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1. Shard Calculation

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Response:

Calculated Shard: 148 (from laksa053)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa148.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.investopedia.com/terms/c/central_limit_theorem.asp\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.investopedia.com/terms/c/central_limit_theorem.asp\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.investopedia.com\/terms\/c\/central_limit_theorem.asp","crawl_time":1775644352,"first_indexed_time":1510390741,"http_code":200,"src_unparsed":"com,investopedia!www,\/terms\/c\/central_limit_theorem.asp s443","src_root_hash":"7273739457760125548","history_drop_reason":null,"meta_title":"What Is the Central Limit Theorem (CLT)?","meta_descriptions":["The Central Limit Theorem (CLT) relies on multiple independent samples that are randomly selected to predict the activity of a population."],"attrs_boilerpipe_text":"Key Takeaways\nThe CLT shows that with a large enough sample size, the distribution of sample means can accurately reflect the characteristics of a population.\nSample sizes equal to or greater than 30 are often considered sufficient under the central limit theorem.\nInvestors may use CLT to study a random sample of stocks to estimate returns for a portfolio.\nInvestopedia \/ Jiaqi Zhou\nWhat Is the Central Limit Theorem (CLT)?\nThe Central Limit Theorem is useful when analyzing large data sets because it assumes that the sampling distribution of the mean will be normally distributed and typically form a\nbell curve\n. The CLT may be used in conjunction with the law of large numbers, which states that the average obtained from a large group of independent random samples converges to the true value.\nUnderstanding the Central Limit Theorem (CLT)\nAccording to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question as the sample size increases, notwithstanding the actual distribution of the data. The concept can hold true regardless of whether the distribution of the population is normal or skewed.\nAs a general rule, sample sizes of 30 or more are typically deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. In addition, the more samples one takes, the more the graphed results should take the shape of a normal distribution.\nThe central limit theorem is often used in conjunction with the\nlaw of large numbers\n, which states that the average of the sample means will come closer to equaling the population means as the sample size grows. This concept can be extremely useful in accurately predicting the characteristics of very large populations.\nAlthough this concept was first developed by Abraham de Moivre in 1733, it was not formalized until 1920, when the Hungarian mathematician George Pólya dubbed it the central limit theorem.\nKey Components of the Central Limit Theorem\nThe central limit theorem has several key components. They largely revolve around sampling technique.\nSampling is random\n:\nAll samples must be selected at random so that they have the same statistical possibility of being selected.\nSamples should be independent\n: The selections or results from one sample should have no bearing on future samples or other sample results.\nLarge sample size\n: As the sample size increases, the sampling distribution should come ever closer to the\nnormal distribution\n.\nSamples come from identical distributions:\nSamples must be drawn under the same conditions and have the same underlying characteristics.\nThe Central Limit Theorem in Finance and Investing\nThe CLT can help examine the returns of an individual stock or broader stock indices because the analysis is simple, due to the relative ease of generating the necessary financial data. Consequently, investors often rely on the CLT to analyze stock returns, construct portfolios, and manage risk.\nSuppose, for example, that an investor wishes to analyze the overall return for a stock index that consists of 1,000 different equities. In this scenario, the investor may simply study a random sample of stocks to arrive at an estimated return for the total index. To be safe in this instance, at least 30 to 50 randomly selected stocks across various sectors should be sampled for the central limit theorem to hold.\nExplain It Like I'm 5\nPicture a big jar filled with a variety of hard candy—some big, some small, some round, and some square. You want to know the average size, but you can’t measure every single one. So, scoop out a handful, measure it, and note the average. Then you do it again. Each handful gives you a slightly different average. But if you keep taking random handfuls and chart those averages, your chart will start to form a bell curve — tall in the middle, shorter on the sides.\nEven though the candies aren’t evenly sized, the central limit theorem (CLT) shows that when you take enough random samples, the averages start to behave predictably and cluster around the actual average. That’s why statisticians and investors can study samples and still make accurate predictions about large groups.\nWhy Is the Central Limit Theorem Useful?\nThe central limit theorem is useful when analyzing large data sets because it allows one to assume that the sampling distribution of the mean will be normally distributed in most cases. This allows for easier statistical analysis and inference. For example, investors can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over some time.\nWhat Is the Formula for Central Limit Theorem?\nThe central limit theorem doesn't have a formula used in its practical application. Its principle is simply applied. With a sufficiently large sample size, the sample distribution will approximate a normal distribution, and the sample mean will approach the population mean. It suggests that if we have a sample size of at least 30, we can begin to analyze the data as if it fit a normal distribution.\nWhy Is the Central Limit Theorem's Minimum Sample Size 30?\nA sample size of 30 or more is fairly common across statistics as the minimum for applying the central limit theorem. The greater your sample size, the more likely the sample will be representative of your population set.\nWhat Is the Law of Large Numbers?\nIn probability theory and statistics, the law of large numbers states that the larger the sample size, the more likely its mean is to reflect the mean of the entire population.\nIn business, the law of large numbers can have a different meaning, specifically that as a company grows in size, maintaining its rate of growth in percentage terms becomes more difficult.\nThe Bottom Line\nThe central limit theorem (CLT) holds that as a sample size gets larger, its mean will increasingly approximate the mean in a normal distribution. This concept can be useful in many applications, such as analyzing investment returns, because it requires only a sufficient sample size (generally interpreted as 30 or more data points) rather than the entire population.","attrs_markdown":"\n\n[Skip to content](https:\/\/www.investopedia.com\/terms\/c\/central_limit_theorem.asp#main)\n\n[Investopedia](https:\/\/www.investopedia.com\/)\n\n[Trade](https:\/\/www.investopedia.com\/simulator\/?inv_to_sim=global_nav)\n\n- Please fill out this field.\n- [Newsletters](https:\/\/www.investopedia.com\/terms\/c\/central_limit_theorem.asp)\n\nPlease fill out this field.\n\n- [News](https:\/\/www.investopedia.com\/news-4427706)\n  News\n  \n  - [Markets](https:\/\/www.investopedia.com\/markets-news-4427704)\n  - [Companies](https:\/\/www.investopedia.com\/company-news-4427705)\n  - [Earnings](https:\/\/www.investopedia.com\/earnings-reports-5218419)\n  - [CD Rates](https:\/\/www.investopedia.com\/certificates-deposit-news-7111770)\n  - [Mortgage Rates](https:\/\/www.investopedia.com\/mortgage-rates-5184598)\n  - 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[Best Life Insurance](https:\/\/www.investopedia.com\/the-best-life-insurance-companies-8763666)\n  - [Best Mortgage Rates](https:\/\/www.investopedia.com\/mortgage-rates-5094943)\n  - [Best Robo-Advisors](https:\/\/www.investopedia.com\/the-best-robo-advisors-8764849)\n  - [Best Personal Loans](https:\/\/www.investopedia.com\/the-best-personal-loans-8761582)\n  - [Best Debt Relief Companies](https:\/\/www.investopedia.com\/the-best-debt-relief-companies-8763179)\n  - [View All](https:\/\/www.investopedia.com\/financial-product-reviews-7110783)\n\n- Top Stories\n- [Average Home Value for 35- to 44-Year Olds](https:\/\/www.investopedia.com\/how-does-your-home-value-stack-up-against-the-average-35-to-44-year-old-s-home-value-11942776)\n- [\\$10,000 a Month for Long-Term Care Is Becoming the Norm—What That Means for Families](https:\/\/www.investopedia.com\/10k-a-month-for-long-term-care-is-becoming-the-norm-what-that-means-for-families-11943223)\n- [Investing In Bonds Isn't Your Only Choice If You're Over 50](https:\/\/www.investopedia.com\/investing-in-bonds-is-not-your-only-choice-if-over-50-11942303)\n- [This Dallas Suburb Is an Affordable Destination for Retirees](https:\/\/www.investopedia.com\/this-dallas-suburb-is-an-affordable-destination-for-active-retirees-11942788)\n\nTable of Contents\n\nExpand\n\nTable of Contents\n\n- [Central Limit Theorem (CLT)](https:\/\/www.investopedia.com\/terms\/c\/central_limit_theorem.asp#toc-what-is-the-central-limit-theorem-clt)\n- [Understanding the CLT](https:\/\/www.investopedia.com\/terms\/c\/central_limit_theorem.asp#toc-understanding-the-central-limit-theorem-clt)\n- [Key Components](https:\/\/www.investopedia.com\/terms\/c\/central_limit_theorem.asp#toc-key-components-of-the-central-limit-theorem)\n- [CLT in Finance and Investing](https:\/\/www.investopedia.com\/terms\/c\/central_limit_theorem.asp#toc-the-central-limit-theorem-in-finance-and-investing)\n- [Explain It Like I'm 5](https:\/\/www.investopedia.com\/terms\/c\/central_limit_theorem.asp#toc-explain-it-like-im-5)\n- [FAQs](https:\/\/www.investopedia.com\/terms\/c\/central_limit_theorem.asp#toc-why-is-the-central-limit-theorem-useful)\n- [The Bottom Line](https:\/\/www.investopedia.com\/terms\/c\/central_limit_theorem.asp#toc-the-bottom-line)\n\n# What Is the Central Limit Theorem (CLT)?\nBy [Akhilesh Ganti](https:\/\/www.investopedia.com\/akhilesh-ganti-4590113)\n\nUpdated March 23, 2026\n\nReviewed by\n\n[JeFreda R. Brown](https:\/\/www.investopedia.com\/jefreda-brown-5188074)\n\n![Dr. Brown]()\n\n![Dr. Brown](https:\/\/www.investopedia.com\/thmb\/4h0l6rpZs0PO1z6tj2Z2Okq5NuM=\/200x200\/filters:no_upscale\$\$:max_bytes\$150000\$:strip_icc\$\$\/Dr.JeFredaR.Brownheadshot-JeFredaBrown-1e8af368a1ea4533a21868d8a951895a.jpg)\n\nReviewed by JeFreda R. Brown\n\n[Full Bio](https:\/\/www.investopedia.com\/jefreda-brown-5188074)\n\nDr. JeFreda R. Brown is a financial consultant, Certified Financial Education Instructor, and researcher who has assisted thousands of clients over a more than two-decade career. She is the CEO of Xaris Financial Enterprises and a course facilitator for Cornell University.\n\nLearn about our  [Financial Review Board](https:\/\/www.investopedia.com\/investopedia-financial-review-board-5076269)\n\nFact checked by\n\n[Michael Rosenston](https:\/\/www.investopedia.com\/michael-rosenston-5198792)\n\n![Michael Rosenston]()\n\n![Michael Rosenston](https:\/\/www.investopedia.com\/thmb\/rY4DYrReXuS1aKW9UHN1vwV76GU=\/200x200\/filters:no_upscale\$\$:max_bytes\$150000\$:strip_icc\$\$\/MichaelRosenston-dc9ea36078684eb9a17e5bbbed31dbad.jpg)\n\nFact checked by Michael Rosenston\n\n[Full Bio](https:\/\/www.investopedia.com\/michael-rosenston-5198792)\n\nMichael Rosenston is a fact-checker and researcher with expertise in business, finance, and insurance.\n\nLearn about our  [editorial policies](https:\/\/www.investopedia.com\/legal-4768893#editorial-policy)\n\nDefinition\n\nThe Central Limit Theorem (CLT) describes how sample means from a population, regardless of the population's distribution, tend to form a normal distribution as the sample size increases.\n\n### Key Takeaways\n- The CLT shows that with a large enough sample size, the distribution of sample means can accurately reflect the characteristics of a population.\n- Sample sizes equal to or greater than 30 are often considered sufficient under the central limit theorem.\n- Investors may use CLT to study a random sample of stocks to estimate returns for a portfolio.\n\n![Central Limit Theorem (CLT)]()\n\n![Central Limit Theorem (CLT)](https:\/\/www.investopedia.com\/thmb\/LNfBuNluEZCLs5qM_QLpj5NISOk=\/1500x0\/filters:no_upscale\$\$:max_bytes\$150000\$:strip_icc\$\$\/Central-Limit-Theorum-390b2def452a4103ab5714b833b5a477.png)\n\nInvestopedia \/ Jiaqi Zhou\n## What Is the Central Limit Theorem (CLT)?\nThe Central Limit Theorem is useful when analyzing large data sets because it assumes that the sampling distribution of the mean will be normally distributed and typically form a [bell curve](https:\/\/www.investopedia.com\/terms\/b\/bell-curve.asp). The CLT may be used in conjunction with the law of large numbers, which states that the average obtained from a large group of independent random samples converges to the true value.\n\n## Understanding the Central Limit Theorem (CLT)\nAccording to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question as the sample size increases, notwithstanding the actual distribution of the data. The concept can hold true regardless of whether the distribution of the population is normal or skewed.\n\nAs a general rule, sample sizes of 30 or more are typically deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. In addition, the more samples one takes, the more the graphed results should take the shape of a normal distribution.\n\nThe central limit theorem is often used in conjunction with the [law of large numbers](https:\/\/www.investopedia.com\/terms\/l\/lawoflargenumbers.asp), which states that the average of the sample means will come closer to equaling the population means as the sample size grows. This concept can be extremely useful in accurately predicting the characteristics of very large populations.\n\nAlthough this concept was first developed by Abraham de Moivre in 1733, it was not formalized until 1920, when the Hungarian mathematician George Pólya dubbed it the central limit theorem.\n\n## Key Components of the Central Limit Theorem\nThe central limit theorem has several key components. They largely revolve around sampling technique.\n\n1. **Sampling is random**:All samples must be selected at random so that they have the same statistical possibility of being selected.\n2. **Samples should be independent**: The selections or results from one sample should have no bearing on future samples or other sample results.\n3. **Large sample size**: As the sample size increases, the sampling distribution should come ever closer to the [normal distribution](https:\/\/www.investopedia.com\/terms\/n\/normaldistribution.asp).\n4. **Samples come from identical distributions:** Samples must be drawn under the same conditions and have the same underlying characteristics.\n## The Central Limit Theorem in Finance and Investing\nThe CLT can help examine the returns of an individual stock or broader stock indices because the analysis is simple, due to the relative ease of generating the necessary financial data. Consequently, investors often rely on the CLT to analyze stock returns, construct portfolios, and manage risk.\n\nSuppose, for example, that an investor wishes to analyze the overall return for a stock index that consists of 1,000 different equities. In this scenario, the investor may simply study a random sample of stocks to arrive at an estimated return for the total index. To be safe in this instance, at least 30 to 50 randomly selected stocks across various sectors should be sampled for the central limit theorem to hold.\n\n## Explain It Like I'm 5\nPicture a big jar filled with a variety of hard candy—some big, some small, some round, and some square. You want to know the average size, but you can’t measure every single one. So, scoop out a handful, measure it, and note the average. Then you do it again. Each handful gives you a slightly different average. But if you keep taking random handfuls and chart those averages, your chart will start to form a bell curve — tall in the middle, shorter on the sides.\n\nEven though the candies aren’t evenly sized, the central limit theorem (CLT) shows that when you take enough random samples, the averages start to behave predictably and cluster around the actual average. That’s why statisticians and investors can study samples and still make accurate predictions about large groups.\n\n## Why Is the Central Limit Theorem Useful?\nThe central limit theorem is useful when analyzing large data sets because it allows one to assume that the sampling distribution of the mean will be normally distributed in most cases. This allows for easier statistical analysis and inference. For example, investors can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over some time.\n\n## What Is the Formula for Central Limit Theorem?\nThe central limit theorem doesn't have a formula used in its practical application. Its principle is simply applied. With a sufficiently large sample size, the sample distribution will approximate a normal distribution, and the sample mean will approach the population mean. It suggests that if we have a sample size of at least 30, we can begin to analyze the data as if it fit a normal distribution.\n\n## Why Is the Central Limit Theorem's Minimum Sample Size 30?\nA sample size of 30 or more is fairly common across statistics as the minimum for applying the central limit theorem. The greater your sample size, the more likely the sample will be representative of your population set.\n\n## What Is the Law of Large Numbers?\nIn probability theory and statistics, the law of large numbers states that the larger the sample size, the more likely its mean is to reflect the mean of the entire population.\n\nIn business, the law of large numbers can have a different meaning, specifically that as a company grows in size, maintaining its rate of growth in percentage terms becomes more difficult.\n## The Bottom Line\nThe central limit theorem (CLT) holds that as a sample size gets larger, its mean will increasingly approximate the mean in a normal distribution. This concept can be useful in many applications, such as analyzing investment returns, because it requires only a sufficient sample size (generally interpreted as 30 or more data points) rather than the entire population.\n\nArticle Sources\n\nInvestopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our [editorial policy.](https:\/\/www.investopedia.com\/legal-4768893#EditorialPolicy)\n\n1. Boston University School of Public Health. \"[Central Limit Theorem](https:\/\/sphweb.bumc.bu.edu\/otlt\/mph-modules\/bs\/bs704_probability\/BS704_Probability12.html).\"\n2. Hans Fischer. \"[A History of the Central Limit Theorem](https:\/\/www.medicine.mcgill.ca\/epidemiology\/hanley\/bios601\/GaussianModel\/HistoryCentralLimitTheorem.pdf),\" Page 1.\n3. The Mathematics Enthusiast. \"[Studying Moments of the Central Limit Theorem, Benjamin A. Stark](https:\/\/scholarworks.umt.edu\/cgi\/viewcontent.cgi?article=1388&context=tme).\"\n4. University of Massachusetts Amherst. \"[What Is Central Limit Theorem? Properties, Best Practices, Examples & Everything to Know](https:\/\/bootcamp.umass.edu\/blog\/quality-management\/central-limit-theorem).\"\n5. Emory University. \"[Final Summary: The Central Limit Theorem](https:\/\/psychology.emory.edu\/clinical\/bliwise\/Tutorials\/CLT\/CLT\/fsummary.htm).\"\n6. Mascha, Edward J., and Thomas R. Vetter. \"[Significance, Errors, Power, and Sample Size: The Blocking and Tackling of Statistics](https:\/\/pubmed.ncbi.nlm.nih.gov\/29346210\/).\" *Anesthesia & Analgesia,* vol 126, no. 2*,* February 2018, pp. 691-698.\n\nRead more\n\n- [Business](https:\/\/www.investopedia.com\/business-4689640)\n- [Corporate Finance](https:\/\/www.investopedia.com\/corporate-finance-and-accounting-4689821)\n- [Financial Analysis](https:\/\/www.investopedia.com\/financial-analysis-4427788)\n\nPartner Links\n\n## Related Articles\n[![Chartered Accountant (CA)]() ![Chartered Accountant (CA)](https:\/\/www.investopedia.com\/thmb\/K8kCOyVnDnsJ5Newh8fLHMcKogU=\/282x188\/filters:no_upscale\$\$:max_bytes\$150000\$:strip_icc\$\$\/Term-Definitions_Chartered-accountant-8ef5808fce2b4fd0bebaf63ab148395b.jpg) Chartered Accountant (CA): Roles, Responsibilities, and Qualifications](https:\/\/www.investopedia.com\/terms\/c\/ca.asp)\n\n[![Capitalization (Cap) Table]() ![Capitalization (Cap) 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[V](https:\/\/www.investopedia.com\/terms-beginning-with-v-4769372)\n- [W](https:\/\/www.investopedia.com\/terms-beginning-with-w-4769373)\n- [X](https:\/\/www.investopedia.com\/terms-beginning-with-x-4769374)\n- [Y](https:\/\/www.investopedia.com\/terms-beginning-with-y-4769375)\n- [Z](https:\/\/www.investopedia.com\/terms-beginning-with-z-4769376)\n\nInvestopedia is part of the [People Inc.](https:\/\/www.people.inc\/) publishing family.\n\nNewsletter Sign Up\n\nNewsletter Sign Up","attrs_readable_markdown":"### Key Takeaways\n- The CLT shows that with a large enough sample size, the distribution of sample means can accurately reflect the characteristics of a population.\n- Sample sizes equal to or greater than 30 are often considered sufficient under the central limit theorem.\n- Investors may use CLT to study a random sample of stocks to estimate returns for a portfolio.\n\nInvestopedia \/ Jiaqi Zhou\n\n## What Is the Central Limit Theorem (CLT)?\nThe Central Limit Theorem is useful when analyzing large data sets because it assumes that the sampling distribution of the mean will be normally distributed and typically form a [bell curve](https:\/\/www.investopedia.com\/terms\/b\/bell-curve.asp). The CLT may be used in conjunction with the law of large numbers, which states that the average obtained from a large group of independent random samples converges to the true value.\n\n## Understanding the Central Limit Theorem (CLT)\nAccording to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question as the sample size increases, notwithstanding the actual distribution of the data. The concept can hold true regardless of whether the distribution of the population is normal or skewed.\n\nAs a general rule, sample sizes of 30 or more are typically deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. In addition, the more samples one takes, the more the graphed results should take the shape of a normal distribution.\n\nThe central limit theorem is often used in conjunction with the [law of large numbers](https:\/\/www.investopedia.com\/terms\/l\/lawoflargenumbers.asp), which states that the average of the sample means will come closer to equaling the population means as the sample size grows. This concept can be extremely useful in accurately predicting the characteristics of very large populations.\n\nAlthough this concept was first developed by Abraham de Moivre in 1733, it was not formalized until 1920, when the Hungarian mathematician George Pólya dubbed it the central limit theorem.\n\n## Key Components of the Central Limit Theorem\nThe central limit theorem has several key components. They largely revolve around sampling technique.\n\n1. **Sampling is random**:All samples must be selected at random so that they have the same statistical possibility of being selected.\n2. **Samples should be independent**: The selections or results from one sample should have no bearing on future samples or other sample results.\n3. **Large sample size**: As the sample size increases, the sampling distribution should come ever closer to the [normal distribution](https:\/\/www.investopedia.com\/terms\/n\/normaldistribution.asp).\n4. **Samples come from identical distributions:** Samples must be drawn under the same conditions and have the same underlying characteristics.\n## The Central Limit Theorem in Finance and Investing\nThe CLT can help examine the returns of an individual stock or broader stock indices because the analysis is simple, due to the relative ease of generating the necessary financial data. Consequently, investors often rely on the CLT to analyze stock returns, construct portfolios, and manage risk.\n\nSuppose, for example, that an investor wishes to analyze the overall return for a stock index that consists of 1,000 different equities. In this scenario, the investor may simply study a random sample of stocks to arrive at an estimated return for the total index. To be safe in this instance, at least 30 to 50 randomly selected stocks across various sectors should be sampled for the central limit theorem to hold.\n\n## Explain It Like I'm 5\nPicture a big jar filled with a variety of hard candy—some big, some small, some round, and some square. You want to know the average size, but you can’t measure every single one. So, scoop out a handful, measure it, and note the average. Then you do it again. Each handful gives you a slightly different average. But if you keep taking random handfuls and chart those averages, your chart will start to form a bell curve — tall in the middle, shorter on the sides.\n\nEven though the candies aren’t evenly sized, the central limit theorem (CLT) shows that when you take enough random samples, the averages start to behave predictably and cluster around the actual average. That’s why statisticians and investors can study samples and still make accurate predictions about large groups.\n\n## Why Is the Central Limit Theorem Useful?\nThe central limit theorem is useful when analyzing large data sets because it allows one to assume that the sampling distribution of the mean will be normally distributed in most cases. This allows for easier statistical analysis and inference. For example, investors can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over some time.\n\n## What Is the Formula for Central Limit Theorem?\nThe central limit theorem doesn't have a formula used in its practical application. Its principle is simply applied. With a sufficiently large sample size, the sample distribution will approximate a normal distribution, and the sample mean will approach the population mean. It suggests that if we have a sample size of at least 30, we can begin to analyze the data as if it fit a normal distribution.\n\n## Why Is the Central Limit Theorem's Minimum Sample Size 30?\nA sample size of 30 or more is fairly common across statistics as the minimum for applying the central limit theorem. The greater your sample size, the more likely the sample will be representative of your population set.\n\n## What Is the Law of Large Numbers?\nIn probability theory and statistics, the law of large numbers states that the larger the sample size, the more likely its mean is to reflect the mean of the entire population.\n\nIn business, the law of large numbers can have a different meaning, specifically that as a company grows in size, maintaining its rate of growth in percentage terms becomes more difficult.\n## The Bottom Line\nThe central limit theorem (CLT) holds that as a sample size gets larger, its mean will increasingly approximate the mean in a normal distribution. This concept can be useful in many applications, such as analyzing investment returns, because it requires only a sufficient sample size (generally interpreted as 30 or more data points) rather than the entire population.","meta_canonical":null}

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Boilerpipe Text	Key Takeaways The CLT shows that with a large enough sample size, the distribution of sample means can accurately reflect the characteristics of a population. Sample sizes equal to or greater than 30 are often considered sufficient under the central limit theorem. Investors may use CLT to study a random sample of stocks to estimate returns for a portfolio. Investopedia / Jiaqi Zhou What Is the Central Limit Theorem (CLT)? The Central Limit Theorem is useful when analyzing large data sets because it assumes that the sampling distribution of the mean will be normally distributed and typically form a bell curve . The CLT may be used in conjunction with the law of large numbers, which states that the average obtained from a large group of independent random samples converges to the true value. Understanding the Central Limit Theorem (CLT) According to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question as the sample size increases, notwithstanding the actual distribution of the data. The concept can hold true regardless of whether the distribution of the population is normal or skewed. As a general rule, sample sizes of 30 or more are typically deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. In addition, the more samples one takes, the more the graphed results should take the shape of a normal distribution. The central limit theorem is often used in conjunction with the law of large numbers , which states that the average of the sample means will come closer to equaling the population means as the sample size grows. This concept can be extremely useful in accurately predicting the characteristics of very large populations. Although this concept was first developed by Abraham de Moivre in 1733, it was not formalized until 1920, when the Hungarian mathematician George Pólya dubbed it the central limit theorem. Key Components of the Central Limit Theorem The central limit theorem has several key components. They largely revolve around sampling technique. Sampling is random : All samples must be selected at random so that they have the same statistical possibility of being selected. Samples should be independent : The selections or results from one sample should have no bearing on future samples or other sample results. Large sample size : As the sample size increases, the sampling distribution should come ever closer to the normal distribution . Samples come from identical distributions: Samples must be drawn under the same conditions and have the same underlying characteristics. The Central Limit Theorem in Finance and Investing The CLT can help examine the returns of an individual stock or broader stock indices because the analysis is simple, due to the relative ease of generating the necessary financial data. Consequently, investors often rely on the CLT to analyze stock returns, construct portfolios, and manage risk. Suppose, for example, that an investor wishes to analyze the overall return for a stock index that consists of 1,000 different equities. In this scenario, the investor may simply study a random sample of stocks to arrive at an estimated return for the total index. To be safe in this instance, at least 30 to 50 randomly selected stocks across various sectors should be sampled for the central limit theorem to hold. Explain It Like I'm 5 Picture a big jar filled with a variety of hard candy—some big, some small, some round, and some square. You want to know the average size, but you can’t measure every single one. So, scoop out a handful, measure it, and note the average. Then you do it again. Each handful gives you a slightly different average. But if you keep taking random handfuls and chart those averages, your chart will start to form a bell curve — tall in the middle, shorter on the sides. Even though the candies aren’t evenly sized, the central limit theorem (CLT) shows that when you take enough random samples, the averages start to behave predictably and cluster around the actual average. That’s why statisticians and investors can study samples and still make accurate predictions about large groups. Why Is the Central Limit Theorem Useful? The central limit theorem is useful when analyzing large data sets because it allows one to assume that the sampling distribution of the mean will be normally distributed in most cases. This allows for easier statistical analysis and inference. For example, investors can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over some time. What Is the Formula for Central Limit Theorem? The central limit theorem doesn't have a formula used in its practical application. Its principle is simply applied. With a sufficiently large sample size, the sample distribution will approximate a normal distribution, and the sample mean will approach the population mean. It suggests that if we have a sample size of at least 30, we can begin to analyze the data as if it fit a normal distribution. Why Is the Central Limit Theorem's Minimum Sample Size 30? A sample size of 30 or more is fairly common across statistics as the minimum for applying the central limit theorem. The greater your sample size, the more likely the sample will be representative of your population set. What Is the Law of Large Numbers? In probability theory and statistics, the law of large numbers states that the larger the sample size, the more likely its mean is to reflect the mean of the entire population. In business, the law of large numbers can have a different meaning, specifically that as a company grows in size, maintaining its rate of growth in percentage terms becomes more difficult. The Bottom Line The central limit theorem (CLT) holds that as a sample size gets larger, its mean will increasingly approximate the mean in a normal distribution. This concept can be useful in many applications, such as analyzing investment returns, because it requires only a sufficient sample size (generally interpreted as 30 or more data points) rather than the entire population.
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[This Dallas Suburb Is an Affordable Destination for Retirees](https://www.investopedia.com/this-dallas-suburb-is-an-affordable-destination-for-active-retirees-11942788) Table of Contents Expand Table of Contents - [Central Limit Theorem (CLT)](https://www.investopedia.com/terms/c/central_limit_theorem.asp#toc-what-is-the-central-limit-theorem-clt) - [Understanding the CLT](https://www.investopedia.com/terms/c/central_limit_theorem.asp#toc-understanding-the-central-limit-theorem-clt) - [Key Components](https://www.investopedia.com/terms/c/central_limit_theorem.asp#toc-key-components-of-the-central-limit-theorem) - [CLT in Finance and Investing](https://www.investopedia.com/terms/c/central_limit_theorem.asp#toc-the-central-limit-theorem-in-finance-and-investing) - [Explain It Like I'm 5](https://www.investopedia.com/terms/c/central_limit_theorem.asp#toc-explain-it-like-im-5) - [FAQs](https://www.investopedia.com/terms/c/central_limit_theorem.asp#toc-why-is-the-central-limit-theorem-useful) - [The Bottom Line](https://www.investopedia.com/terms/c/central_limit_theorem.asp#toc-the-bottom-line) # What Is the Central Limit Theorem (CLT)? By [Akhilesh Ganti](https://www.investopedia.com/akhilesh-ganti-4590113) Updated March 23, 2026 Reviewed by [JeFreda R. Brown](https://www.investopedia.com/jefreda-brown-5188074) ![Dr. Brown]() ![Dr. Brown](https://www.investopedia.com/thmb/4h0l6rpZs0PO1z6tj2Z2Okq5NuM=/200x200/filters:no_upscale:max_bytes$150000$:strip_icc/Dr.JeFredaR.Brownheadshot-JeFredaBrown-1e8af368a1ea4533a21868d8a951895a.jpg) Reviewed by JeFreda R. Brown [Full Bio](https://www.investopedia.com/jefreda-brown-5188074) Dr. JeFreda R. Brown is a financial consultant, Certified Financial Education Instructor, and researcher who has assisted thousands of clients over a more than two-decade career. She is the CEO of Xaris Financial Enterprises and a course facilitator for Cornell University. 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Learn about our [editorial policies](https://www.investopedia.com/legal-4768893#editorial-policy) Definition The Central Limit Theorem (CLT) describes how sample means from a population, regardless of the population's distribution, tend to form a normal distribution as the sample size increases. ### Key Takeaways - The CLT shows that with a large enough sample size, the distribution of sample means can accurately reflect the characteristics of a population. - Sample sizes equal to or greater than 30 are often considered sufficient under the central limit theorem. - Investors may use CLT to study a random sample of stocks to estimate returns for a portfolio. ![Central Limit Theorem (CLT)]() ![Central Limit Theorem (CLT)](https://www.investopedia.com/thmb/LNfBuNluEZCLs5qM_QLpj5NISOk=/1500x0/filters:no_upscale:max_bytes$150000$:strip_icc/Central-Limit-Theorum-390b2def452a4103ab5714b833b5a477.png) Investopedia / Jiaqi Zhou ## What Is the Central Limit Theorem (CLT)? The Central Limit Theorem is useful when analyzing large data sets because it assumes that the sampling distribution of the mean will be normally distributed and typically form a [bell curve](https://www.investopedia.com/terms/b/bell-curve.asp). The CLT may be used in conjunction with the law of large numbers, which states that the average obtained from a large group of independent random samples converges to the true value. ## Understanding the Central Limit Theorem (CLT) According to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question as the sample size increases, notwithstanding the actual distribution of the data. The concept can hold true regardless of whether the distribution of the population is normal or skewed. As a general rule, sample sizes of 30 or more are typically deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. In addition, the more samples one takes, the more the graphed results should take the shape of a normal distribution. The central limit theorem is often used in conjunction with the [law of large numbers](https://www.investopedia.com/terms/l/lawoflargenumbers.asp), which states that the average of the sample means will come closer to equaling the population means as the sample size grows. This concept can be extremely useful in accurately predicting the characteristics of very large populations. Although this concept was first developed by Abraham de Moivre in 1733, it was not formalized until 1920, when the Hungarian mathematician George Pólya dubbed it the central limit theorem. ## Key Components of the Central Limit Theorem The central limit theorem has several key components. They largely revolve around sampling technique. 1. Sampling is random:All samples must be selected at random so that they have the same statistical possibility of being selected. 2. Samples should be independent: The selections or results from one sample should have no bearing on future samples or other sample results. 3. Large sample size: As the sample size increases, the sampling distribution should come ever closer to the [normal distribution](https://www.investopedia.com/terms/n/normaldistribution.asp). 4. Samples come from identical distributions: Samples must be drawn under the same conditions and have the same underlying characteristics. ## The Central Limit Theorem in Finance and Investing The CLT can help examine the returns of an individual stock or broader stock indices because the analysis is simple, due to the relative ease of generating the necessary financial data. Consequently, investors often rely on the CLT to analyze stock returns, construct portfolios, and manage risk. Suppose, for example, that an investor wishes to analyze the overall return for a stock index that consists of 1,000 different equities. In this scenario, the investor may simply study a random sample of stocks to arrive at an estimated return for the total index. To be safe in this instance, at least 30 to 50 randomly selected stocks across various sectors should be sampled for the central limit theorem to hold. ## Explain It Like I'm 5 Picture a big jar filled with a variety of hard candy—some big, some small, some round, and some square. You want to know the average size, but you can’t measure every single one. So, scoop out a handful, measure it, and note the average. Then you do it again. Each handful gives you a slightly different average. But if you keep taking random handfuls and chart those averages, your chart will start to form a bell curve — tall in the middle, shorter on the sides. Even though the candies aren’t evenly sized, the central limit theorem (CLT) shows that when you take enough random samples, the averages start to behave predictably and cluster around the actual average. That’s why statisticians and investors can study samples and still make accurate predictions about large groups. ## Why Is the Central Limit Theorem Useful? The central limit theorem is useful when analyzing large data sets because it allows one to assume that the sampling distribution of the mean will be normally distributed in most cases. This allows for easier statistical analysis and inference. For example, investors can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over some time. ## What Is the Formula for Central Limit Theorem? The central limit theorem doesn't have a formula used in its practical application. Its principle is simply applied. With a sufficiently large sample size, the sample distribution will approximate a normal distribution, and the sample mean will approach the population mean. It suggests that if we have a sample size of at least 30, we can begin to analyze the data as if it fit a normal distribution. ## Why Is the Central Limit Theorem's Minimum Sample Size 30? A sample size of 30 or more is fairly common across statistics as the minimum for applying the central limit theorem. The greater your sample size, the more likely the sample will be representative of your population set. ## What Is the Law of Large Numbers? In probability theory and statistics, the law of large numbers states that the larger the sample size, the more likely its mean is to reflect the mean of the entire population. In business, the law of large numbers can have a different meaning, specifically that as a company grows in size, maintaining its rate of growth in percentage terms becomes more difficult. ## The Bottom Line The central limit theorem (CLT) holds that as a sample size gets larger, its mean will increasingly approximate the mean in a normal distribution. This concept can be useful in many applications, such as analyzing investment returns, because it requires only a sufficient sample size (generally interpreted as 30 or more data points) rather than the entire population. Article Sources Investopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our [editorial policy.](https://www.investopedia.com/legal-4768893#EditorialPolicy) 1. Boston University School of Public Health. "[Central Limit Theorem](https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_probability/BS704_Probability12.html)." 2. Hans Fischer. "[A History of the Central Limit Theorem](https://www.medicine.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/HistoryCentralLimitTheorem.pdf)," Page 1. 3. The Mathematics Enthusiast. "[Studying Moments of the Central Limit Theorem, Benjamin A. Stark](https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1388&context=tme)." 4. University of Massachusetts Amherst. "[What Is Central Limit Theorem? Properties, Best Practices, Examples & Everything to Know](https://bootcamp.umass.edu/blog/quality-management/central-limit-theorem)." 5. Emory University. "[Final Summary: The Central Limit Theorem](https://psychology.emory.edu/clinical/bliwise/Tutorials/CLT/CLT/fsummary.htm)." 6. Mascha, Edward J., and Thomas R. Vetter. "[Significance, Errors, Power, and Sample Size: The Blocking and Tackling of Statistics](https://pubmed.ncbi.nlm.nih.gov/29346210/)." Anesthesia & Analgesia, vol 126, no. 2, February 2018, pp. 691-698. 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Readable Markdown	### Key Takeaways - The CLT shows that with a large enough sample size, the distribution of sample means can accurately reflect the characteristics of a population. - Sample sizes equal to or greater than 30 are often considered sufficient under the central limit theorem. - Investors may use CLT to study a random sample of stocks to estimate returns for a portfolio. Investopedia / Jiaqi Zhou ## What Is the Central Limit Theorem (CLT)? The Central Limit Theorem is useful when analyzing large data sets because it assumes that the sampling distribution of the mean will be normally distributed and typically form a [bell curve](https://www.investopedia.com/terms/b/bell-curve.asp). The CLT may be used in conjunction with the law of large numbers, which states that the average obtained from a large group of independent random samples converges to the true value. ## Understanding the Central Limit Theorem (CLT) According to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question as the sample size increases, notwithstanding the actual distribution of the data. The concept can hold true regardless of whether the distribution of the population is normal or skewed. As a general rule, sample sizes of 30 or more are typically deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. In addition, the more samples one takes, the more the graphed results should take the shape of a normal distribution. The central limit theorem is often used in conjunction with the [law of large numbers](https://www.investopedia.com/terms/l/lawoflargenumbers.asp), which states that the average of the sample means will come closer to equaling the population means as the sample size grows. This concept can be extremely useful in accurately predicting the characteristics of very large populations. Although this concept was first developed by Abraham de Moivre in 1733, it was not formalized until 1920, when the Hungarian mathematician George Pólya dubbed it the central limit theorem. ## Key Components of the Central Limit Theorem The central limit theorem has several key components. They largely revolve around sampling technique. 1. Sampling is random:All samples must be selected at random so that they have the same statistical possibility of being selected. 2. Samples should be independent: The selections or results from one sample should have no bearing on future samples or other sample results. 3. Large sample size: As the sample size increases, the sampling distribution should come ever closer to the [normal distribution](https://www.investopedia.com/terms/n/normaldistribution.asp). 4. Samples come from identical distributions: Samples must be drawn under the same conditions and have the same underlying characteristics. ## The Central Limit Theorem in Finance and Investing The CLT can help examine the returns of an individual stock or broader stock indices because the analysis is simple, due to the relative ease of generating the necessary financial data. Consequently, investors often rely on the CLT to analyze stock returns, construct portfolios, and manage risk. Suppose, for example, that an investor wishes to analyze the overall return for a stock index that consists of 1,000 different equities. In this scenario, the investor may simply study a random sample of stocks to arrive at an estimated return for the total index. To be safe in this instance, at least 30 to 50 randomly selected stocks across various sectors should be sampled for the central limit theorem to hold. ## Explain It Like I'm 5 Picture a big jar filled with a variety of hard candy—some big, some small, some round, and some square. You want to know the average size, but you can’t measure every single one. So, scoop out a handful, measure it, and note the average. Then you do it again. Each handful gives you a slightly different average. But if you keep taking random handfuls and chart those averages, your chart will start to form a bell curve — tall in the middle, shorter on the sides. Even though the candies aren’t evenly sized, the central limit theorem (CLT) shows that when you take enough random samples, the averages start to behave predictably and cluster around the actual average. That’s why statisticians and investors can study samples and still make accurate predictions about large groups. ## Why Is the Central Limit Theorem Useful? The central limit theorem is useful when analyzing large data sets because it allows one to assume that the sampling distribution of the mean will be normally distributed in most cases. This allows for easier statistical analysis and inference. For example, investors can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over some time. ## What Is the Formula for Central Limit Theorem? The central limit theorem doesn't have a formula used in its practical application. Its principle is simply applied. With a sufficiently large sample size, the sample distribution will approximate a normal distribution, and the sample mean will approach the population mean. It suggests that if we have a sample size of at least 30, we can begin to analyze the data as if it fit a normal distribution. ## Why Is the Central Limit Theorem's Minimum Sample Size 30? A sample size of 30 or more is fairly common across statistics as the minimum for applying the central limit theorem. The greater your sample size, the more likely the sample will be representative of your population set. ## What Is the Law of Large Numbers? In probability theory and statistics, the law of large numbers states that the larger the sample size, the more likely its mean is to reflect the mean of the entire population. In business, the law of large numbers can have a different meaning, specifically that as a company grows in size, maintaining its rate of growth in percentage terms becomes more difficult. ## The Bottom Line The central limit theorem (CLT) holds that as a sample size gets larger, its mean will increasingly approximate the mean in a normal distribution. This concept can be useful in many applications, such as analyzing investment returns, because it requires only a sufficient sample size (generally interpreted as 30 or more data points) rather than the entire population.
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