sample-t-interval-paired-data) 12. [Interpreting a confidence interval for a mean](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/a/interpret-one-sample-t-interval-mean) 13. [Sample size for a given margin of error for a mean](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/v/sample-size-for-a-given-margin-of-error-for-a-mean) 14. [Sample size and margin of error in a confidence interval for a mean](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/e/sample-size-margin-error-one-sample-t-interval-mean) ### More confidence interval videos 1. [T-statistic confidence interval](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/old-confidence-interval-videos/v/t-statistic-confidence-interval) 2. [Small sample size confidence intervals](https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/old-confidence-interval-videos/v/small-sample-size-confidence-intervals) [Unit 12: Significance tests (hypothesis testing)](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample) Unit mastery: 0% ### The idea of significance tests 1. [Simple hypothesis testing](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/v/simple-hypothesis-testing) 2. [Idea behind hypothesis testing](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/v/idea-behind-hypothesis-testing) 3. [Simple hypothesis testing](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/e/hypothesis-testing-with-simulations) 4. [Examples of null and alternative hypotheses](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/v/examples-of-null-and-alternative-hypotheses) 5. [Writing null and alternative hypotheses](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/e/writing-null-and-alternative-hypotheses-informal) 6. [P-values and significance tests](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/v/p-values-and-significance-tests) 7. [Comparing P-values to different significance levels](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/v/comparing-p-value-diff-significance-levels) 8. [Estimating a P-value from a simulation](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/v/estimating-p-value-from-simulation) 9. [Estimating P-values from simulations](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/e/estimating-p-values-and-making-conclusions) 10. [Using P-values to make conclusions](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/a/p-value-conclusions) ### Error probabilities and power 1. [Introduction to Type I and Type II errors](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/error-probabilities-and-power/v/introduction-to-type-i-and-type-ii-errors) 2. [Type 1 errors](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/error-probabilities-and-power/v/type-1-errors) 3. [Examples identifying Type I and Type II errors](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/error-probabilities-and-power/v/examples-identifying-type-i-and-type-ii-errors) 4. [Type I vs Type II error](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/error-probabilities-and-power/e/type-i-error-type-ii-error-power) 5. [Introduction to power in significance tests](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/error-probabilities-and-power/v/introduction-to-power-in-significance-tests) 6. [Examples thinking about power in significance tests](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/error-probabilities-and-power/v/examples-thinking-about-power-in-significance-tests) 7. [Error probabilities and power](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/error-probabilities-and-power/e/error-probabilities-power) 8. [Consequences of errors and significance](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/error-probabilities-and-power/a/consequences-errors-significance) ### Tests about a population proportion 1. [Constructing hypotheses for a significance test about a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/v/constructing-hypotheses-for-a-significance-test) 2. [Writing hypotheses for a test about a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/e/writing-hypotheses-one-sample-z-test-proportion) 3. [Conditions for a z test about a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/v/conditions-for-a-z-test-about-a-proportion) 4. [Reference: Conditions for inference on a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/a/conditions-inference-one-proportion) 5. [Conditions for a z test about a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/e/checking-conditions-one-sample-z-test-proportion) 6. [Calculating a z statistic in a test about a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/v/calculating-a-z-statistic-in-a-significance-test) 7. [Calculating the test statistic in a z test for a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/e/test-stat-one-sample-z-test-p) 8. [Calculating a P-value given a z statistic](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/v/calculating-p-value-from-z-table) 9. [Calculating the P-value in a z test for a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/e/p-value-one-sample-z-test-p) 10. [Making conclusions in a test about a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/v/comparing-p-value-to-significance-level-example) 11. [Making conclusions in a z test for a proportion](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-proportion/e/conclusions-one-sample-z-test-proportion) ### Tests about a population mean 1. [Writing hypotheses for a significance test about a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/writing-hypotheses-for-significance-test-about-means) 2. [Writing hypotheses for a test about a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/e/writing-hypotheses-one-sample-t-test-mean) 3. [Conditions for a t test about a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/conditions-t-test-mean) 4. [Reference: Conditions for inference on a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/a/reference-conditions-inference-one-mean) 5. [Conditions for a t test about a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/e/checking-conditions-one-sample-t-test-mean) 6. [When to use z or t statistics in significance tests](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/when-to-use-z-or-t-statistics-in-significance-tests) 7. [Example calculating t statistic for a test about a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/example-calculating-t-statistic-for-signficance-test) 8. [Calculating the test statistic in a t test for a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/e/test-statistic-t-test) 9. [Using TI calculator for P-value from t statistic](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/using-ti-calculator-for-p-value-from-t-statistic) 10. [Using a table to estimate P-value from t statistic](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/calculating-p-value-from-t-statistic) 11. [Calculating the P-value in a t test for a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/e/p-value-t-test-mean) 12. [Comparing P-value from t statistic to significance level](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/comparing-p-value-from-t-statistic-to-significance-level) 13. [Making conclusions in a t test for a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/e/conclusions-one-sample-t-test-mean) 14. [Free response example: Significance test for a mean](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/ap-statistics-free-response-on-significance-test-for-mean) ### More significance testing videos 1. [Hypothesis testing and p-values](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/more-significance-testing-videos/v/hypothesis-testing-and-p-values) 2. [One-tailed and two-tailed tests](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/more-significance-testing-videos/v/one-tailed-and-two-tailed-tests) 3. [Z-statistics vs. T-statistics](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/more-significance-testing-videos/v/z-statistics-vs-t-statistics) 4. [Small sample hypothesis test](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/more-significance-testing-videos/v/small-sample-hypothesis-test) 5. [Large sample proportion hypothesis testing](https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/more-significance-testing-videos/v/large-sample-proportion-hypothesis-testing) [Unit 13: Two-sample inference for the difference between groups](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples) Mastery unavailable ### Comparing two proportions 1. [Comparing population proportions 1](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples/comparing-two-proportions/v/comparing-population-proportions-1) 2. [Comparing population proportions 2](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples/comparing-two-proportions/v/comparing-population-proportions-2) 3. [Hypothesis test comparing population proportions](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples/comparing-two-proportions/v/hypothesis-test-comparing-population-proportions) ### Comparing two means 1. [Statistical significance of experiment](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples/comparing-two-means/v/statistical-significance-experiment) 2. [Statistical significance on bus speeds](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples/comparing-two-means/v/statistical-significance-on-bus-speeds) 3. [Hypothesis testing in experiments](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples/comparing-two-means/e/hypothesis-testing-in-experiments) 4. [Difference of sample means distribution](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples/comparing-two-means/v/difference-of-sample-means-distribution) 5. [Confidence interval of difference of means](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples/comparing-two-means/v/confidence-interval-of-difference-of-means) 6. [Clarification of confidence interval of difference of means](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples/comparing-two-means/v/clarification-of-confidence-interval-of-difference-of-means) 7. [Hypothesis test for difference of means](https://www.khanacademy.org/math/statistics-probability/significance-tests-confidence-intervals-two-samples/comparing-two-means/v/hypothesis-test-for-difference-of-means) [Unit 14: Inference for categorical data (chi-square tests)](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests) Unit mastery: 0% ### Chi-square goodness-of-fit tests 1. [Chi-square distribution introduction](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/v/chi-square-distribution-introduction) 2. [Pearson's chi square test (goodness of fit)](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/v/pearson-s-chi-square-test-goodness-of-fit) 3. [Chi-square statistic for hypothesis testing](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/v/chi-square-statistic) 4. [Chi-square goodness-of-fit example](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/v/goodness-of-fit-example) 5. [Expected counts in a goodness-of-fit test](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/e/expected-counts-chi-square-goodness-of-fit-test) 6. [Conditions for a goodness-of-fit test](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/e/checking-conditions-chi-square-goodness-of-fit-test) 7. [Test statistic and P-value in a goodness-of-fit test](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/e/test-statistic-p-value-chi-square-goodness-of-fit-test) 8. [Conclusions in a goodness-of-fit test](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-goodness-of-fit-tests/e/conclusions-chi-square-goodness-of-fit-test) ### Chi-square tests for relationships 1. [Filling out frequency table for independent events](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/v/frequency-table-independent-events) 2. [Contingency table chi-square test](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/v/contingency-table-chi-square-test) 3. [Introduction to the chi-square test for homogeneity](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/v/chi-square-test-homogeneity) 4. [Chi-square test for association (independence)](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/v/chi-square-test-association-independence) 5. [Expected counts in chi-squared tests with two-way tables](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/e/expected-counts-chi-square-test-two-way-tables) 6. [Test statistic and P-value in chi-square tests with two-way tables](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/e/test-statistics-p-values-chi-square-tests-two-way-tables) 7. [Making conclusions in chi-square tests for two-way tables](https://www.khanacademy.org/math/statistics-probability/inference-categorical-data-chi-square-tests/chi-square-tests-for-homogeneity-and-association-independence/e/conclusions-chi-square-tests-for-two-way-tables) [Unit 15: Advanced regression (inference and transforming)](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming) Mastery unavailable ### Inference about slope 1. [Introduction to inference about slope in linear regression](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/inference-on-slope/v/intro-inference-slope) 2. [Conditions for inference on slope](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/inference-on-slope/v/conditions-inference-slope) 3. [Confidence interval for the slope of a regression line](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/inference-on-slope/v/confidence-interval-slope) 4. [Confidence interval for slope](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/inference-on-slope/e/confidence-interval-slope) 5. [Calculating t statistic for slope of regression line](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/inference-on-slope/v/t-statistic-slope) 6. [Test statistic for slope](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/inference-on-slope/e/test-statistic-p-value-t-test-slope) 7. [Using a P-value to make conclusions in a test about slope](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/inference-on-slope/v/making-conclusions-for-regression-slope-hypothesis-test) 8. [Using a confidence interval to test slope](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/inference-on-slope/v/interval-test-slope) 9. [Making conclusions about slope](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/inference-on-slope/e/conclusions-t-test-slope) ### Nonlinear regression 1. [Comparing models to fit data example](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/nonlinear-regression/v/comparing-models-to-fit-data) 2. [Fitting quadratic and exponential functions to scatter plots](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/nonlinear-regression/e/fitting-functions-to-scatter-plots) 3. [Transforming nonlinear data](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/nonlinear-regression/v/transforming-nonlinear-data) 4. [Worked example of linear regression using transformed data](https://www.khanacademy.org/math/statistics-probability/advanced-regression-inference-transforming/nonlinear-regression/v/transformed-linear-regression-example) [Unit 16: Analysis of variance (ANOVA)](https://www.khanacademy.org/math/statistics-probability/analysis-of-variance-anova-library) Mastery unavailable ### Analysis of variance (ANOVA) 1. [ANOVA 1: Calculating SST (total sum of squares)](https://www.khanacademy.org/math/statistics-probability/analysis-of-variance-anova-library/analysis-of-variance-anova/v/anova-1-calculating-sst-total-sum-of-squares) 2. [ANOVA 2: Calculating SSW and SSB (total sum of squares within and between)](https://www.khanacademy.org/math/statistics-probability/analysis-of-variance-anova-library/analysis-of-variance-anova/v/anova-2-calculating-ssw-and-ssb-total-sum-of-squares-within-and-between-avi) 3. [ANOVA 3: Hypothesis test with F-statistic](https://www.khanacademy.org/math/statistics-probability/analysis-of-variance-anova-library/analysis-of-variance-anova/v/anova-3-hypothesis-test-with-f-statistic) Up next for you:Course challengeTest your knowledge of the skills in this course.[Start Course challenge](https://www.khanacademy.org/math/statistics-probability/test/x83aef204:course-challenge) [Community questions](https://www.khanacademy.org/math/statistics-probability/d) Our mission is to provide a free, world-class education to anyone, anywhere. Khan Academy is a 501(c)(3) nonprofit organization. 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