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Calculated Shard: 11 (from laksa140)

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curl -X POST \
  'http://laksa011.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.scribbr.com/statistics/central-limit-theorem/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.scribbr.com/statistics/central-limit-theorem/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/","crawl_time":1775340085,"first_indexed_time":1657547981,"http_code":200,"src_unparsed":"com,scribbr!www,\/statistics\/central-limit-theorem\/ s443","src_root_hash":"9175914091137272011","history_drop_reason":null,"meta_title":"Central Limit Theorem | Formula, Definition & Examples","meta_descriptions":["The central limit theorem states that if you take sufficiently large samples from a population, the samples’ means will be normally distributed."],"attrs_boilerpipe_text":"Published on\n    July 6, 2022\n    by\nShaun Turney\n.\n                \n        Revised on\n        February 25, 2026.\nThe\ncentral limit theorem\nstates that if you take sufficiently large samples from a population, the samples’ means will be\nnormally distributed\n, even if the population isn’t normally distributed.\nExample: Central limit theorem\nA\npopulation\nfollows a\nPoisson distribution\n(left image). If we take 10,000\nsamples\nfrom the population, each with a sample size of 50, the sample means follow a normal distribution, as predicted by the\ncentral limit theorem\n(right image).\nTable of contents\nWhat is the central limit theorem?\nCentral limit theorem formula\nSample size and the central limit theorem\nConditions of the central limit theorem\nImportance of the central limit theorem\nCentral limit theorem examples\nPractice questions\nOther interesting articles\nWhat is the central limit theorem?\nThe central limit theorem relies on the concept of a\nsampling distribution\n, which is the\nprobability distribution\nof a\nstatistic\nfor a large number of\nsamples\ntaken from a population.\nImagining an experiment may help you to understand sampling distributions:\nSuppose that you draw a\nrandom sample\nfrom a population and calculate a\nstatistic\nfor the sample, such as the mean.\nNow you draw another random sample of the same size, and again calculate the\nmean\n.\nYou repeat this process many times, and end up with a large number of means, one for each sample.\nThe distribution of the sample means is an example of a\nsampling distribution.\nThe central limit theorem says that the sampling distribution of the mean will always be\nnormally distributed\n, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal.\nA normal distribution is a symmetrical, bell-shaped distribution, with increasingly fewer observations the further from the center of the distribution.\nReceive feedback on language, structure, and formatting\nProfessional editors proofread and edit your paper by focusing on:\nAcademic style\nVague sentences\nGrammar\nStyle consistency\nSee an example\nCentral limit theorem formula\nFortunately, you don’t need to actually repeatedly sample a population to know the shape of the sampling distribution. The\nparameters\nof the sampling distribution of the mean are determined by the parameters of the population:\nThe\nmean\nof the sampling distribution is the mean of the population.\n \n \nThe\nstandard deviation\nof the sampling distribution is the standard deviation of the population divided by the square root of the sample size.\n \n \nWe can describe the sampling distribution of the mean using this notation:\n \n \nWhere:\nX̄ is the sampling distribution of the sample means\n~ means “follows the distribution”\nN\nis the\nnormal distribution\nµ is the mean of the population\nσ is the standard deviation of the population\nn\nis the sample size\nSample size and the central limit theorem\nThe\nsample size\n(\nn\n) is the number of observations drawn from the population for each sample. The sample size is the same for all samples.\nThe sample size affects the sampling distribution of the mean in two ways.\n1. Sample size and normality\nThe larger the sample size, the more closely the sampling distribution will follow a\nnormal distribution\n.\nWhen the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.”\nBy convention, we consider a sample size of 30 to be “sufficiently large.”\nWhen\nn\n< 30\n, the central limit theorem doesn’t apply. The sampling distribution will follow a similar distribution to the population. Therefore, the sampling distribution will only be normal if the population is normal.\nWhen\nn\n≥ 30\n, the central limit theorem applies. The sampling distribution will approximately follow a normal distribution.\n2. Sample size and standard deviations\nThe sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the\nvariability\nor spread of the distribution (i.e., how wide or narrow it is).\nWhen\nn\nis low\n, the standard deviation is high. There’s a lot of spread in the samples’ means because they aren’t precise estimates of the population’s mean.\nWhen\nn\nis high\n, the\nstandard deviation\nis low. There’s not much spread in the samples’ means because they’re precise estimates of the population’s mean.\nConditions of the central limit theorem\nThe central limit theorem states that the sampling distribution of the mean will always follow a\nnormal distribution\nunder the following conditions:\nThe sample size is\nsufficiently large\n. This condition is usually met if the sample size is\nn\n≥ 30.\nThe samples are\nindependent and identically distributed (i.i.d.) random variables\n. This condition is usually met if the\nsampling is random\n.\nThe population’s distribution has\nfinite\nvariance\n. Central limit theorem doesn’t apply to distributions with infinite variance, such as the Cauchy distribution. Most distributions have finite variance.\nImportance of the central limit theorem\nThe central limit theorem is one of the most fundamental statistical theorems. In fact, the “central” in “central limit theorem” refers to the importance of the theorem.\nNote\nParametric tests\n, such as\nt\ntests\n,\nANOVAs\n, and\nlinear regression\n, have more statistical power than most\nnon-parametric tests\n. Their\nstatistical power\ncomes from assumptions about populations’ distributions that are based on the central limit theorem.\nCentral limit theorem examples\nApplying the central limit theorem to real distributions may help you to better understand how it works.\nContinuous distribution\nSuppose that you’re interested in the age that people retire in the United States. The\npopulation\nis all retired Americans, and the distribution of the population might look something like this:\nAge at retirement follows a\nleft-skewed\ndistribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years.\nImagine that you take a small\nsample\nof the population. You randomly select five retirees and ask them what age they retired.\nExample: Central limit theorem; sample of\nn\n= 5\n68\n73\n70\n62\n63\nThe mean of the sample is an\nestimate\nof the population mean. It might not be a very precise estimate, since the sample size is only 5.\nExample: Central limit theorem; mean of a small sample\nmean = (68 + 73 + 70 + 62 + 63) \/ 5\nmean = 67.2 years\nSuppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a\nsampling distribution of the mean\n.\nExample: Central limit theorem; means of 10 small samples\n60.8\n57.8\n62.2\n68.6\n67.4\n67.8\n68.3\n65.6\n66.5\n62.1\nIf you repeat the procedure many more times, a histogram of the sample means will look something like this:\nAlthough this sampling distribution is more normally distributed than the population, it still has a bit of a\nleft skew\n.\nNotice also that the spread of the sampling distribution is less than the spread of the population.\nThe\ncentral limit theorem\nsays that the sampling distribution of the mean will always follow a normal distribution when the sample size is sufficiently large. This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large.\nNow, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired.\nExample: Central limit theorem; sample of\nn\n= 50\n73\n49\n62\n68\n72\n71\n65\n60\n69\n61\n62\n75\n66\n63\n66\n68\n76\n68\n54\n74\n68\n60\n72\n63\n57\n64\n65\n59\n72\n52\n52\n72\n69\n62\n68\n64\n60\n65\n53\n69\n59\n68\n67\n71\n69\n70\n52\n62\n64\n68\nThe mean of the sample is an\nestimate\nof the population mean. It’s a precise estimate, because the sample size is large.\nExample: Central limit theorem; mean of a large sample\nmean = 64.8 years\nAgain, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample:\nIn the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem.\nThe standard deviation of this sampling distribution is 0.85 years, which is less than the spread of the small sample sampling distribution, and much less than the spread of the population. If you were to increase the sample size further, the spread would decrease even more.\nWe can use the central limit theorem formula to describe the sampling distribution:\nµ = 65\nσ = 6\nn\n= 50\nDiscrete distribution\nApproximately 10% of people are left-handed. If we assign a value of 1 to left-handedness and a value of 0 to right-handedness, the\nprobability distribution\nof left-handedness for the\npopulation\nof all humans looks like this:\nThe population mean is the proportion of people who are left-handed (0.1). The population standard deviation is 0.3.\nImagine that you take a\nrandom sample\nof five people and ask them whether they’re left-handed.\nExample: Central limit theorem; sample of\nn\n= 5\n0\n0\n0\n1\n0\nThe mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5.\nExample: Central limit theorem; mean of a small sample\nmean = (0 + 0 + 0 + 1 + 0) \/ 5\nmean = 0.2\nImagine you repeat this process 10 times, randomly sampling five people and calculating the mean of the sample. This is a\nsampling distribution of the mean\n.\nExample: Central limit theorem; means of 10 small samples\n0\n0\n0.4\n0.2\n0.2\n0\n0.4\n0\nIf you repeat this process many more times, the distribution will look something like this:\nThe sampling distribution isn’t normally distributed because the sample size isn’t sufficiently large for the central limit theorem to apply.\nAs the sample size increases, the sampling distribution looks increasingly similar to a normal distribution, and the spread decreases:\nThe sampling distribution of the mean for samples with\nn\n= 30 approaches normality. When the sample size is increased further to\nn\n= 100, the sampling distribution follows a normal distribution.\nWe can use the central limit theorem formula to describe the sampling distribution for\nn\n= 100.\nµ = 0.1\nσ = 0.3\nn\n= 100\nPractice questions\nOther interesting articles\nIf you want to know more about\nstatistics\n,\nmethodology\n, or\nresearch bias\n, make sure to check out some of our other articles with explanations and examples.\nStatistics\nConfidence interval\nKurtosis\nDescriptive statistics\nMeasures of central tendency\nCorrelation coefficient\np\nvalue\nFAQ article\nCite this Scribbr article\nIf you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.\nTurney, S. \n                    (2026, February 25).\nCentral Limit Theorem | Formula, Definition & Examples.\nScribbr. \n        Retrieved April 2, 2026, \n        from https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/\nYou have already voted. 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[Mode](https:\/\/www.scribbr.com\/statistics\/mode\/ \"How to Find the Mode | Definition, Examples & Calculator\")\n    - [Median](https:\/\/www.scribbr.com\/statistics\/median\/ \"How to Find the Median | Definition, Examples & Calculator\")\n    - [Mean](https:\/\/www.scribbr.com\/statistics\/mean\/ \"How to Find the Mean | Definition, Examples & Calculator\")\n    - [Geometric mean](https:\/\/www.scribbr.com\/statistics\/geometric-mean\/ \"How to Find the Geometric Mean | Calculator & Formula\")\n  - [Measures of variability](https:\/\/www.scribbr.com\/statistics\/variability\/ \"Variability | Calculating Range, IQR, Variance, Standard Deviation\")\n    - [Overview](https:\/\/www.scribbr.com\/statistics\/variability\/ \"Variability | Calculating Range, IQR, Variance, Standard Deviation\")\n    - [Range](https:\/\/www.scribbr.com\/statistics\/range\/ \"How to Find the Range of a Data Set | Calculator & Formula\")\n    - [Interquartile range](https:\/\/www.scribbr.com\/statistics\/interquartile-range\/ \"How to Find Interquartile Range (IQR) | Calculator & Examples\")\n    - [Standard deviation](https:\/\/www.scribbr.com\/statistics\/standard-deviation\/ \"How to Calculate Standard Deviation (Guide) | Calculator & Examples\")\n    - [Variance](https:\/\/www.scribbr.com\/statistics\/variance\/ \"How to Calculate Variance | Calculator, Analysis & Examples\")\n  - [Frequency distribution](https:\/\/www.scribbr.com\/statistics\/frequency-distributions\/ \"Frequency Distribution | Tables, Types & Examples\")\n    - [Frequency distribution](https:\/\/www.scribbr.com\/statistics\/frequency-distributions\/ \"Frequency Distribution | Tables, Types & Examples\")\n    - [Quartiles & quantiles](https:\/\/www.scribbr.com\/statistics\/quartiles-quantiles\/ \"Quartiles & Quantiles | Calculation, Definition & Interpretation\")\n- [Probability (distributions)](https:\/\/www.scribbr.com\/statistics\/probability-distributions\/ \"Probability Distribution | Formula, Types, & Examples\")\n  - [Probability distributions](https:\/\/www.scribbr.com\/statistics\/probability-distributions\/ \"Probability Distribution | Formula, Types, & Examples\")\n    - [Overview](https:\/\/www.scribbr.com\/statistics\/probability-distributions\/ \"Probability Distribution | Formula, Types, & Examples\")\n    - [Normal distribution](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/ \"Normal Distribution | Examples, Formulas, & Uses\")\n    - [Standard normal distribution](https:\/\/www.scribbr.com\/statistics\/standard-normal-distribution\/ \"The Standard Normal Distribution | Calculator, Examples & Uses\")\n    - [Poisson distribution](https:\/\/www.scribbr.com\/statistics\/poisson-distribution\/ \"Poisson Distributions | Definition, Formula & Examples\")\n    - [Chi-square distribution](https:\/\/www.scribbr.com\/statistics\/chi-square-distributions\/ \"Chi-Square (Χ²) Distributions | Definition & Examples\")\n    - [Chi-square table](https:\/\/www.scribbr.com\/statistics\/chi-square-distribution-table\/ \"Chi-Square (Χ²) Table | Examples & Downloadable Table\")\n    - [*t* distribution](https:\/\/www.scribbr.com\/statistics\/t-distribution\/ \"T-distribution: What it is and how to use it\")\n    - [*t* table](https:\/\/www.scribbr.com\/statistics\/students-t-table\/ \"Student’s t Table (Free Download) | Guide & Examples\")\n- [Inferential statistics](https:\/\/www.scribbr.com\/statistics\/inferential-statistics\/ \"Inferential Statistics | An Easy Introduction & Examples\")\n  - [Overview inferential statistics](https:\/\/www.scribbr.com\/statistics\/inferential-statistics\/ \"Inferential Statistics | An Easy Introduction & Examples\")\n  - [Degrees of freedom](https:\/\/www.scribbr.com\/statistics\/degrees-of-freedom\/ \"How to Find Degrees of Freedom | Definition & Formula\")\n  - [Central limit theorem](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/ \"Central Limit Theorem | Formula, Definition & Examples\")\n  - [Parameters & test statistics](https:\/\/www.scribbr.com\/statistics\/parameter-vs-statistic\/ \"Parameter vs Statistic | Definitions, Differences & Examples\")\n    - [Parameters vs. test statistics](https:\/\/www.scribbr.com\/statistics\/parameter-vs-statistic\/ \"Parameter vs Statistic | Definitions, Differences & Examples\")\n    - [Test statistics](https:\/\/www.scribbr.com\/statistics\/test-statistic\/ \"Test statistics | Definition, Interpretation, and Examples\")\n  - [Estimation](https:\/\/www.scribbr.com\/statistics\/standard-error\/ \"What Is Standard Error? | How to Calculate (Guide with Examples)\")\n    - [Standard error](https:\/\/www.scribbr.com\/statistics\/standard-error\/ \"What Is Standard Error? | How to Calculate (Guide with Examples)\")\n    - [Confidence intervals](https:\/\/www.scribbr.com\/statistics\/confidence-interval\/ \"Understanding Confidence Intervals | Easy Examples & Formulas\")\n  - [Hypothesis testing](https:\/\/www.scribbr.com\/statistics\/hypothesis-testing\/ \"Hypothesis Testing | A Step-by-Step Guide with Easy Examples\")\n    - [Hypothesis testing guide](https:\/\/www.scribbr.com\/statistics\/hypothesis-testing\/ \"Hypothesis Testing | A Step-by-Step Guide with Easy Examples\")\n    - [Null vs. alternative hypotheses](https:\/\/www.scribbr.com\/statistics\/null-and-alternative-hypotheses\/ \"Null and Alternative Hypotheses | Definitions & Examples\")\n    - [Statistical significance](https:\/\/www.scribbr.com\/statistics\/statistical-significance\/ \"An Easy Introduction to Statistical Significance (With Examples)\")\n    - [*p* value](https:\/\/www.scribbr.com\/statistics\/p-value\/ \"Understanding P values | Definition and Examples\")\n    - [Type I & Type II errors](https:\/\/www.scribbr.com\/statistics\/type-i-and-type-ii-errors\/ \"Type I & Type II Errors | Differences, Examples, Visualizations\")\n    - [Statistical power](https:\/\/www.scribbr.com\/statistics\/statistical-power\/ \"Statistical Power and Why It Matters | A Simple Introduction\")\n- [Statistical tests](https:\/\/www.scribbr.com\/statistics\/statistical-tests\/ \"Choosing the Right Statistical Test | Types & Examples\")\n  - [Choosing the right test](https:\/\/www.scribbr.com\/statistics\/statistical-tests\/ \"Choosing the Right Statistical Test | Types & Examples\")\n  - [Assumptions for hypothesis testing](https:\/\/www.scribbr.com\/statistics\/skewness\/ \"Skewness | Definition, Examples & Formula\")\n    - [Skewness](https:\/\/www.scribbr.com\/statistics\/skewness\/ \"Skewness | Definition, Examples & Formula\")\n    - [Kurtosis](https:\/\/www.scribbr.com\/statistics\/kurtosis\/ \"What Is Kurtosis? | Definition, Examples & Formula\")\n  - [Correlation](https:\/\/www.scribbr.com\/methodology\/correlation-vs-causation\/ \"Correlation vs. Causation | Difference, Designs & Examples\")\n    - [Correlation vs. causation](https:\/\/www.scribbr.com\/methodology\/correlation-vs-causation\/ \"Correlation vs. Causation | Difference, Designs & Examples\")\n    - [Correlation coefficient](https:\/\/www.scribbr.com\/statistics\/correlation-coefficient\/ \"Correlation Coefficient | Types, Formulas & Examples\")\n    - [Pearson correlation](https:\/\/www.scribbr.com\/statistics\/pearson-correlation-coefficient\/ \"Pearson Correlation Coefficient (r) | Guide & Examples\")\n  - [Regression analysis](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/ \"Simple Linear Regression | An Easy Introduction & Examples\")\n    - [Simple linear regression](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/ \"Simple Linear Regression | An Easy Introduction & Examples\")\n    - [Multiple linear regression](https:\/\/www.scribbr.com\/statistics\/multiple-linear-regression\/ \"Multiple Linear Regression | A Quick Guide (Examples)\")\n    - [Linear regression in R](https:\/\/www.scribbr.com\/statistics\/linear-regression-in-r\/ \"Linear Regression in R | A Step-by-Step Guide & Examples\")\n  - [*t* tests](https:\/\/www.scribbr.com\/statistics\/t-test\/ \"An Introduction to t Tests | Definitions, Formula and Examples\")\n  - [ANOVAs](https:\/\/www.scribbr.com\/statistics\/one-way-anova\/ \"One-way ANOVA | When and How to Use It (With Examples)\")\n    - [One-way ANOVA](https:\/\/www.scribbr.com\/statistics\/one-way-anova\/ \"One-way ANOVA | When and How to Use It (With Examples)\")\n    - [Two-way ANOVA](https:\/\/www.scribbr.com\/statistics\/two-way-anova\/ \"Two-Way ANOVA | Examples & When To Use It\")\n    - [ANOVA in R](https:\/\/www.scribbr.com\/statistics\/anova-in-r\/ \"ANOVA in R | A Complete Step-by-Step Guide with Examples\")\n  - [Chi-square](https:\/\/www.scribbr.com\/statistics\/chi-square-tests\/ \"Chi-Square (Χ²) Tests | Types, Formula & Examples\")\n    - [Overview chi-square tests](https:\/\/www.scribbr.com\/statistics\/chi-square-tests\/ \"Chi-Square (Χ²) Tests | Types, Formula & Examples\")\n    - [Chi-square goodness of fit test](https:\/\/www.scribbr.com\/statistics\/chi-square-goodness-of-fit\/ \"Chi-Square Goodness of Fit Test | Formula, Guide & Examples\")\n    - [Chi-square test of independence](https:\/\/www.scribbr.com\/statistics\/chi-square-test-of-independence\/ \"Chi-Square Test of Independence | Formula, Guide & Examples\")\n  - [Effect size](https:\/\/www.scribbr.com\/statistics\/effect-size\/ \"What is Effect Size and Why Does It Matter? (Examples)\")\n    - [Overview of effect sizes](https:\/\/www.scribbr.com\/statistics\/effect-size\/ \"What is Effect Size and Why Does It Matter? (Examples)\")\n    - [Coefficient of determination](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/ \"Coefficient of Determination (R²) | Calculation & Interpretation\")\n  - [Model selection](https:\/\/www.scribbr.com\/statistics\/akaike-information-criterion\/ \"Akaike Information Criterion | When & How to Use It (Example)\")\n    - [Akaike information criterion](https:\/\/www.scribbr.com\/statistics\/akaike-information-criterion\/ \"Akaike Information Criterion | When & How to Use It (Example)\")\n- [Reporting statistics in APA](https:\/\/www.scribbr.com\/apa-style\/numbers-and-statistics\/ \"Reporting Statistics in APA Style | Guidelines & Examples\")\n\n### Interesting topics\n- [Parts of speech](https:\/\/www.scribbr.com\/category\/parts-of-speech\/ \"View all articles about Parts of speech\")\n- [Working with sources](https:\/\/www.scribbr.com\/category\/working-with-sources\/ \"View all articles about Working with sources\")\n- [IEEE](https:\/\/www.scribbr.com\/category\/ieee\/ \"View all articles about IEEE\")\n- [Commonly confused words](https:\/\/www.scribbr.com\/category\/commonly-confused-words\/ \"View all articles about Commonly confused words\")\n- [Commas](https:\/\/www.scribbr.com\/category\/commas\/ \"View all articles about Commas\")\n- [Definitions](https:\/\/www.scribbr.com\/category\/definitions\/ \"View all articles about Definitions\")\n- [UK vs. US English](https:\/\/www.scribbr.com\/category\/us-vs-uk\/ \"View all articles about UK vs. US English\")\n- [Research bias](https:\/\/www.scribbr.com\/category\/research-bias\/ \"View all articles about Research bias\")\n- [Nouns and pronouns](https:\/\/www.scribbr.com\/category\/nouns-and-pronouns\/ \"View all articles about Nouns and pronouns\")\n- [AMA style](https:\/\/www.scribbr.com\/category\/ama\/ \"View all articles about AMA style\")\n- [College essay](https:\/\/www.scribbr.com\/category\/college-essay\/ \"View all articles about College essay\")\n- [Sentence structure](https:\/\/www.scribbr.com\/category\/sentence-structure\/ \"View all articles about Sentence structure\")\n- [Verbs](https:\/\/www.scribbr.com\/category\/verbs\/ \"View all articles about Verbs\")\n- [Common mistakes](https:\/\/www.scribbr.com\/category\/common-mistakes\/ \"View all articles about Common mistakes\")\n- [Effective communication](https:\/\/www.scribbr.com\/category\/effective-communication\/ \"View all articles about Effective communication\")\n- [Using AI tools](https:\/\/www.scribbr.com\/category\/ai-tools\/ \"View all articles about Using AI tools\")\n- [Fallacies](https:\/\/www.scribbr.com\/category\/fallacies\/ \"View all articles about Fallacies\")\n- [Rhetoric](https:\/\/www.scribbr.com\/category\/rhetoric\/ \"View all articles about Rhetoric\")\n- [Plurals](https:\/\/www.scribbr.com\/category\/plurals\/ \"View all articles about Plurals\")\n- [APA Style 6th edition](https:\/\/www.scribbr.com\/category\/apa-style\/6th-edition\/ \"View all articles about APA Style 6th edition\")\n- [Applying to graduate school](https:\/\/www.scribbr.com\/category\/graduate-school\/ \"View all articles about Applying to graduate school\")\n- [Statistics](https:\/\/www.scribbr.com\/category\/statistics\/ \"View all articles about Statistics\")\n- [Chicago Style](https:\/\/www.scribbr.com\/category\/chicago-style\/ \"View all articles about Chicago Style\")\n- [Language rules](https:\/\/www.scribbr.com\/category\/language-rules\/ \"View all articles about Language rules\")\n- [Methodology](https:\/\/www.scribbr.com\/category\/methodology\/ \"View all articles about Methodology\")\n- [MLA Style](https:\/\/www.scribbr.com\/category\/mla\/ \"View all articles about MLA Style\")\n- [Research paper](https:\/\/www.scribbr.com\/category\/research-paper\/ \"View all articles about Research paper\")\n- [Academic writing](https:\/\/www.scribbr.com\/category\/academic-writing\/ \"View all articles about Academic writing\")\n- [Starting the research process](https:\/\/www.scribbr.com\/category\/research-process\/ \"View all articles about Starting the research process\")\n- [Dissertation](https:\/\/www.scribbr.com\/category\/dissertation\/ \"View all articles about Dissertation\")\n- [Essay](https:\/\/www.scribbr.com\/category\/academic-essay\/ \"View all articles about Essay\")\n- [Tips](https:\/\/www.scribbr.com\/category\/tips\/ \"View all articles about Tips\")\n- [APA Style 7th edition](https:\/\/www.scribbr.com\/category\/apa-style\/ \"View all articles about APA Style 7th edition\")\n- [APA citation examples](https:\/\/www.scribbr.com\/category\/apa-examples\/ \"View all articles about APA citation examples\")\n- [Citing sources](https:\/\/www.scribbr.com\/category\/citing-sources\/ \"View all articles about Citing sources\")\n- [Plagiarism](https:\/\/www.scribbr.com\/category\/plagiarism\/ \"View all articles about Plagiarism\")\n\n##### Try our other services\n[![](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2032%2032'%3E%3C\/svg%3E) ![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2024\/09\/human-proofreading.svg) Proofreading & Editing Have a human editor polish your writing to ensure your arguments are judged on merit, not grammar errors. Get expert writing help](https:\/\/www.scribbr.com\/proofreading-editing\/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar)\n\n[![](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2032%2032'%3E%3C\/svg%3E) ![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2024\/09\/ai-proofreading.svg) AI Proofreader Get unlimited proofreading for 30 days Try for free](https:\/\/www.scribbr.com\/ai-proofreader\/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar)\n\n[![](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2032%2032'%3E%3C\/svg%3E) ![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2024\/09\/plagiarism-checker.svg) Plagiarism Checker Compare your paper to billions of pages and articles with Scribbr’s plagiarism checker. Run a free check](https:\/\/www.scribbr.com\/plagiarism-checker\/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar)\n\n[![](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2032%2032'%3E%3C\/svg%3E) ![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2024\/09\/citation-generator.svg) Citation Generator Generate accurate APA, MLA, and Chicago citations for free with Scribbr's Citation Generator. Start citing](https:\/\/www.scribbr.com\/citation\/generator\/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar)\n\n[![](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2032%2032'%3E%3C\/svg%3E) ![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2024\/09\/paraphrasing.svg) Paraphrasing Tool Rewrite and paraphrase texts instantly with our AI-powered paraphrasing tool. Try for free](https:\/\/www.scribbr.com\/paraphrasing-tool\/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar)\n\n[![](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2032%2032'%3E%3C\/svg%3E) ![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2024\/09\/grammer-checker.svg) Grammar Checker Eliminate grammar errors and improve your writing with our free AI-powered grammar checker. Try for free](https:\/\/www.scribbr.com\/grammar-checker\/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar)\n\n# Central Limit Theorem \\| Formula, Definition & Examples\nPublished on July 6, 2022 by  [Shaun Turney](https:\/\/www.scribbr.com\/author\/shaunt\/ \"All articles by Shaun Turney\"). Revised on February 25, 2026.\n\nThe **central limit theorem** states that if you take sufficiently large samples from a population, the samples’ means will be [normally distributed](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/) , even if the population isn’t normally distributed.\n\nExample: Central limit theorem\n\nA **population** follows a [**Poisson distribution**](https:\/\/www.scribbr.com\/statistics\/poisson-distribution\/) (left image). If we take 10,000 **samples** from the population, each with a sample size of 50, the sample means follow a normal distribution, as predicted by the **central limit theorem** (right image).\n\n![Central Limit Theorem](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%200'%3E%3C\/svg%3E)![Central Limit Theorem](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/Central-limit-theorem.webp)\n\n## Table of contents\n1. [What is the central limit theorem?](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#central-limit-theorem)\n2. [Central limit theorem formula](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#theorem-formula)\n3. [Sample size and the central limit theorem](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#theorem)\n4. [Conditions of the central limit theorem](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#conditions)\n5. [Importance of the central limit theorem](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#importance)\n6. [Central limit theorem examples](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#examples)\n7. [Practice questions](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#practice-questions)\n8. [Other interesting articles](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#other)\n## What is the central limit theorem?\nThe central limit theorem relies on the concept of a **sampling distribution** , which is the [probability distribution](https:\/\/www.scribbr.com\/statistics\/probability-distributions\/) of a **statistic** for a large number of [samples](https:\/\/www.scribbr.com\/methodology\/population-vs-sample\/) taken from a population.\n\nImagining an experiment may help you to understand sampling distributions:\n\n- Suppose that you draw a [random sample](https:\/\/www.scribbr.com\/methodology\/simple-random-sampling\/) from a population and calculate a [statistic](https:\/\/www.scribbr.com\/statistics\/parameter-vs-statistic\/) for the sample, such as the mean.\n- Now you draw another random sample of the same size, and again calculate the [mean](https:\/\/www.scribbr.com\/statistics\/mean\/) .\n- You repeat this process many times, and end up with a large number of means, one for each sample.\n\nThe distribution of the sample means is an example of a **sampling distribution.**\n\nThe central limit theorem says that the sampling distribution of the mean will always be **normally distributed**, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal.\n\nA normal distribution is a symmetrical, bell-shaped distribution, with increasingly fewer observations the further from the center of the distribution.\n\n### Receive feedback on language, structure, and formatting\nProfessional editors proofread and edit your paper by focusing on:\n\n- Academic style\n- Vague sentences\n- Grammar\n- Style consistency\n\n[See an example](https:\/\/www.scribbr.com\/proofreading-editing\/paper-editing-service\/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=feedback+language+structure+layout)\n\n![](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20830%20660'%3E%3C\/svg%3E)\n\n![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2018\/01\/dissertation-proofreading-service.png)\n## Central limit theorem formula\nFortunately, you don’t need to actually repeatedly sample a population to know the shape of the sampling distribution. The [parameters](https:\/\/www.scribbr.com\/statistics\/parameter-vs-statistic\/) of the sampling distribution of the mean are determined by the parameters of the population:\n\n- The [mean](https:\/\/www.scribbr.com\/statistics\/mean\/) of the sampling distribution is the mean of the population.\n\n![\\\\begin{equation\\*}\\\\mu\\_{\\\\bar{x}}=\\\\mu\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2057%2012'%3E%3C\/svg%3E)![\\\\begin{equation\\*}\\\\mu\\_{\\\\bar{x}}=\\\\mu\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-7f19c78f3ab7a25bf9edd19b8ba36367_l3.png)\n\n- The [standard deviation](https:\/\/www.scribbr.com\/statistics\/standard-deviation\/) of the sampling distribution is the standard deviation of the population divided by the square root of the sample size.\n\n![\\\\begin{equation\\*}\\\\sigma\\_{\\\\bar{x}} = \\\\dfrac{\\\\sigma}{\\\\sqrt{n}}\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2074%2038'%3E%3C\/svg%3E)![\\\\begin{equation\\*}\\\\sigma\\_{\\\\bar{x}} = \\\\dfrac{\\\\sigma}{\\\\sqrt{n}}\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-6443a361247d9c6f5764e4503e390634_l3.png)\n\nWe can describe the sampling distribution of the mean using this notation:\n\n![\\\\begin{equation\\*}\\\\bar{X}\\\\sim N (\\\\mu,\\\\dfrac{\\\\sigma}{\\\\sqrt{n}})\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20123%2038'%3E%3C\/svg%3E)![\\\\begin{equation\\*}\\\\bar{X}\\\\sim N (\\\\mu,\\\\dfrac{\\\\sigma}{\\\\sqrt{n}})\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-34ffb4548c9499a6e0972093a9d2b1ae_l3.png)\n\nWhere:\n\n- X̄ is the sampling distribution of the sample means\n- ~ means “follows the distribution”\n- *N* is the [normal distribution](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/)\n- µ is the mean of the population\n- σ is the standard deviation of the population\n- *n* is the sample size\n## Sample size and the central limit theorem\nThe **sample size** (*n*) is the number of observations drawn from the population for each sample. The sample size is the same for all samples.\n\nThe sample size affects the sampling distribution of the mean in two ways.\n\n### 1\\. Sample size and normality\nThe larger the sample size, the more closely the sampling distribution will follow a [normal distribution](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/).\n\nWhen the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.”\n\nBy convention, we consider a sample size of 30 to be “sufficiently large.”\n\n- **When** ***n*** **\\< 30**, the central limit theorem doesn’t apply. The sampling distribution will follow a similar distribution to the population. Therefore, the sampling distribution will only be normal if the population is normal.\n\n- **When** ***n*** **≥ 30**, the central limit theorem applies. The sampling distribution will approximately follow a normal distribution.\n### 2\\. Sample size and standard deviations\nThe sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the [variability](https:\/\/www.scribbr.com\/statistics\/variability\/) or spread of the distribution (i.e., how wide or narrow it is).\n\n- **When** ***n*** **is low**, the standard deviation is high. There’s a lot of spread in the samples’ means because they aren’t precise estimates of the population’s mean.\n\n- **When** ***n*** **is high**, the [standard deviation](https:\/\/www.scribbr.com\/statistics\/standard-deviation\/) is low. There’s not much spread in the samples’ means because they’re precise estimates of the population’s mean.\n## Conditions of the central limit theorem\nThe central limit theorem states that the sampling distribution of the mean will always follow a [normal distribution](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/) under the following conditions:\n\n1. The sample size is **sufficiently large**. This condition is usually met if the sample size is *n* ≥ 30.\n\n1. The samples are **independent and identically distributed (i.i.d.) random variables**. This condition is usually met if the [sampling is random](https:\/\/www.scribbr.com\/methodology\/simple-random-sampling\/).\n\n1. The population’s distribution has **finite** [**variance**](https:\/\/www.scribbr.com\/statistics\/variance\/). Central limit theorem doesn’t apply to distributions with infinite variance, such as the Cauchy distribution. Most distributions have finite variance.\n\n## Here's why students love Scribbr's proofreading services\n[Trustpilot](https:\/\/www.trustpilot.com\/review\/www.scribbr.com)\n\n[Discover proofreading & editing](https:\/\/www.scribbr.com\/proofreading-editing\/paper-editing-service\/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=trustpilot)\n## Importance of the central limit theorem\nThe central limit theorem is one of the most fundamental statistical theorems. In fact, the “central” in “central limit theorem” refers to the importance of the theorem.\n\nNote\n\n[**Parametric tests**](https:\/\/www.scribbr.com\/statistics\/statistical-tests\/#parametric), such as [*t* tests](https:\/\/www.scribbr.com\/statistics\/t-test\/), [ANOVAs](https:\/\/www.scribbr.com\/statistics\/one-way-anova\/), and [linear regression](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/), have more statistical power than most [non-parametric tests](https:\/\/www.scribbr.com\/statistics\/statistical-tests\/#nonparametric). Their [statistical power](https:\/\/www.scribbr.com\/statistics\/statistical-power\/) comes from assumptions about populations’ distributions that are based on the central limit theorem.\n## Central limit theorem examples\nApplying the central limit theorem to real distributions may help you to better understand how it works.\n\n### Continuous distribution\nSuppose that you’re interested in the age that people retire in the United States. The [**population**](https:\/\/www.scribbr.com\/methodology\/population-vs-sample\/) is all retired Americans, and the distribution of the population might look something like this:\n\n![Central Limit Theorem - Continuous distribution](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%200'%3E%3C\/svg%3E)![Central Limit Theorem - Continuous distribution](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/continuous-distribution.png)\n\nAge at retirement follows a [left-skewed](https:\/\/www.scribbr.com\/statistics\/skewness\/#left-skew) distribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years.\n\nImagine that you take a small **sample** of the population. You randomly select five retirees and ask them what age they retired.\n\nExample: Central limit theorem; sample of *n* = 5\n\n|   |   |   |   |   |\n|---|---|---|---|---|\n| 68 | 73 | 70 | 62 | 63 |\n\nThe mean of the sample is an [estimate](https:\/\/www.scribbr.com\/statistics\/parameter-vs-statistic\/#estimating-parameters-from-statistics) of the population mean. It might not be a very precise estimate, since the sample size is only 5.\n\nExample: Central limit theorem; mean of a small sample\n\nmean = (68 + 73 + 70 + 62 + 63) \/ 5\n\nmean = 67.2 years\n\nSuppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a **sampling distribution of the mean**.\n\nExample: Central limit theorem; means of 10 small samples\n\n|   |   |   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|---|---|\n| 60\\.8 | 57\\.8 | 62\\.2 | 68\\.6 | 67\\.4 | 67\\.8 | 68\\.3 | 65\\.6 | 66\\.5 | 62\\.1 |\n\nIf you repeat the procedure many more times, a histogram of the sample means will look something like this:\n\n![Central Limit Theorem - Sampling distribution](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%200'%3E%3C\/svg%3E)![Central Limit Theorem - Sampling distribution](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/Sampling-distribution-of-the-mean-1.webp)\n\nAlthough this sampling distribution is more normally distributed than the population, it still has a bit of a [left skew](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/).\n\nNotice also that the spread of the sampling distribution is less than the spread of the population.\n\nThe **central limit theorem** says that the sampling distribution of the mean will always follow a normal distribution when the sample size is sufficiently large. This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large.\n\nNow, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired.\n\nExample: Central limit theorem; sample of *n* = 50\n\n|   |   |   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|---|---|\n| 73 | 49 | 62 | 68 | 72 | 71 | 65 | 60 | 69 | 61 |\n| 62 | 75 | 66 | 63 | 66 | 68 | 76 | 68 | 54 | 74 |\n| 68 | 60 | 72 | 63 | 57 | 64 | 65 | 59 | 72 | 52 |\n| 52 | 72 | 69 | 62 | 68 | 64 | 60 | 65 | 53 | 69 |\n| 59 | 68 | 67 | 71 | 69 | 70 | 52 | 62 | 64 | 68 |\n\nThe mean of the sample is an [estimate](https:\/\/www.scribbr.com\/statistics\/parameter-vs-statistic\/#estimating-parameters-from-statistics) of the population mean. It’s a precise estimate, because the sample size is large.\n\nExample: Central limit theorem; mean of a large sample\n\nmean = 64.8 years\n\nAgain, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample:\n\n![Central Limit Theorem - Mean-of-a-large-sample](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%200'%3E%3C\/svg%3E)![Central Limit Theorem - Mean-of-a-large-sample](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/Mean-of-a-large-sample-1.webp)\n\nIn the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem.\n\nThe standard deviation of this sampling distribution is 0.85 years, which is less than the spread of the small sample sampling distribution, and much less than the spread of the population. If you were to increase the sample size further, the spread would decrease even more.\n\nWe can use the central limit theorem formula to describe the sampling distribution:\n\n![\\\\bar{X} \\\\sim N (\\\\mu,\\\\dfrac{\\\\sigma}{\\\\sqrt{n}})](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20123%2038'%3E%3C\/svg%3E)![\\\\bar{X} \\\\sim N (\\\\mu,\\\\dfrac{\\\\sigma}{\\\\sqrt{n}})](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-8a8299367f3bafc9ded6b4b151f2cbc6_l3.png)\n\nµ = 65\n\nσ = 6\n\n*n* = 50\n\n![\\\\bar{X} \\\\sim N (65,\\\\dfrac{6}{\\\\sqrt{50}})](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20138%2043'%3E%3C\/svg%3E)![\\\\bar{X} \\\\sim N (65,\\\\dfrac{6}{\\\\sqrt{50}})](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-c749b7f11e70e99d290bf4118c7ca85d_l3.png)\n\n![\\\\bar{X} \\\\sim N (65,0.85)](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20132%2021'%3E%3C\/svg%3E)![\\\\bar{X} \\\\sim N (65,0.85)](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-3cad0ef92bdfaa9dbd48b7b269a6bf4e_l3.png)\n\n### Discrete distribution\nApproximately 10% of people are left-handed. If we assign a value of 1 to left-handedness and a value of 0 to right-handedness, the [probability distribution](https:\/\/www.scribbr.com\/statistics\/probability-distributions\/) of left-handedness for the **population** of all humans looks like this:\n\n![Central Limit Theorem - Theorem-discrete-distribution](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%200'%3E%3C\/svg%3E)![Central Limit Theorem - Theorem-discrete-distribution](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/central-limit-theorem-discrete-distribution.png)\n\nThe population mean is the proportion of people who are left-handed (0.1). The population standard deviation is 0.3.\n\nImagine that you take a [random sample](https:\/\/www.scribbr.com\/methodology\/simple-random-sampling\/) of five people and ask them whether they’re left-handed.\n\nExample: Central limit theorem; sample of *n* = 5\n\n|   |   |   |   |   |\n|---|---|---|---|---|\n| 0 | 0 | 0 | 1 | 0 |\n\nThe mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5.\n\nExample: Central limit theorem; mean of a small sample\n\nmean = (0 + 0 + 0 + 1 + 0) \/ 5\n\nmean = 0.2\n\nImagine you repeat this process 10 times, randomly sampling five people and calculating the mean of the sample. This is a **sampling distribution of the mean** .\n\nExample: Central limit theorem; means of 10 small samples\n\n|   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|\n| 0 | 0 | 0\\.4 | 0\\.2 | 0\\.2 | 0 | 0\\.4 | 0 |\n\nIf you repeat this process many more times, the distribution will look something like this:\n\n![Central Limit Theorem - Theorem-discrete-distribution](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%200'%3E%3C\/svg%3E)![Central Limit Theorem - Theorem-discrete-distribution](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/central-limit-theorem-small-samples.png)\n\nThe sampling distribution isn’t normally distributed because the sample size isn’t sufficiently large for the central limit theorem to apply.\n\nAs the sample size increases, the sampling distribution looks increasingly similar to a normal distribution, and the spread decreases:\n\n- [*n* = 10](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/)\n- [*n* = 20](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/)\n- [*n* = 30](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/)\n- [*n* = 100](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/)\n\n![Central Limit Theorem - n=10](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%20418'%3E%3C\/svg%3E)\n\n![Central Limit Theorem - n=10](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/central-limit-theorem-n10.png)\n\n![Central Limit Theorem - n=20](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%20418'%3E%3C\/svg%3E)\n\n![Central Limit Theorem - n=20](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/central-limit-theorem-n20.png)\n\n![Central Limit Theorem - n=30](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%20418'%3E%3C\/svg%3E)\n\n![Central Limit Theorem - n=30](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/central-limit-theorem-n30.png)\n\n![Central Limit Theorem - n=100](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%20418'%3E%3C\/svg%3E)\n\n![Central Limit Theorem - n=100](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/central-limit-theorem-n100.png)\n\nThe sampling distribution of the mean for samples with *n* = 30 approaches normality. When the sample size is increased further to *n* = 100, the sampling distribution follows a normal distribution.\n\nWe can use the central limit theorem formula to describe the sampling distribution for *n* = 100.\n\n![\\\\bar{X} \\\\sim N (\\\\mu,\\\\dfrac{\\\\sigma}{\\\\sqrt{n}})](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20123%2038'%3E%3C\/svg%3E) ![\\\\bar{X} \\\\sim N (\\\\mu,\\\\dfrac{\\\\sigma}{\\\\sqrt{n}})](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-8a8299367f3bafc9ded6b4b151f2cbc6_l3.png)\n\nµ = 0.1\n\nσ = 0.3\n\n*n* = 100\n\n![\\\\bar{X} \\\\sim N (0.1,\\\\dfrac{0.3}{\\\\sqrt{100}})](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20153%2043'%3E%3C\/svg%3E) ![\\\\bar{X} \\\\sim N (0.1,\\\\dfrac{0.3}{\\\\sqrt{100}})](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-d9b84c5604bb23b5830a2deee4872381_l3.png)\n\n![\\\\bar{X} \\\\sim N (0.1,0.03)](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20137%2021'%3E%3C\/svg%3E) ![\\\\bar{X} \\\\sim N (0.1,0.03)](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-3e8ab9abb7207f6044a9873b22db8681_l3.png)\n\n## Practice questions\n## Other interesting articles\nIf you want to know more about [statistics](https:\/\/www.scribbr.com\/category\/statistics\/) , [methodology](https:\/\/www.scribbr.com\/category\/methodology\/) , or [research bias](https:\/\/www.scribbr.com\/faq-category\/research-bias\/) , make sure to check out some of our other articles with explanations and examples.\n\n**Statistics**\n\n- [Confidence interval](https:\/\/www.scribbr.com\/statistics\/confidence-interval\/)\n- [Kurtosis](https:\/\/www.scribbr.com\/statistics\/kurtosis\/)\n- [Descriptive statistics](https:\/\/www.scribbr.com\/statistics\/descriptive-statistics\/)\n- [Measures of central tendency](https:\/\/www.scribbr.com\/statistics\/central-tendency\/)\n- [Correlation coefficient](https:\/\/www.scribbr.com\/statistics\/correlation-coefficient\/)\n- [*p* value](https:\/\/www.scribbr.com\/statistics\/p-value\/)\n\n**Methodology**\n\n- [Cluster sampling](https:\/\/www.scribbr.com\/methodology\/cluster-sampling\/)\n- [Stratified sampling](https:\/\/www.scribbr.com\/methodology\/stratified-sampling\/)\n- [Types of interviews](https:\/\/www.scribbr.com\/methodology\/interviews-research\/)\n- [Case study](https:\/\/www.scribbr.com\/methodology\/case-study\/)\n- [Cohort study](https:\/\/www.scribbr.com\/methodology\/cohort-study\/)\n- [Thematic analysis](https:\/\/www.scribbr.com\/methodology\/thematic-analysis\/)\n\n**Research bias**\n\n- [Implicit bias](https:\/\/www.scribbr.com\/research-bias\/implicit-bias\/)\n- [Cognitive bias](https:\/\/www.scribbr.com\/research-bias\/cognitive-bias\/)\n- [Survivorship bias](https:\/\/www.scribbr.com\/research-bias\/survivorship-bias\/)\n- [Availability heuristic](https:\/\/www.scribbr.com\/research-bias\/availability-heuristic\/)\n- [Nonresponse bias](https:\/\/www.scribbr.com\/research-bias\/nonresponse-bias\/)\n- [Regression to the mean](https:\/\/www.scribbr.com\/research-bias\/regression-to-the-mean\/)\n\nFAQ article\n\n#### Cite this Scribbr article\nIf you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.\n\n> Turney, S. 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Revised on February 25, 2026.\n\nThe **central limit theorem** states that if you take sufficiently large samples from a population, the samples’ means will be [normally distributed](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/) , even if the population isn’t normally distributed.\n\nExample: Central limit theorem\n\nA **population** follows a [**Poisson distribution**](https:\/\/www.scribbr.com\/statistics\/poisson-distribution\/) (left image). If we take 10,000 **samples** from the population, each with a sample size of 50, the sample means follow a normal distribution, as predicted by the **central limit theorem** (right image).\n\n![Central Limit Theorem](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/Central-limit-theorem.webp)\n\n## Table of contents\n1. [What is the central limit theorem?](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#central-limit-theorem)\n2. [Central limit theorem formula](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#theorem-formula)\n3. [Sample size and the central limit theorem](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#theorem)\n4. [Conditions of the central limit theorem](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#conditions)\n5. [Importance of the central limit theorem](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#importance)\n6. [Central limit theorem examples](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#examples)\n7. [Practice questions](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#practice-questions)\n8. [Other interesting articles](https:\/\/www.scribbr.com\/statistics\/central-limit-theorem\/#other)\n## What is the central limit theorem?\nThe central limit theorem relies on the concept of a **sampling distribution** , which is the [probability distribution](https:\/\/www.scribbr.com\/statistics\/probability-distributions\/) of a **statistic** for a large number of [samples](https:\/\/www.scribbr.com\/methodology\/population-vs-sample\/) taken from a population.\n\nImagining an experiment may help you to understand sampling distributions:\n\n- Suppose that you draw a [random sample](https:\/\/www.scribbr.com\/methodology\/simple-random-sampling\/) from a population and calculate a [statistic](https:\/\/www.scribbr.com\/statistics\/parameter-vs-statistic\/) for the sample, such as the mean.\n- Now you draw another random sample of the same size, and again calculate the [mean](https:\/\/www.scribbr.com\/statistics\/mean\/) .\n- You repeat this process many times, and end up with a large number of means, one for each sample.\n\nThe distribution of the sample means is an example of a **sampling distribution.**\n\nThe central limit theorem says that the sampling distribution of the mean will always be **normally distributed**, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal.\n\nA normal distribution is a symmetrical, bell-shaped distribution, with increasingly fewer observations the further from the center of the distribution.\n\n### Receive feedback on language, structure, and formatting\nProfessional editors proofread and edit your paper by focusing on:\n\n- Academic style\n- Vague sentences\n- Grammar\n- Style consistency\n\n[See an example](https:\/\/www.scribbr.com\/proofreading-editing\/paper-editing-service\/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=feedback+language+structure+layout)\n\n![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2018\/01\/dissertation-proofreading-service.png)\n## Central limit theorem formula\nFortunately, you don’t need to actually repeatedly sample a population to know the shape of the sampling distribution. The [parameters](https:\/\/www.scribbr.com\/statistics\/parameter-vs-statistic\/) of the sampling distribution of the mean are determined by the parameters of the population:\n\n- The [mean](https:\/\/www.scribbr.com\/statistics\/mean\/) of the sampling distribution is the mean of the population.\n\n![\\\\begin{equation\\*}\\\\mu\\_{\\\\bar{x}}=\\\\mu\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-7f19c78f3ab7a25bf9edd19b8ba36367_l3.png)\n\n- The [standard deviation](https:\/\/www.scribbr.com\/statistics\/standard-deviation\/) of the sampling distribution is the standard deviation of the population divided by the square root of the sample size.\n\n![\\\\begin{equation\\*}\\\\sigma\\_{\\\\bar{x}} = \\\\dfrac{\\\\sigma}{\\\\sqrt{n}}\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-6443a361247d9c6f5764e4503e390634_l3.png)\n\nWe can describe the sampling distribution of the mean using this notation:\n\n![\\\\begin{equation\\*}\\\\bar{X}\\\\sim N (\\\\mu,\\\\dfrac{\\\\sigma}{\\\\sqrt{n}})\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-34ffb4548c9499a6e0972093a9d2b1ae_l3.png)\n\nWhere:\n\n- X̄ is the sampling distribution of the sample means\n- ~ means “follows the distribution”\n- *N* is the [normal distribution](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/)\n- µ is the mean of the population\n- σ is the standard deviation of the population\n- *n* is the sample size\n## Sample size and the central limit theorem\nThe **sample size** (*n*) is the number of observations drawn from the population for each sample. The sample size is the same for all samples.\n\nThe sample size affects the sampling distribution of the mean in two ways.\n\n### 1\\. Sample size and normality\nThe larger the sample size, the more closely the sampling distribution will follow a [normal distribution](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/).\n\nWhen the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.”\n\nBy convention, we consider a sample size of 30 to be “sufficiently large.”\n\n- **When** ***n*** **\\< 30**, the central limit theorem doesn’t apply. The sampling distribution will follow a similar distribution to the population. Therefore, the sampling distribution will only be normal if the population is normal.\n\n- **When** ***n*** **≥ 30**, the central limit theorem applies. The sampling distribution will approximately follow a normal distribution.\n### 2\\. Sample size and standard deviations\nThe sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the [variability](https:\/\/www.scribbr.com\/statistics\/variability\/) or spread of the distribution (i.e., how wide or narrow it is).\n\n- **When** ***n*** **is low**, the standard deviation is high. There’s a lot of spread in the samples’ means because they aren’t precise estimates of the population’s mean.\n\n- **When** ***n*** **is high**, the [standard deviation](https:\/\/www.scribbr.com\/statistics\/standard-deviation\/) is low. There’s not much spread in the samples’ means because they’re precise estimates of the population’s mean.\n## Conditions of the central limit theorem\nThe central limit theorem states that the sampling distribution of the mean will always follow a [normal distribution](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/) under the following conditions:\n\n1. The sample size is **sufficiently large**. This condition is usually met if the sample size is *n* ≥ 30.\n\n1. The samples are **independent and identically distributed (i.i.d.) random variables**. This condition is usually met if the [sampling is random](https:\/\/www.scribbr.com\/methodology\/simple-random-sampling\/).\n\n1. The population’s distribution has **finite** [**variance**](https:\/\/www.scribbr.com\/statistics\/variance\/). Central limit theorem doesn’t apply to distributions with infinite variance, such as the Cauchy distribution. Most distributions have finite variance.\n## Importance of the central limit theorem\nThe central limit theorem is one of the most fundamental statistical theorems. In fact, the “central” in “central limit theorem” refers to the importance of the theorem.\n\nNote\n\n[**Parametric tests**](https:\/\/www.scribbr.com\/statistics\/statistical-tests\/#parametric), such as [*t* tests](https:\/\/www.scribbr.com\/statistics\/t-test\/), [ANOVAs](https:\/\/www.scribbr.com\/statistics\/one-way-anova\/), and [linear regression](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/), have more statistical power than most [non-parametric tests](https:\/\/www.scribbr.com\/statistics\/statistical-tests\/#nonparametric). Their [statistical power](https:\/\/www.scribbr.com\/statistics\/statistical-power\/) comes from assumptions about populations’ distributions that are based on the central limit theorem.\n## Central limit theorem examples\nApplying the central limit theorem to real distributions may help you to better understand how it works.\n\n### Continuous distribution\nSuppose that you’re interested in the age that people retire in the United States. The [**population**](https:\/\/www.scribbr.com\/methodology\/population-vs-sample\/) is all retired Americans, and the distribution of the population might look something like this:\n\n![Central Limit Theorem - Continuous distribution](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/continuous-distribution.png)\n\nAge at retirement follows a [left-skewed](https:\/\/www.scribbr.com\/statistics\/skewness\/#left-skew) distribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years.\n\nImagine that you take a small **sample** of the population. You randomly select five retirees and ask them what age they retired.\n\nExample: Central limit theorem; sample of *n* = 5\n\n|   |   |   |   |   |\n|---|---|---|---|---|\n| 68 | 73 | 70 | 62 | 63 |\n\nThe mean of the sample is an [estimate](https:\/\/www.scribbr.com\/statistics\/parameter-vs-statistic\/#estimating-parameters-from-statistics) of the population mean. It might not be a very precise estimate, since the sample size is only 5.\n\nExample: Central limit theorem; mean of a small sample\n\nmean = (68 + 73 + 70 + 62 + 63) \/ 5\n\nmean = 67.2 years\n\nSuppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a **sampling distribution of the mean**.\n\nExample: Central limit theorem; means of 10 small samples\n\n|   |   |   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|---|---|\n| 60\\.8 | 57\\.8 | 62\\.2 | 68\\.6 | 67\\.4 | 67\\.8 | 68\\.3 | 65\\.6 | 66\\.5 | 62\\.1 |\n\nIf you repeat the procedure many more times, a histogram of the sample means will look something like this:\n\n![Central Limit Theorem - Sampling distribution](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/Sampling-distribution-of-the-mean-1.webp)\n\nAlthough this sampling distribution is more normally distributed than the population, it still has a bit of a [left skew](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/).\n\nNotice also that the spread of the sampling distribution is less than the spread of the population.\n\nThe **central limit theorem** says that the sampling distribution of the mean will always follow a normal distribution when the sample size is sufficiently large. This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large.\n\nNow, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired.\n\nExample: Central limit theorem; sample of *n* = 50\n\n|   |   |   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|---|---|\n| 73 | 49 | 62 | 68 | 72 | 71 | 65 | 60 | 69 | 61 |\n| 62 | 75 | 66 | 63 | 66 | 68 | 76 | 68 | 54 | 74 |\n| 68 | 60 | 72 | 63 | 57 | 64 | 65 | 59 | 72 | 52 |\n| 52 | 72 | 69 | 62 | 68 | 64 | 60 | 65 | 53 | 69 |\n| 59 | 68 | 67 | 71 | 69 | 70 | 52 | 62 | 64 | 68 |\n\nThe mean of the sample is an [estimate](https:\/\/www.scribbr.com\/statistics\/parameter-vs-statistic\/#estimating-parameters-from-statistics) of the population mean. It’s a precise estimate, because the sample size is large.\n\nExample: Central limit theorem; mean of a large sample\n\nmean = 64.8 years\n\nAgain, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample:\n\n![Central Limit Theorem - Mean-of-a-large-sample](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/Mean-of-a-large-sample-1.webp)\n\nIn the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem.\n\nThe standard deviation of this sampling distribution is 0.85 years, which is less than the spread of the small sample sampling distribution, and much less than the spread of the population. If you were to increase the sample size further, the spread would decrease even more.\n\nWe can use the central limit theorem formula to describe the sampling distribution:\n\n![\\\\bar{X} \\\\sim N (\\\\mu,\\\\dfrac{\\\\sigma}{\\\\sqrt{n}})](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-8a8299367f3bafc9ded6b4b151f2cbc6_l3.png)\n\nµ = 65\n\nσ = 6\n\n*n* = 50\n\n![\\\\bar{X} \\\\sim N (65,\\\\dfrac{6}{\\\\sqrt{50}})](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-c749b7f11e70e99d290bf4118c7ca85d_l3.png)\n\n![\\\\bar{X} \\\\sim N (65,0.85)](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-3cad0ef92bdfaa9dbd48b7b269a6bf4e_l3.png)\n\n### Discrete distribution\nApproximately 10% of people are left-handed. If we assign a value of 1 to left-handedness and a value of 0 to right-handedness, the [probability distribution](https:\/\/www.scribbr.com\/statistics\/probability-distributions\/) of left-handedness for the **population** of all humans looks like this:\n\n![Central Limit Theorem - Theorem-discrete-distribution](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/central-limit-theorem-discrete-distribution.png)\n\nThe population mean is the proportion of people who are left-handed (0.1). The population standard deviation is 0.3.\n\nImagine that you take a [random sample](https:\/\/www.scribbr.com\/methodology\/simple-random-sampling\/) of five people and ask them whether they’re left-handed.\n\nExample: Central limit theorem; sample of *n* = 5\n\n|   |   |   |   |   |\n|---|---|---|---|---|\n| 0 | 0 | 0 | 1 | 0 |\n\nThe mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5.\n\nExample: Central limit theorem; mean of a small sample\n\nmean = (0 + 0 + 0 + 1 + 0) \/ 5\n\nmean = 0.2\n\nImagine you repeat this process 10 times, randomly sampling five people and calculating the mean of the sample. This is a **sampling distribution of the mean** .\n\nExample: Central limit theorem; means of 10 small samples\n\n|   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|\n| 0 | 0 | 0\\.4 | 0\\.2 | 0\\.2 | 0 | 0\\.4 | 0 |\n\nIf you repeat this process many more times, the distribution will look something like this:\n\n![Central Limit Theorem - Theorem-discrete-distribution](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/07\/central-limit-theorem-small-samples.png)\n\nThe sampling distribution isn’t normally distributed because the sample size isn’t sufficiently large for the central limit theorem to apply.\n\nAs the sample size increases, the sampling distribution looks increasingly similar to a normal distribution, and the spread decreases:\n\nThe sampling distribution of the mean for samples with *n* = 30 approaches normality. When the sample size is increased further to *n* = 100, the sampling distribution follows a normal distribution.\n\nWe can use the central limit theorem formula to describe the sampling distribution for *n* = 100.\n\n![\\\\bar{X} \\\\sim N (\\\\mu,\\\\dfrac{\\\\sigma}{\\\\sqrt{n}})](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-8a8299367f3bafc9ded6b4b151f2cbc6_l3.png)\n\nµ = 0.1\n\nσ = 0.3\n\n*n* = 100\n\n![\\\\bar{X} \\\\sim N (0.1,\\\\dfrac{0.3}{\\\\sqrt{100}})](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-d9b84c5604bb23b5830a2deee4872381_l3.png)\n\n![\\\\bar{X} \\\\sim N (0.1,0.03)](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-3e8ab9abb7207f6044a9873b22db8681_l3.png)\n\n## Practice questions\n## Other interesting articles\nIf you want to know more about [statistics](https:\/\/www.scribbr.com\/category\/statistics\/) , [methodology](https:\/\/www.scribbr.com\/category\/methodology\/) , or [research bias](https:\/\/www.scribbr.com\/faq-category\/research-bias\/) , make sure to check out some of our other articles with explanations and examples.\n\n**Statistics**\n\n- [Confidence interval](https:\/\/www.scribbr.com\/statistics\/confidence-interval\/)\n- [Kurtosis](https:\/\/www.scribbr.com\/statistics\/kurtosis\/)\n- [Descriptive statistics](https:\/\/www.scribbr.com\/statistics\/descriptive-statistics\/)\n- [Measures of central tendency](https:\/\/www.scribbr.com\/statistics\/central-tendency\/)\n- [Correlation coefficient](https:\/\/www.scribbr.com\/statistics\/correlation-coefficient\/)\n- [*p* value](https:\/\/www.scribbr.com\/statistics\/p-value\/)\n\nFAQ article\n\n#### Cite this Scribbr article\nIf you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.\n\n> Turney, S. 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Boilerpipe Text	Published on July 6, 2022 by Shaun Turney . Revised on February 25, 2026. The central limit theorem states that if you take sufficiently large samples from a population, the samples’ means will be normally distributed , even if the population isn’t normally distributed. Example: Central limit theorem A population follows a Poisson distribution (left image). If we take 10,000 samples from the population, each with a sample size of 50, the sample means follow a normal distribution, as predicted by the central limit theorem (right image). Table of contents What is the central limit theorem? Central limit theorem formula Sample size and the central limit theorem Conditions of the central limit theorem Importance of the central limit theorem Central limit theorem examples Practice questions Other interesting articles What is the central limit theorem? The central limit theorem relies on the concept of a sampling distribution , which is the probability distribution of a statistic for a large number of samples taken from a population. Imagining an experiment may help you to understand sampling distributions: Suppose that you draw a random sample from a population and calculate a statistic for the sample, such as the mean. Now you draw another random sample of the same size, and again calculate the mean . You repeat this process many times, and end up with a large number of means, one for each sample. The distribution of the sample means is an example of a sampling distribution. The central limit theorem says that the sampling distribution of the mean will always be normally distributed , as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal. A normal distribution is a symmetrical, bell-shaped distribution, with increasingly fewer observations the further from the center of the distribution. Receive feedback on language, structure, and formatting Professional editors proofread and edit your paper by focusing on: Academic style Vague sentences Grammar Style consistency See an example Central limit theorem formula Fortunately, you don’t need to actually repeatedly sample a population to know the shape of the sampling distribution. The parameters of the sampling distribution of the mean are determined by the parameters of the population: The mean of the sampling distribution is the mean of the population. The standard deviation of the sampling distribution is the standard deviation of the population divided by the square root of the sample size. We can describe the sampling distribution of the mean using this notation: Where: X̄ is the sampling distribution of the sample means ~ means “follows the distribution” N is the normal distribution µ is the mean of the population σ is the standard deviation of the population n is the sample size Sample size and the central limit theorem The sample size ( n ) is the number of observations drawn from the population for each sample. The sample size is the same for all samples. The sample size affects the sampling distribution of the mean in two ways. 1. Sample size and normality The larger the sample size, the more closely the sampling distribution will follow a normal distribution . When the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.” By convention, we consider a sample size of 30 to be “sufficiently large.” When n < 30 , the central limit theorem doesn’t apply. The sampling distribution will follow a similar distribution to the population. Therefore, the sampling distribution will only be normal if the population is normal. When n ≥ 30 , the central limit theorem applies. The sampling distribution will approximately follow a normal distribution. 2. Sample size and standard deviations The sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the variability or spread of the distribution (i.e., how wide or narrow it is). When n is low , the standard deviation is high. There’s a lot of spread in the samples’ means because they aren’t precise estimates of the population’s mean. When n is high , the standard deviation is low. There’s not much spread in the samples’ means because they’re precise estimates of the population’s mean. Conditions of the central limit theorem The central limit theorem states that the sampling distribution of the mean will always follow a normal distribution under the following conditions: The sample size is sufficiently large . This condition is usually met if the sample size is n ≥ 30. The samples are independent and identically distributed (i.i.d.) random variables . This condition is usually met if the sampling is random . The population’s distribution has finite variance . Central limit theorem doesn’t apply to distributions with infinite variance, such as the Cauchy distribution. Most distributions have finite variance. Importance of the central limit theorem The central limit theorem is one of the most fundamental statistical theorems. In fact, the “central” in “central limit theorem” refers to the importance of the theorem. Note Parametric tests , such as t tests , ANOVAs , and linear regression , have more statistical power than most non-parametric tests . Their statistical power comes from assumptions about populations’ distributions that are based on the central limit theorem. Central limit theorem examples Applying the central limit theorem to real distributions may help you to better understand how it works. Continuous distribution Suppose that you’re interested in the age that people retire in the United States. The population is all retired Americans, and the distribution of the population might look something like this: Age at retirement follows a left-skewed distribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years. Imagine that you take a small sample of the population. You randomly select five retirees and ask them what age they retired. Example: Central limit theorem; sample of n = 5 68 73 70 62 63 The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5. Example: Central limit theorem; mean of a small sample mean = (68 + 73 + 70 + 62 + 63) / 5 mean = 67.2 years Suppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a sampling distribution of the mean . Example: Central limit theorem; means of 10 small samples 60.8 57.8 62.2 68.6 67.4 67.8 68.3 65.6 66.5 62.1 If you repeat the procedure many more times, a histogram of the sample means will look something like this: Although this sampling distribution is more normally distributed than the population, it still has a bit of a left skew . Notice also that the spread of the sampling distribution is less than the spread of the population. The central limit theorem says that the sampling distribution of the mean will always follow a normal distribution when the sample size is sufficiently large. This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large. Now, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired. Example: Central limit theorem; sample of n = 50 73 49 62 68 72 71 65 60 69 61 62 75 66 63 66 68 76 68 54 74 68 60 72 63 57 64 65 59 72 52 52 72 69 62 68 64 60 65 53 69 59 68 67 71 69 70 52 62 64 68 The mean of the sample is an estimate of the population mean. It’s a precise estimate, because the sample size is large. Example: Central limit theorem; mean of a large sample mean = 64.8 years Again, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample: In the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem. The standard deviation of this sampling distribution is 0.85 years, which is less than the spread of the small sample sampling distribution, and much less than the spread of the population. If you were to increase the sample size further, the spread would decrease even more. We can use the central limit theorem formula to describe the sampling distribution: µ = 65 σ = 6 n = 50 Discrete distribution Approximately 10% of people are left-handed. If we assign a value of 1 to left-handedness and a value of 0 to right-handedness, the probability distribution of left-handedness for the population of all humans looks like this: The population mean is the proportion of people who are left-handed (0.1). The population standard deviation is 0.3. Imagine that you take a random sample of five people and ask them whether they’re left-handed. Example: Central limit theorem; sample of n = 5 0 0 0 1 0 The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5. Example: Central limit theorem; mean of a small sample mean = (0 + 0 + 0 + 1 + 0) / 5 mean = 0.2 Imagine you repeat this process 10 times, randomly sampling five people and calculating the mean of the sample. This is a sampling distribution of the mean . Example: Central limit theorem; means of 10 small samples 0 0 0.4 0.2 0.2 0 0.4 0 If you repeat this process many more times, the distribution will look something like this: The sampling distribution isn’t normally distributed because the sample size isn’t sufficiently large for the central limit theorem to apply. As the sample size increases, the sampling distribution looks increasingly similar to a normal distribution, and the spread decreases: The sampling distribution of the mean for samples with n = 30 approaches normality. When the sample size is increased further to n = 100, the sampling distribution follows a normal distribution. We can use the central limit theorem formula to describe the sampling distribution for n = 100. µ = 0.1 σ = 0.3 n = 100 Practice questions Other interesting articles If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples. Statistics Confidence interval Kurtosis Descriptive statistics Measures of central tendency Correlation coefficient p value FAQ article Cite this Scribbr article If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator. Turney, S. (2026, February 25). Central Limit Theorem \| Formula, Definition & Examples. Scribbr. Retrieved April 2, 2026, from https://www.scribbr.com/statistics/central-limit-theorem/ You have already voted. Thanks :-) Your vote is saved :-) Processing your vote...
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Try for free](https://www.scribbr.com/paraphrasing-tool/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar) [![](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2032%2032'%3E%3C/svg%3E) ![](https://www.scribbr.com/wp-content/uploads/2024/09/grammer-checker.svg) Grammar Checker Eliminate grammar errors and improve your writing with our free AI-powered grammar checker. Try for free](https://www.scribbr.com/grammar-checker/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar) # Central Limit Theorem \\| Formula, Definition & Examples Published on July 6, 2022 by [Shaun Turney](https://www.scribbr.com/author/shaunt/ "All articles by Shaun Turney"). Revised on February 25, 2026. The central limit theorem states that if you take sufficiently large samples from a population, the samples’ means will be [normally distributed](https://www.scribbr.com/statistics/normal-distribution/) , even if the population isn’t normally distributed. Example: Central limit theorem A population follows a [Poisson distribution](https://www.scribbr.com/statistics/poisson-distribution/) (left image). If we take 10,000 samples from the population, each with a sample size of 50, the sample means follow a normal distribution, as predicted by the central limit theorem (right image). ![Central Limit Theorem](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%200'%3E%3C/svg%3E)![Central Limit Theorem](https://www.scribbr.com/wp-content/uploads/2022/07/Central-limit-theorem.webp) ## Table of contents 1. [What is the central limit theorem?](https://www.scribbr.com/statistics/central-limit-theorem/#central-limit-theorem) 2. [Central limit theorem formula](https://www.scribbr.com/statistics/central-limit-theorem/#theorem-formula) 3. [Sample size and the central limit theorem](https://www.scribbr.com/statistics/central-limit-theorem/#theorem) 4. [Conditions of the central limit theorem](https://www.scribbr.com/statistics/central-limit-theorem/#conditions) 5. [Importance of the central limit theorem](https://www.scribbr.com/statistics/central-limit-theorem/#importance) 6. [Central limit theorem examples](https://www.scribbr.com/statistics/central-limit-theorem/#examples) 7. [Practice questions](https://www.scribbr.com/statistics/central-limit-theorem/#practice-questions) 8. [Other interesting articles](https://www.scribbr.com/statistics/central-limit-theorem/#other) ## What is the central limit theorem? The central limit theorem relies on the concept of a sampling distribution , which is the [probability distribution](https://www.scribbr.com/statistics/probability-distributions/) of a statistic for a large number of [samples](https://www.scribbr.com/methodology/population-vs-sample/) taken from a population. Imagining an experiment may help you to understand sampling distributions: - Suppose that you draw a [random sample](https://www.scribbr.com/methodology/simple-random-sampling/) from a population and calculate a [statistic](https://www.scribbr.com/statistics/parameter-vs-statistic/) for the sample, such as the mean. - Now you draw another random sample of the same size, and again calculate the [mean](https://www.scribbr.com/statistics/mean/) . - You repeat this process many times, and end up with a large number of means, one for each sample. The distribution of the sample means is an example of a sampling distribution. The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal. A normal distribution is a symmetrical, bell-shaped distribution, with increasingly fewer observations the further from the center of the distribution. ### Receive feedback on language, structure, and formatting Professional editors proofread and edit your paper by focusing on: - Academic style - Vague sentences - Grammar - Style consistency [See an example](https://www.scribbr.com/proofreading-editing/paper-editing-service/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=feedback+language+structure+layout) ![](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20830%20660'%3E%3C/svg%3E) ![](https://www.scribbr.com/wp-content/uploads/2018/01/dissertation-proofreading-service.png) ## Central limit theorem formula Fortunately, you don’t need to actually repeatedly sample a population to know the shape of the sampling distribution. The [parameters](https://www.scribbr.com/statistics/parameter-vs-statistic/) of the sampling distribution of the mean are determined by the parameters of the population: - The [mean](https://www.scribbr.com/statistics/mean/) of the sampling distribution is the mean of the population. ![\\begin{equation\}\\mu\_{\\bar{x}}=\\mu\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2057%2012'%3E%3C/svg%3E)![\\begin{equation\}\\mu\_{\\bar{x}}=\\mu\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-7f19c78f3ab7a25bf9edd19b8ba36367_l3.png) - The [standard deviation](https://www.scribbr.com/statistics/standard-deviation/) of the sampling distribution is the standard deviation of the population divided by the square root of the sample size. ![\\begin{equation\}\\sigma\_{\\bar{x}} = \\dfrac{\\sigma}{\\sqrt{n}}\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2074%2038'%3E%3C/svg%3E)![\\begin{equation\}\\sigma\_{\\bar{x}} = \\dfrac{\\sigma}{\\sqrt{n}}\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-6443a361247d9c6f5764e4503e390634_l3.png) We can describe the sampling distribution of the mean using this notation: ![\\begin{equation\}\\bar{X}\\sim N (\\mu,\\dfrac{\\sigma}{\\sqrt{n}})\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20123%2038'%3E%3C/svg%3E)![\\begin{equation\}\\bar{X}\\sim N (\\mu,\\dfrac{\\sigma}{\\sqrt{n}})\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-34ffb4548c9499a6e0972093a9d2b1ae_l3.png) Where: - X̄ is the sampling distribution of the sample means - ~ means “follows the distribution” - N is the [normal distribution](https://www.scribbr.com/statistics/normal-distribution/) - µ is the mean of the population - σ is the standard deviation of the population - n is the sample size ## Sample size and the central limit theorem The sample size (n) is the number of observations drawn from the population for each sample. The sample size is the same for all samples. The sample size affects the sampling distribution of the mean in two ways. ### 1\. Sample size and normality The larger the sample size, the more closely the sampling distribution will follow a [normal distribution](https://www.scribbr.com/statistics/normal-distribution/). When the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.” By convention, we consider a sample size of 30 to be “sufficiently large.” - When *n* \< 30, the central limit theorem doesn’t apply. The sampling distribution will follow a similar distribution to the population. Therefore, the sampling distribution will only be normal if the population is normal. - When *n* ≥ 30, the central limit theorem applies. The sampling distribution will approximately follow a normal distribution. ### 2\. Sample size and standard deviations The sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the [variability](https://www.scribbr.com/statistics/variability/) or spread of the distribution (i.e., how wide or narrow it is). - When *n* is low, the standard deviation is high. There’s a lot of spread in the samples’ means because they aren’t precise estimates of the population’s mean. - When *n* is high, the [standard deviation](https://www.scribbr.com/statistics/standard-deviation/) is low. There’s not much spread in the samples’ means because they’re precise estimates of the population’s mean. ## Conditions of the central limit theorem The central limit theorem states that the sampling distribution of the mean will always follow a [normal distribution](https://www.scribbr.com/statistics/normal-distribution/) under the following conditions: 1. The sample size is sufficiently large. This condition is usually met if the sample size is n ≥ 30. 1. The samples are independent and identically distributed (i.i.d.) random variables. This condition is usually met if the [sampling is random](https://www.scribbr.com/methodology/simple-random-sampling/). 1. The population’s distribution has finite [variance](https://www.scribbr.com/statistics/variance/). Central limit theorem doesn’t apply to distributions with infinite variance, such as the Cauchy distribution. Most distributions have finite variance. ## Here's why students love Scribbr's proofreading services [Trustpilot](https://www.trustpilot.com/review/www.scribbr.com) [Discover proofreading & editing](https://www.scribbr.com/proofreading-editing/paper-editing-service/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=trustpilot) ## Importance of the central limit theorem The central limit theorem is one of the most fundamental statistical theorems. In fact, the “central” in “central limit theorem” refers to the importance of the theorem. Note [Parametric tests](https://www.scribbr.com/statistics/statistical-tests/#parametric), such as [t tests](https://www.scribbr.com/statistics/t-test/), [ANOVAs](https://www.scribbr.com/statistics/one-way-anova/), and [linear regression](https://www.scribbr.com/statistics/simple-linear-regression/), have more statistical power than most [non-parametric tests](https://www.scribbr.com/statistics/statistical-tests/#nonparametric). Their [statistical power](https://www.scribbr.com/statistics/statistical-power/) comes from assumptions about populations’ distributions that are based on the central limit theorem. ## Central limit theorem examples Applying the central limit theorem to real distributions may help you to better understand how it works. ### Continuous distribution Suppose that you’re interested in the age that people retire in the United States. The [population](https://www.scribbr.com/methodology/population-vs-sample/) is all retired Americans, and the distribution of the population might look something like this: ![Central Limit Theorem - Continuous distribution](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%200'%3E%3C/svg%3E)![Central Limit Theorem - Continuous distribution](https://www.scribbr.com/wp-content/uploads/2022/07/continuous-distribution.png) Age at retirement follows a [left-skewed](https://www.scribbr.com/statistics/skewness/#left-skew) distribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years. Imagine that you take a small sample of the population. You randomly select five retirees and ask them what age they retired. Example: Central limit theorem; sample of n = 5 \| \| \| \| \| \| \|---\|---\|---\|---\|---\| \| 68 \| 73 \| 70 \| 62 \| 63 \| The mean of the sample is an [estimate](https://www.scribbr.com/statistics/parameter-vs-statistic/#estimating-parameters-from-statistics) of the population mean. It might not be a very precise estimate, since the sample size is only 5. Example: Central limit theorem; mean of a small sample mean = (68 + 73 + 70 + 62 + 63) / 5 mean = 67.2 years Suppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a sampling distribution of the mean. Example: Central limit theorem; means of 10 small samples \| \| \| \| \| \| \| \| \| \| \| \|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\| \| 60\.8 \| 57\.8 \| 62\.2 \| 68\.6 \| 67\.4 \| 67\.8 \| 68\.3 \| 65\.6 \| 66\.5 \| 62\.1 \| If you repeat the procedure many more times, a histogram of the sample means will look something like this: ![Central Limit Theorem - Sampling distribution](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%200'%3E%3C/svg%3E)![Central Limit Theorem - Sampling distribution](https://www.scribbr.com/wp-content/uploads/2022/07/Sampling-distribution-of-the-mean-1.webp) Although this sampling distribution is more normally distributed than the population, it still has a bit of a [left skew](https://www.scribbr.com/statistics/normal-distribution/). Notice also that the spread of the sampling distribution is less than the spread of the population. The central limit theorem says that the sampling distribution of the mean will always follow a normal distribution when the sample size is sufficiently large. This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large. Now, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired. Example: Central limit theorem; sample of n = 50 \| \| \| \| \| \| \| \| \| \| \| \|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\| \| 73 \| 49 \| 62 \| 68 \| 72 \| 71 \| 65 \| 60 \| 69 \| 61 \| \| 62 \| 75 \| 66 \| 63 \| 66 \| 68 \| 76 \| 68 \| 54 \| 74 \| \| 68 \| 60 \| 72 \| 63 \| 57 \| 64 \| 65 \| 59 \| 72 \| 52 \| \| 52 \| 72 \| 69 \| 62 \| 68 \| 64 \| 60 \| 65 \| 53 \| 69 \| \| 59 \| 68 \| 67 \| 71 \| 69 \| 70 \| 52 \| 62 \| 64 \| 68 \| The mean of the sample is an [estimate](https://www.scribbr.com/statistics/parameter-vs-statistic/#estimating-parameters-from-statistics) of the population mean. It’s a precise estimate, because the sample size is large. Example: Central limit theorem; mean of a large sample mean = 64.8 years Again, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample: ![Central Limit Theorem - Mean-of-a-large-sample](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%200'%3E%3C/svg%3E)![Central Limit Theorem - Mean-of-a-large-sample](https://www.scribbr.com/wp-content/uploads/2022/07/Mean-of-a-large-sample-1.webp) In the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem. The standard deviation of this sampling distribution is 0.85 years, which is less than the spread of the small sample sampling distribution, and much less than the spread of the population. If you were to increase the sample size further, the spread would decrease even more. We can use the central limit theorem formula to describe the sampling distribution: ![\\bar{X} \\sim N (\\mu,\\dfrac{\\sigma}{\\sqrt{n}})](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20123%2038'%3E%3C/svg%3E)![\\bar{X} \\sim N (\\mu,\\dfrac{\\sigma}{\\sqrt{n}})](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-8a8299367f3bafc9ded6b4b151f2cbc6_l3.png) µ = 65 σ = 6 n = 50 ![\\bar{X} \\sim N (65,\\dfrac{6}{\\sqrt{50}})](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20138%2043'%3E%3C/svg%3E)![\\bar{X} \\sim N (65,\\dfrac{6}{\\sqrt{50}})](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-c749b7f11e70e99d290bf4118c7ca85d_l3.png) ![\\bar{X} \\sim N (65,0.85)](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20132%2021'%3E%3C/svg%3E)![\\bar{X} \\sim N (65,0.85)](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-3cad0ef92bdfaa9dbd48b7b269a6bf4e_l3.png) ### Discrete distribution Approximately 10% of people are left-handed. If we assign a value of 1 to left-handedness and a value of 0 to right-handedness, the [probability distribution](https://www.scribbr.com/statistics/probability-distributions/) of left-handedness for the population of all humans looks like this: ![Central Limit Theorem - Theorem-discrete-distribution](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%200'%3E%3C/svg%3E)![Central Limit Theorem - Theorem-discrete-distribution](https://www.scribbr.com/wp-content/uploads/2022/07/central-limit-theorem-discrete-distribution.png) The population mean is the proportion of people who are left-handed (0.1). The population standard deviation is 0.3. Imagine that you take a [random sample](https://www.scribbr.com/methodology/simple-random-sampling/) of five people and ask them whether they’re left-handed. Example: Central limit theorem; sample of n = 5 \| \| \| \| \| \| \|---\|---\|---\|---\|---\| \| 0 \| 0 \| 0 \| 1 \| 0 \| The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5. Example: Central limit theorem; mean of a small sample mean = (0 + 0 + 0 + 1 + 0) / 5 mean = 0.2 Imagine you repeat this process 10 times, randomly sampling five people and calculating the mean of the sample. This is a sampling distribution of the mean . Example: Central limit theorem; means of 10 small samples \| \| \| \| \| \| \| \| \| \|---\|---\|---\|---\|---\|---\|---\|---\| \| 0 \| 0 \| 0\.4 \| 0\.2 \| 0\.2 \| 0 \| 0\.4 \| 0 \| If you repeat this process many more times, the distribution will look something like this: ![Central Limit Theorem - Theorem-discrete-distribution](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%200'%3E%3C/svg%3E)![Central Limit Theorem - Theorem-discrete-distribution](https://www.scribbr.com/wp-content/uploads/2022/07/central-limit-theorem-small-samples.png) The sampling distribution isn’t normally distributed because the sample size isn’t sufficiently large for the central limit theorem to apply. As the sample size increases, the sampling distribution looks increasingly similar to a normal distribution, and the spread decreases: - [n = 10](https://www.scribbr.com/statistics/central-limit-theorem/) - [n = 20](https://www.scribbr.com/statistics/central-limit-theorem/) - [n = 30](https://www.scribbr.com/statistics/central-limit-theorem/) - [n = 100](https://www.scribbr.com/statistics/central-limit-theorem/) ![Central Limit Theorem - n=10](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%20418'%3E%3C/svg%3E) ![Central Limit Theorem - n=10](https://www.scribbr.com/wp-content/uploads/2022/07/central-limit-theorem-n10.png) ![Central Limit Theorem - n=20](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%20418'%3E%3C/svg%3E) ![Central Limit Theorem - n=20](https://www.scribbr.com/wp-content/uploads/2022/07/central-limit-theorem-n20.png) ![Central Limit Theorem - n=30](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%20418'%3E%3C/svg%3E) ![Central Limit Theorem - n=30](https://www.scribbr.com/wp-content/uploads/2022/07/central-limit-theorem-n30.png) ![Central Limit Theorem - n=100](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%20418'%3E%3C/svg%3E) ![Central Limit Theorem - n=100](https://www.scribbr.com/wp-content/uploads/2022/07/central-limit-theorem-n100.png) The sampling distribution of the mean for samples with n = 30 approaches normality. When the sample size is increased further to n = 100, the sampling distribution follows a normal distribution. We can use the central limit theorem formula to describe the sampling distribution for n = 100. ![\\bar{X} \\sim N (\\mu,\\dfrac{\\sigma}{\\sqrt{n}})](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20123%2038'%3E%3C/svg%3E) ![\\bar{X} \\sim N (\\mu,\\dfrac{\\sigma}{\\sqrt{n}})](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-8a8299367f3bafc9ded6b4b151f2cbc6_l3.png) µ = 0.1 σ = 0.3 n = 100 ![\\bar{X} \\sim N (0.1,\\dfrac{0.3}{\\sqrt{100}})](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20153%2043'%3E%3C/svg%3E) ![\\bar{X} \\sim N (0.1,\\dfrac{0.3}{\\sqrt{100}})](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-d9b84c5604bb23b5830a2deee4872381_l3.png) ![\\bar{X} \\sim N (0.1,0.03)](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20137%2021'%3E%3C/svg%3E) ![\\bar{X} \\sim N (0.1,0.03)](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-3e8ab9abb7207f6044a9873b22db8681_l3.png) ## Practice questions ## Other interesting articles If you want to know more about [statistics](https://www.scribbr.com/category/statistics/) , [methodology](https://www.scribbr.com/category/methodology/) , or [research bias](https://www.scribbr.com/faq-category/research-bias/) , make sure to check out some of our other articles with explanations and examples. 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Readable Markdown	Published on July 6, 2022 by [Shaun Turney](https://www.scribbr.com/author/shaunt/ "All articles by Shaun Turney"). Revised on February 25, 2026. The central limit theorem states that if you take sufficiently large samples from a population, the samples’ means will be [normally distributed](https://www.scribbr.com/statistics/normal-distribution/) , even if the population isn’t normally distributed. Example: Central limit theorem A population follows a [Poisson distribution](https://www.scribbr.com/statistics/poisson-distribution/) (left image). If we take 10,000 samples from the population, each with a sample size of 50, the sample means follow a normal distribution, as predicted by the central limit theorem (right image). ![Central Limit Theorem](https://www.scribbr.com/wp-content/uploads/2022/07/Central-limit-theorem.webp) ## Table of contents 1. [What is the central limit theorem?](https://www.scribbr.com/statistics/central-limit-theorem/#central-limit-theorem) 2. [Central limit theorem formula](https://www.scribbr.com/statistics/central-limit-theorem/#theorem-formula) 3. [Sample size and the central limit theorem](https://www.scribbr.com/statistics/central-limit-theorem/#theorem) 4. [Conditions of the central limit theorem](https://www.scribbr.com/statistics/central-limit-theorem/#conditions) 5. [Importance of the central limit theorem](https://www.scribbr.com/statistics/central-limit-theorem/#importance) 6. [Central limit theorem examples](https://www.scribbr.com/statistics/central-limit-theorem/#examples) 7. [Practice questions](https://www.scribbr.com/statistics/central-limit-theorem/#practice-questions) 8. [Other interesting articles](https://www.scribbr.com/statistics/central-limit-theorem/#other) ## What is the central limit theorem? The central limit theorem relies on the concept of a sampling distribution , which is the [probability distribution](https://www.scribbr.com/statistics/probability-distributions/) of a statistic for a large number of [samples](https://www.scribbr.com/methodology/population-vs-sample/) taken from a population. Imagining an experiment may help you to understand sampling distributions: - Suppose that you draw a [random sample](https://www.scribbr.com/methodology/simple-random-sampling/) from a population and calculate a [statistic](https://www.scribbr.com/statistics/parameter-vs-statistic/) for the sample, such as the mean. - Now you draw another random sample of the same size, and again calculate the [mean](https://www.scribbr.com/statistics/mean/) . - You repeat this process many times, and end up with a large number of means, one for each sample. The distribution of the sample means is an example of a sampling distribution. The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal. A normal distribution is a symmetrical, bell-shaped distribution, with increasingly fewer observations the further from the center of the distribution. ### Receive feedback on language, structure, and formatting Professional editors proofread and edit your paper by focusing on: - Academic style - Vague sentences - Grammar - Style consistency [See an example](https://www.scribbr.com/proofreading-editing/paper-editing-service/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=feedback+language+structure+layout) ![](https://www.scribbr.com/wp-content/uploads/2018/01/dissertation-proofreading-service.png) ## Central limit theorem formula Fortunately, you don’t need to actually repeatedly sample a population to know the shape of the sampling distribution. The [parameters](https://www.scribbr.com/statistics/parameter-vs-statistic/) of the sampling distribution of the mean are determined by the parameters of the population: - The [mean](https://www.scribbr.com/statistics/mean/) of the sampling distribution is the mean of the population. ![\\begin{equation\}\\mu\_{\\bar{x}}=\\mu\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-7f19c78f3ab7a25bf9edd19b8ba36367_l3.png) - The [standard deviation](https://www.scribbr.com/statistics/standard-deviation/) of the sampling distribution is the standard deviation of the population divided by the square root of the sample size. ![\\begin{equation\}\\sigma\_{\\bar{x}} = \\dfrac{\\sigma}{\\sqrt{n}}\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-6443a361247d9c6f5764e4503e390634_l3.png) We can describe the sampling distribution of the mean using this notation: ![\\begin{equation\}\\bar{X}\\sim N (\\mu,\\dfrac{\\sigma}{\\sqrt{n}})\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-34ffb4548c9499a6e0972093a9d2b1ae_l3.png) Where: - X̄ is the sampling distribution of the sample means - ~ means “follows the distribution” - N is the [normal distribution](https://www.scribbr.com/statistics/normal-distribution/) - µ is the mean of the population - σ is the standard deviation of the population - n is the sample size ## Sample size and the central limit theorem The sample size (n) is the number of observations drawn from the population for each sample. The sample size is the same for all samples. The sample size affects the sampling distribution of the mean in two ways. ### 1\. Sample size and normality The larger the sample size, the more closely the sampling distribution will follow a [normal distribution](https://www.scribbr.com/statistics/normal-distribution/). When the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.” By convention, we consider a sample size of 30 to be “sufficiently large.” - When *n* \< 30, the central limit theorem doesn’t apply. The sampling distribution will follow a similar distribution to the population. Therefore, the sampling distribution will only be normal if the population is normal. - When *n* ≥ 30, the central limit theorem applies. The sampling distribution will approximately follow a normal distribution. ### 2\. Sample size and standard deviations The sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the [variability](https://www.scribbr.com/statistics/variability/) or spread of the distribution (i.e., how wide or narrow it is). - When *n* is low, the standard deviation is high. There’s a lot of spread in the samples’ means because they aren’t precise estimates of the population’s mean. - When *n* is high, the [standard deviation](https://www.scribbr.com/statistics/standard-deviation/) is low. There’s not much spread in the samples’ means because they’re precise estimates of the population’s mean. ## Conditions of the central limit theorem The central limit theorem states that the sampling distribution of the mean will always follow a [normal distribution](https://www.scribbr.com/statistics/normal-distribution/) under the following conditions: 1. The sample size is sufficiently large. This condition is usually met if the sample size is n ≥ 30. 1. The samples are independent and identically distributed (i.i.d.) random variables. This condition is usually met if the [sampling is random](https://www.scribbr.com/methodology/simple-random-sampling/). 1. The population’s distribution has finite [variance](https://www.scribbr.com/statistics/variance/). Central limit theorem doesn’t apply to distributions with infinite variance, such as the Cauchy distribution. Most distributions have finite variance. ## Importance of the central limit theorem The central limit theorem is one of the most fundamental statistical theorems. In fact, the “central” in “central limit theorem” refers to the importance of the theorem. Note [Parametric tests](https://www.scribbr.com/statistics/statistical-tests/#parametric), such as [t tests](https://www.scribbr.com/statistics/t-test/), [ANOVAs](https://www.scribbr.com/statistics/one-way-anova/), and [linear regression](https://www.scribbr.com/statistics/simple-linear-regression/), have more statistical power than most [non-parametric tests](https://www.scribbr.com/statistics/statistical-tests/#nonparametric). Their [statistical power](https://www.scribbr.com/statistics/statistical-power/) comes from assumptions about populations’ distributions that are based on the central limit theorem. ## Central limit theorem examples Applying the central limit theorem to real distributions may help you to better understand how it works. ### Continuous distribution Suppose that you’re interested in the age that people retire in the United States. The [population](https://www.scribbr.com/methodology/population-vs-sample/) is all retired Americans, and the distribution of the population might look something like this: ![Central Limit Theorem - Continuous distribution](https://www.scribbr.com/wp-content/uploads/2022/07/continuous-distribution.png) Age at retirement follows a [left-skewed](https://www.scribbr.com/statistics/skewness/#left-skew) distribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years. Imagine that you take a small sample of the population. You randomly select five retirees and ask them what age they retired. Example: Central limit theorem; sample of n = 5 \| \| \| \| \| \| \|---\|---\|---\|---\|---\| \| 68 \| 73 \| 70 \| 62 \| 63 \| The mean of the sample is an [estimate](https://www.scribbr.com/statistics/parameter-vs-statistic/#estimating-parameters-from-statistics) of the population mean. It might not be a very precise estimate, since the sample size is only 5. Example: Central limit theorem; mean of a small sample mean = (68 + 73 + 70 + 62 + 63) / 5 mean = 67.2 years Suppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a sampling distribution of the mean. Example: Central limit theorem; means of 10 small samples \| \| \| \| \| \| \| \| \| \| \| \|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\| \| 60\.8 \| 57\.8 \| 62\.2 \| 68\.6 \| 67\.4 \| 67\.8 \| 68\.3 \| 65\.6 \| 66\.5 \| 62\.1 \| If you repeat the procedure many more times, a histogram of the sample means will look something like this: ![Central Limit Theorem - Sampling distribution](https://www.scribbr.com/wp-content/uploads/2022/07/Sampling-distribution-of-the-mean-1.webp) Although this sampling distribution is more normally distributed than the population, it still has a bit of a [left skew](https://www.scribbr.com/statistics/normal-distribution/). Notice also that the spread of the sampling distribution is less than the spread of the population. The central limit theorem says that the sampling distribution of the mean will always follow a normal distribution when the sample size is sufficiently large. This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large. Now, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired. Example: Central limit theorem; sample of n = 50 \| \| \| \| \| \| \| \| \| \| \| \|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\| \| 73 \| 49 \| 62 \| 68 \| 72 \| 71 \| 65 \| 60 \| 69 \| 61 \| \| 62 \| 75 \| 66 \| 63 \| 66 \| 68 \| 76 \| 68 \| 54 \| 74 \| \| 68 \| 60 \| 72 \| 63 \| 57 \| 64 \| 65 \| 59 \| 72 \| 52 \| \| 52 \| 72 \| 69 \| 62 \| 68 \| 64 \| 60 \| 65 \| 53 \| 69 \| \| 59 \| 68 \| 67 \| 71 \| 69 \| 70 \| 52 \| 62 \| 64 \| 68 \| The mean of the sample is an [estimate](https://www.scribbr.com/statistics/parameter-vs-statistic/#estimating-parameters-from-statistics) of the population mean. It’s a precise estimate, because the sample size is large. Example: Central limit theorem; mean of a large sample mean = 64.8 years Again, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample: ![Central Limit Theorem - Mean-of-a-large-sample](https://www.scribbr.com/wp-content/uploads/2022/07/Mean-of-a-large-sample-1.webp) In the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem. The standard deviation of this sampling distribution is 0.85 years, which is less than the spread of the small sample sampling distribution, and much less than the spread of the population. If you were to increase the sample size further, the spread would decrease even more. We can use the central limit theorem formula to describe the sampling distribution: ![\\bar{X} \\sim N (\\mu,\\dfrac{\\sigma}{\\sqrt{n}})](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-8a8299367f3bafc9ded6b4b151f2cbc6_l3.png) µ = 65 σ = 6 n = 50 ![\\bar{X} \\sim N (65,\\dfrac{6}{\\sqrt{50}})](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-c749b7f11e70e99d290bf4118c7ca85d_l3.png) ![\\bar{X} \\sim N (65,0.85)](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-3cad0ef92bdfaa9dbd48b7b269a6bf4e_l3.png) ### Discrete distribution Approximately 10% of people are left-handed. If we assign a value of 1 to left-handedness and a value of 0 to right-handedness, the [probability distribution](https://www.scribbr.com/statistics/probability-distributions/) of left-handedness for the population of all humans looks like this: ![Central Limit Theorem - Theorem-discrete-distribution](https://www.scribbr.com/wp-content/uploads/2022/07/central-limit-theorem-discrete-distribution.png) The population mean is the proportion of people who are left-handed (0.1). The population standard deviation is 0.3. Imagine that you take a [random sample](https://www.scribbr.com/methodology/simple-random-sampling/) of five people and ask them whether they’re left-handed. Example: Central limit theorem; sample of n = 5 \| \| \| \| \| \| \|---\|---\|---\|---\|---\| \| 0 \| 0 \| 0 \| 1 \| 0 \| The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5. Example: Central limit theorem; mean of a small sample mean = (0 + 0 + 0 + 1 + 0) / 5 mean = 0.2 Imagine you repeat this process 10 times, randomly sampling five people and calculating the mean of the sample. This is a sampling distribution of the mean . Example: Central limit theorem; means of 10 small samples \| \| \| \| \| \| \| \| \| \|---\|---\|---\|---\|---\|---\|---\|---\| \| 0 \| 0 \| 0\.4 \| 0\.2 \| 0\.2 \| 0 \| 0\.4 \| 0 \| If you repeat this process many more times, the distribution will look something like this: ![Central Limit Theorem - Theorem-discrete-distribution](https://www.scribbr.com/wp-content/uploads/2022/07/central-limit-theorem-small-samples.png) The sampling distribution isn’t normally distributed because the sample size isn’t sufficiently large for the central limit theorem to apply. As the sample size increases, the sampling distribution looks increasingly similar to a normal distribution, and the spread decreases: The sampling distribution of the mean for samples with n = 30 approaches normality. When the sample size is increased further to n = 100, the sampling distribution follows a normal distribution. We can use the central limit theorem formula to describe the sampling distribution for n = 100. ![\\bar{X} \\sim N (\\mu,\\dfrac{\\sigma}{\\sqrt{n}})](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-8a8299367f3bafc9ded6b4b151f2cbc6_l3.png) µ = 0.1 σ = 0.3 n = 100 ![\\bar{X} \\sim N (0.1,\\dfrac{0.3}{\\sqrt{100}})](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-d9b84c5604bb23b5830a2deee4872381_l3.png) ![\\bar{X} \\sim N (0.1,0.03)](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-3e8ab9abb7207f6044a9873b22db8681_l3.png) ## Practice questions ## Other interesting articles If you want to know more about [statistics](https://www.scribbr.com/category/statistics/) , [methodology](https://www.scribbr.com/category/methodology/) , or [research bias](https://www.scribbr.com/faq-category/research-bias/) , make sure to check out some of our other articles with explanations and examples. 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