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Types and ExamplesLesson - 10](https://www.simplilearn.com/tutorials/statistics-tutorial/hypothesis-testing-in-statistics) [Understanding the Fundamentals of Arithmetic and Geometric ProgressionLesson - 11](https://www.simplilearn.com/tutorials/statistics-tutorial/arithmetic-and-geometric-progression) [The Definitive Guide to Understand Spearman’s Rank CorrelationLesson - 12](https://www.simplilearn.com/tutorials/statistics-tutorial/spearmans-rank-correlation) [Mean Squared Error: Overview, Examples, Concepts and MoreLesson - 13](https://www.simplilearn.com/tutorials/statistics-tutorial/mean-squared-error) [All You Need to Know About the Empirical Rule in StatisticsLesson - 14](https://www.simplilearn.com/tutorials/statistics-tutorial/all-about-the-empirical-rule-in-statistics) [The Complete Guide to Skewness and KurtosisLesson - 15](https://www.simplilearn.com/tutorials/statistics-tutorial/skewness-and-kurtosis) [A Holistic Look at Bernoulli DistributionLesson - 16](https://www.simplilearn.com/tutorials/data-science-tutorial/bernoulli-distribution) [All You Need to Know About Bias in StatisticsLesson - 17](https://www.simplilearn.com/tutorials/statistics-tutorial/bias-in-statistics) [A Complete Guide to Get a Grasp of Time Series AnalysisLesson - 18](https://www.simplilearn.com/tutorials/statistics-tutorial/what-is-time-series-analysis) [The Key Differences Between Z-Test Vs. T-TestLesson - 19](https://www.simplilearn.com/tutorials/statistics-tutorial/z-test-vs-t-test) [The Complete Guide to Understand Pearson's CorrelationLesson - 20](https://www.simplilearn.com/tutorials/statistics-tutorial/pearson-correlation-coefficient-in-statistics) [A Complete Guide on the Types of Statistical StudiesLesson - 21](https://www.simplilearn.com/tutorials/statistics-tutorial/types-of-statistical-studies) [Everything You Need to Know About Poisson DistributionLesson - 22](https://www.simplilearn.com/tutorials/data-science-tutorial/poisson-distribution) [Your Best Guide to Understand Correlation vs. RegressionLesson - 23](https://www.simplilearn.com/tutorials/statistics-tutorial/correlation-vs-regression) [The Most Comprehensive Guide for Beginners on What Is CorrelationLesson - 24](https://www.simplilearn.com/tutorials/statistics-tutorial/what-is-correlation-in-statistics) # What Is Central Limit Theorem, Its Significance & Uses Lesson 2 of 24[By Avijeet Biswal](https://www.simplilearn.com/authors/avijeet-biswal) Last updated on Aug 24, 202563814  [Previous](https://www.simplilearn.com/tutorials/statistics-tutorial/probability-density-function)[Next](https://www.simplilearn.com/tutorials/data-analytics-tutorial/measures-of-central-tendency) - [Tutorial Playlist](https://www.simplilearn.com/tutorials/statistics-tutorial/central-limit-theorem) [Statistics TutorialOverview](https://www.simplilearn.com/tutorials/statistics-tutorial) [Everything You Need to Know About the Probability Density Function in StatisticsLesson - 1](https://www.simplilearn.com/tutorials/statistics-tutorial/probability-density-function) [The Best Guide to Understand Central Limit TheoremLesson - 2](https://www.simplilearn.com/tutorials/statistics-tutorial/central-limit-theorem) [Measures of Central Tendency : Mean, Median and ModeLesson - 3](https://www.simplilearn.com/tutorials/data-analytics-tutorial/measures-of-central-tendency) [The Ultimate Guide to Understand Conditional ProbabilityLesson - 4](https://www.simplilearn.com/tutorials/statistics-tutorial/conditional-probability) [Percentile in StatisticsLesson - 5](https://www.simplilearn.com/tutorials/data-analytics-tutorial/percentile-in-statistics) [The Best Guide to Understand Bayes TheoremLesson - 6](https://www.simplilearn.com/tutorials/statistics-tutorial/bayes-theorem) [Everything You Need to Know About the Normal DistributionLesson - 7](https://www.simplilearn.com/tutorials/statistics-tutorial/what-is-normal-distribution) [An In-Depth Explanation of Cumulative Distribution FunctionLesson - 8](https://www.simplilearn.com/tutorials/statistics-tutorial/cumulative-distribution-function) [Chi-Square TestLesson - 9](https://www.simplilearn.com/tutorials/statistics-tutorial/chi-square-test) [What Is Hypothesis Testing in Statistics? Types and ExamplesLesson - 10](https://www.simplilearn.com/tutorials/statistics-tutorial/hypothesis-testing-in-statistics) [Understanding the Fundamentals of Arithmetic and Geometric ProgressionLesson - 11](https://www.simplilearn.com/tutorials/statistics-tutorial/arithmetic-and-geometric-progression) [The Definitive Guide to Understand Spearman’s Rank CorrelationLesson - 12](https://www.simplilearn.com/tutorials/statistics-tutorial/spearmans-rank-correlation) [Mean Squared Error: Overview, Examples, Concepts and MoreLesson - 13](https://www.simplilearn.com/tutorials/statistics-tutorial/mean-squared-error) [All You Need to Know About the Empirical Rule in StatisticsLesson - 14](https://www.simplilearn.com/tutorials/statistics-tutorial/all-about-the-empirical-rule-in-statistics) [The Complete Guide to Skewness and KurtosisLesson - 15](https://www.simplilearn.com/tutorials/statistics-tutorial/skewness-and-kurtosis) [A Holistic Look at Bernoulli DistributionLesson - 16](https://www.simplilearn.com/tutorials/data-science-tutorial/bernoulli-distribution) [All You Need to Know About Bias in StatisticsLesson - 17](https://www.simplilearn.com/tutorials/statistics-tutorial/bias-in-statistics) [A Complete Guide to Get a Grasp of Time Series AnalysisLesson - 18](https://www.simplilearn.com/tutorials/statistics-tutorial/what-is-time-series-analysis) [The Key Differences Between Z-Test Vs. T-TestLesson - 19](https://www.simplilearn.com/tutorials/statistics-tutorial/z-test-vs-t-test) [The Complete Guide to Understand Pearson's CorrelationLesson - 20](https://www.simplilearn.com/tutorials/statistics-tutorial/pearson-correlation-coefficient-in-statistics) [A Complete Guide on the Types of Statistical StudiesLesson - 21](https://www.simplilearn.com/tutorials/statistics-tutorial/types-of-statistical-studies) [Everything You Need to Know About Poisson DistributionLesson - 22](https://www.simplilearn.com/tutorials/data-science-tutorial/poisson-distribution) [Your Best Guide to Understand Correlation vs. RegressionLesson - 23](https://www.simplilearn.com/tutorials/statistics-tutorial/correlation-vs-regression) [The Most Comprehensive Guide for Beginners on What Is CorrelationLesson - 24](https://www.simplilearn.com/tutorials/statistics-tutorial/what-is-correlation-in-statistics) ## Table of Contents [What is the Central Limit Theorem?](https://www.simplilearn.com/tutorials/statistics-tutorial/central-limit-theorem#what_is_the_central_limit_theorem "What is the Central Limit Theorem?") [Significance of Central Limit Theorem](https://www.simplilearn.com/tutorials/statistics-tutorial/central-limit-theorem#significance_of_central_limit_theorem "Significance of Central Limit Theorem") [Assumptions Behind the Central Limit Theorem](https://www.simplilearn.com/tutorials/statistics-tutorial/central-limit-theorem#assumptions_behind_the_central_limit_theorem "Assumptions Behind the Central Limit Theorem") [Why n ≥ 30 Samples?](https://www.simplilearn.com/tutorials/statistics-tutorial/central-limit-theorem#why_n__30_samples "Why n ≥ 30 Samples?") [Mean and Standard Deviation of the Sample](https://www.simplilearn.com/tutorials/statistics-tutorial/central-limit-theorem#mean_and_standard_deviation_of_the_sample "Mean and Standard Deviation of the Sample") View More Central Limit Theorem, also known as the CLT, is a crucial pillar of statistics and [machine learning](https://www.simplilearn.com/tutorials/machine-learning-tutorial/what-is-machine-learning "machine learning"). It is at the heart of hypothesis testing. In this tutorial, you will understand the concept of the CLT and its applications. #### Become a Certified Power BI Developer PL-300 Microsoft Power BI Certification Training[Explore Program](https://www.simplilearn.com/power-bi-certification-training-course?source=GhPreviewCTABanner)  ## What is the Central Limit Theorem? The CLT is a statistical theory that states that - if you take a sufficiently large sample size from a population with a finite level of variance, the mean of all [samples](https://www.simplilearn.com/tutorials/machine-learning-tutorial/population-vs-sample "samples") from that population will be roughly equal to the population mean. Consider there are 15 sections in class X, and each section has 50 students. Our task is to calculate the average marks of students in class X. The standard approach will be to calculate the average simply: - Calculate the total marks of all the students in Class X - Add all the marks - Divide the total marks by the total number of students But what if the [data](https://www.simplilearn.com/what-is-data-article "data") is extremely large? Is this a good approach? No way, calculation marks of all the students will be a tedious and time-consuming process. So, what are the alternatives? Let's take a look at another approach. - To begin, select groups of students from the class at random. This will be referred to as a sample. Create several samples, each with 30 students. - Calculate each sample's individual mean. - Calculate the average of these sample means. - The value will give us the approximate average marks of the students in Class X. - The histogram of the sample means marks of the students will resemble a bell curve or normal distribution. ## Significance of Central Limit Theorem The CLT has several applications. Look at the places where you can use it. - Political/election polling is a great example of how you can use CLT. These polls are used to estimate the number of people who support a specific candidate. You may have seen these results with confidence intervals on news channels. The CLT aids in this calculation. - You use the CLT in various census fields to calculate various population details, such as family income, electricity consumption, individual salaries, and so on. The CLT is useful in a variety of fields. Are there any others that come to mind? Put them in the comments section below this tutorial. ## Assumptions Behind the Central Limit Theorem Before we move on further, it is important to understand the assumptions behind CLT: - The data must adhere to the randomization rule. It needs to be sampled at random. - The samples should be unrelated to one another. One sample should not impact the others. - When taking samples without replacement, the sample size should not exceed 10% of the population. When the population is symmetric, a sample size of 30 is generally considered reasonable. ## Why n ≥ 30 Samples? The sample size of 30 is considered sufficient to see the effect of the CLT. If the population distribution is closer to the normal distribution, you will need fewer samples to demonstrate the central limit theorem. On the other hand, if the population distribution is highly skewed, you will need a large number of samples to understand the CLT. ## Mean and Standard Deviation of the Sample You denote the mean of the sample by    And you denote as the standard deviation of the sample mean as:   That’s the concept and theory behind the CLT. Now, go to the [python](https://www.simplilearn.com/learn-the-basics-of-python-article "python") compiler and understand the working of CLT. #### Take Your Data Scientist Skills to the Next Level With the Data Scientist Master’s Program from IBM[Explore Program](https://www.simplilearn.com/big-data-and-analytics/senior-data-scientist-masters-program-training?source=GhPreviewCTABanner)  ## Implementation Of Central Limit Theorem in Python You can understand the working of the CLT with an example involving the rolling of a die. A die has a different number on each side, ranging from 1 to 6. Each number has a one-in-six chance of appearing on a roll. Given the equal likelihood, the dispersion of the numbers that come up from a dice roll is uniform. You will use the randint() function to generate the random numbers ranging from 1 to 6.  The example will generate and print the sample of 100 dice rolls along with the mean.  You will then repeat the process 1000 times. This will give you the result of 1000 sample means. According to CLT, the result of these sample means will be gaussian. The example below shows the resulting distribution of sample means.  The following graph shows the distribution of sample means.  #### Become a Certified Power BI Developer PL-300 Microsoft Power BI Certification Training[Explore Program](https://www.simplilearn.com/power-bi-certification-training-course?source=GhPreviewCTABanner)  ## Conclusion The central limit theorem is a crucial concept in statistics and, by extension, data science. It's also crucial to learn about central tendency measures like mean, median, mode, and standard deviation. If you want to learn further, you can check the [Data Scientist](https://www.simplilearn.com/data-science-course "Data Scientist") course by Simplilearn. The course gives exposure to key technologies including R, [Python,](https://www.simplilearn.com/why-learn-python-a-guide-to-unlock-your-python-career-article "Python,") [Tableau](https://www.simplilearn.com/learn-tableau-tips-to-start-article "Tableau"), and Spark and will take you from basics to advanced level in learning. If you have any doubts and feedback regarding this article, do let us know in the comments section. ## About the Author [Avijeet Biswal](https://www.simplilearn.com/authors/avijeet-biswal) Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football. 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Central Limit Theorem, also known as the CLT, is a crucial pillar of statistics and [machine learning](https://www.simplilearn.com/tutorials/machine-learning-tutorial/what-is-machine-learning "machine learning"). It is at the heart of hypothesis testing. In this tutorial, you will understand the concept of the CLT and its applications. ## What is the Central Limit Theorem? The CLT is a statistical theory that states that - if you take a sufficiently large sample size from a population with a finite level of variance, the mean of all [samples](https://www.simplilearn.com/tutorials/machine-learning-tutorial/population-vs-sample "samples") from that population will be roughly equal to the population mean. Consider there are 15 sections in class X, and each section has 50 students. Our task is to calculate the average marks of students in class X. The standard approach will be to calculate the average simply: - Calculate the total marks of all the students in Class X - Add all the marks - Divide the total marks by the total number of students But what if the [data](https://www.simplilearn.com/what-is-data-article "data") is extremely large? Is this a good approach? No way, calculation marks of all the students will be a tedious and time-consuming process. So, what are the alternatives? Let's take a look at another approach. - To begin, select groups of students from the class at random. This will be referred to as a sample. Create several samples, each with 30 students. - Calculate each sample's individual mean. - Calculate the average of these sample means. - The value will give us the approximate average marks of the students in Class X. - The histogram of the sample means marks of the students will resemble a bell curve or normal distribution. ## Significance of Central Limit Theorem The CLT has several applications. Look at the places where you can use it. - Political/election polling is a great example of how you can use CLT. These polls are used to estimate the number of people who support a specific candidate. You may have seen these results with confidence intervals on news channels. The CLT aids in this calculation. - You use the CLT in various census fields to calculate various population details, such as family income, electricity consumption, individual salaries, and so on. The CLT is useful in a variety of fields. Are there any others that come to mind? Put them in the comments section below this tutorial. ## Assumptions Behind the Central Limit Theorem Before we move on further, it is important to understand the assumptions behind CLT: - The data must adhere to the randomization rule. It needs to be sampled at random. - The samples should be unrelated to one another. One sample should not impact the others. - When taking samples without replacement, the sample size should not exceed 10% of the population. When the population is symmetric, a sample size of 30 is generally considered reasonable. ## Why n ≥ 30 Samples? The sample size of 30 is considered sufficient to see the effect of the CLT. If the population distribution is closer to the normal distribution, you will need fewer samples to demonstrate the central limit theorem. On the other hand, if the population distribution is highly skewed, you will need a large number of samples to understand the CLT. ## Mean and Standard Deviation of the Sample You denote the mean of the sample by    And you denote as the standard deviation of the sample mean as:   That’s the concept and theory behind the CLT. Now, go to the [python](https://www.simplilearn.com/learn-the-basics-of-python-article "python") compiler and understand the working of CLT. ## Implementation Of Central Limit Theorem in Python You can understand the working of the CLT with an example involving the rolling of a die. A die has a different number on each side, ranging from 1 to 6. Each number has a one-in-six chance of appearing on a roll. Given the equal likelihood, the dispersion of the numbers that come up from a dice roll is uniform. You will use the randint() function to generate the random numbers ranging from 1 to 6.  The example will generate and print the sample of 100 dice rolls along with the mean.  You will then repeat the process 1000 times. This will give you the result of 1000 sample means. According to CLT, the result of these sample means will be gaussian. The example below shows the resulting distribution of sample means.  The following graph shows the distribution of sample means.  ## Conclusion The central limit theorem is a crucial concept in statistics and, by extension, data science. It's also crucial to learn about central tendency measures like mean, median, mode, and standard deviation. If you want to learn further, you can check the [Data Scientist](https://www.simplilearn.com/data-science-course "Data Scientist") course by Simplilearn. The course gives exposure to key technologies including R, [Python,](https://www.simplilearn.com/why-learn-python-a-guide-to-unlock-your-python-career-article "Python,") [Tableau](https://www.simplilearn.com/learn-tableau-tips-to-start-article "Tableau"), and Spark and will take you from basics to advanced level in learning. If you have any doubts and feedback regarding this article, do let us know in the comments section.