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Calculated Shard: 68 (from laksa076)

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Query:

curl -X POST \
  'http://laksa068.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://corporatefinanceinstitute.com/resources/data-science/central-limit-theorem/\')), getAhrefsUnparsedNoserviceFromURL(\'https://corporatefinanceinstitute.com/resources/data-science/central-limit-theorem/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/central-limit-theorem\/","crawl_time":1775587220,"first_indexed_time":1667690478,"http_code":200,"src_unparsed":"com,corporatefinanceinstitute!\/resources\/data-science\/central-limit-theorem\/ s443","src_root_hash":"407738686562150068","history_drop_reason":null,"meta_title":"Central Limit Theorem - Overview, Example, History","meta_descriptions":["The central limit theorem states that the sample mean of a random variable will assume a near normal or normal distribution if the sample size is large"],"attrs_boilerpipe_text":"\"The sample mean of a random variable will assume a near normal or normal distribution if the sample size is large enough\"\nPublished\nMay 14, 2020\nRead Time\n4\n                                                      minutes\nOver 2.8 million + professionals use CFI to learn accounting, financial analysis, modeling and more. Unlock the essentials of corporate finance with our free resources and get an exclusive sneak peek at the first module of each course.\nStart Free\nWhat is the Central Limit Theorem (CLT)?\nThe Central Limit Theorem (CLT) is a statistical concept that states that the sample mean distribution of a random variable will assume a near-normal or normal distribution if the sample size is large enough. In simple terms, the theorem states that the sampling distribution of the\nmean\napproaches a normal distribution as the size of the sample increases, regardless of the shape of the original population distribution.\nAs the user increases the number of samples to 30, 40, 50, etc., the graph of the sample means will move towards a normal distribution. The sample size must be 30 or higher for the central limit theorem to hold.\nOne of the most important components of the theorem is that the mean of the sample will be the mean of the entire population. If you calculate the mean of multiple samples of the population, add them up, and find their average, the result will be the estimate of the population mean.\nThe same applies when using\nstandard deviation\n. If you calculate the standard deviation of all the samples in the population, add them up, and find the average, the result will be the standard deviation of the entire population.\nHow Does the Central Limit Theorem Work?\n \nThe central limit theorem forms the basis of the probability distribution. It makes it easy to understand how population estimates behave when subjected to repeated\nsampling\n. When plotted on a graph, the theorem shows the shape of the distribution formed by means of repeated population samples.\nAs the sample sizes get bigger, the distribution of the means from the repeated samples tends to normalize and resemble a normal distribution. The result remains the same regardless of what the original shape of the distribution was. It can be illustrated in the figure below:\nFrom the figure above, we can deduce that despite the fact that the original shape of the distribution was uniform, it tends towards a normal distribution as the value of n (sample size) increases.\nApart from showing the shape that the sample means will take, the central limit theorem also gives an overview of the mean and variance of the distribution. The sample mean of the distribution is the actual population mean from which the samples were taken.\nThe variance of the sample distribution, on the other hand, is the variance of the population divided by\nn\n. Therefore, the larger the sample size of the distribution, the smaller the variance of the sample mean.\nExample of Central Limit Theorem\nAn investor is interested in estimating the return of ABC stock market index that is comprised of 100,000 stocks. Due to the large size of the\nindex\n, the investor is unable to analyze each stock independently and instead chooses to use random sampling to get an estimate of the overall return of the index.\nThe investor picks random samples of the stocks, with each sample comprising at least 30 stocks. The samples must be random, and any previously selected samples must be replaced in subsequent samples to avoid bias.\nIf the first sample produces an average return of 7.5%, the next sample may produce an average return of 7.8%. With the nature of randomized sampling, each sample will produce a different result. As you increase the size of the sample size with each sample you pick, the sample means will start forming their own distributions.\nThe distribution of the sample means will move toward normal as the value of n increases. The average return of the stocks in the sample index estimates the return of the whole index of 100,000 stocks, and the average return is normally distributed.\nHistory of the Central Limit Theorem\nThe initial version of the central limit theorem was coined by Abraham De Moivre, a French-born mathematician. In an\narticle\npublished in 1733, De Moivre used the normal distribution to find the number of heads resulting from multiple tosses of a coin. The concept was unpopular at the time, and it was forgotten quickly.\nHowever, in 1812, the concept was reintroduced by Pierre-Simon Laplace, another famous French mathematician. Laplace re-introduced the normal distribution concept in his work titled “Théorie Analytique des Probabilités,” where he attempted to approximate binomial distribution with the normal distribution.\nThe mathematician found that the average of independent random variables, when increased in number, tends to follow a normal distribution.  At that time, Laplace’s findings on the central limit theorem attracted attention from other theorists and academicians.\nLater in 1901, the central limit theorem was expanded by Aleksandr Lyapunov, a Russian mathematician. Lyapunov went a step ahead to define the concept in general terms and prove how the concept worked mathematically. The characteristic functions that he used to provide the theorem were adopted in modern probability theory.\nRelated Readings\nThank you for reading CFI’s guide to Central Limit Theorem. To keep learning and advancing your career, the additional CFI resources below will be useful:\nBayes’ Theorem\nCentral Tendency\nLaw of Large Numbers\nTotal Probability Rule\nSee all data science resources","attrs_markdown":"[Corporate Finance Institute](https:\/\/corporatefinanceinstitute.com\/)\n\nMenu\n\n- [Certifications](https:\/\/corporatefinanceinstitute.com\/certifications\/)\n- [Courses](https:\/\/corporatefinanceinstitute.com\/collections\/)\n- [Resources](https:\/\/corporatefinanceinstitute.com\/resources\/)\n- [About](https:\/\/corporatefinanceinstitute.com\/about-cfi\/)\n- [Pricing](https:\/\/corporatefinanceinstitute.com\/pricing\/)\n- [Teams](https:\/\/corporatefinanceinstitute.com\/cfi-for-teams\/)\n\n[My Dashboard](https:\/\/learn.corporatefinanceinstitute.com\/dashboard)\n\n[Log In](https:\/\/learn.corporatefinanceinstitute.com\/auth\/login)\n\n[Start Free](https:\/\/learn.corporatefinanceinstitute.com\/auth\/registration)\n\n- [Certifications](https:\/\/corporatefinanceinstitute.com\/certifications\/)\n- [Specializations](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/central-limit-theorem\/)\n- [Courses](https:\/\/corporatefinanceinstitute.com\/collections\/)\n- [About CFI](https:\/\/corporatefinanceinstitute.com\/about-cfi\/)\n- [Pricing](https:\/\/corporatefinanceinstitute.com\/pricing\/)\n- [Resources](https:\/\/corporatefinanceinstitute.com\/resources\/)\n- [Pricing](https:\/\/corporatefinanceinstitute.com\/?page_id=207142)\n- [Why CFI Certifications?](https:\/\/corporatefinanceinstitute.com\/cfi-for-teams\/certified-professionals\/)\n\n[My Dashboard](https:\/\/learn.corporatefinanceinstitute.com\/dashboard)\n\n[Log In](https:\/\/learn.corporatefinanceinstitute.com\/auth\/login)\n\n[Start Free](https:\/\/learn.corporatefinanceinstitute.com\/auth\/registration)\n\n[Home](https:\/\/corporatefinanceinstitute.com\/ \"Go to Corporate Finance Institute.\")\n\n›\n\n[Resources](https:\/\/corporatefinanceinstitute.com\/resources\/ \"Go to Resources.\")\n\n›\n\n[Data Science](https:\/\/corporatefinanceinstitute.com\/topic\/data-science\/ \"Go to the Data Science Resource Type archives.\")\n\n›\n\nCentral Limit Theorem\n\nTable of Contents\n\n- [What is the Central Limit Theorem (CLT)?](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/central-limit-theorem\/#what-is-the-central-limit-theorem-clt)\n  - [How Does the Central Limit Theorem Work?](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/central-limit-theorem\/#how-does-the-central-limit-theorem-work)\n  - [Example of Central Limit Theorem](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/central-limit-theorem\/#example-of-central-limit-theorem)\n  - [History of the Central Limit Theorem](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/central-limit-theorem\/#history-of-the-central-limit-theorem)\n  - [Related Readings](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/central-limit-theorem\/#related-readings)\n\n# Central Limit Theorem\n\"The sample mean of a random variable will assume a near normal or normal distribution if the sample size is large enough\"\n\nWritten by  [CFI Team](https:\/\/corporatefinanceinstitute.com\/about-cfi\/)\n\nPublished May 14, 2020\n\nRead Time  4 minutes\n\nOver 2.8 million + professionals use CFI to learn accounting, financial analysis, modeling and more. Unlock the essentials of corporate finance with our free resources and get an exclusive sneak peek at the first module of each course.    [Start Free](https:\/\/learn.corporatefinanceinstitute.com\/auth\/registration)\n## What is the Central Limit Theorem (CLT)?\nThe Central Limit Theorem (CLT) is a statistical concept that states that the sample mean distribution of a random variable will assume a near-normal or normal distribution if the sample size is large enough. In simple terms, the theorem states that the sampling distribution of the [mean](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/mean\/) approaches a normal distribution as the size of the sample increases, regardless of the shape of the original population distribution.\n\n![Central Limit Theorem (CLT) Diagram Showing Convergence to Normal Distribution]()\n\nAs the user increases the number of samples to 30, 40, 50, etc., the graph of the sample means will move towards a normal distribution. The sample size must be 30 or higher for the central limit theorem to hold.\n\nOne of the most important components of the theorem is that the mean of the sample will be the mean of the entire population. If you calculate the mean of multiple samples of the population, add them up, and find their average, the result will be the estimate of the population mean.\n\nThe same applies when using [standard deviation](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/standard-deviation\/). If you calculate the standard deviation of all the samples in the population, add them up, and find the average, the result will be the standard deviation of the entire population.\n\n### How Does the Central Limit Theorem Work?\nThe central limit theorem forms the basis of the probability distribution. It makes it easy to understand how population estimates behave when subjected to repeated [sampling](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/type-ii-error\/). When plotted on a graph, the theorem shows the shape of the distribution formed by means of repeated population samples.\n\nAs the sample sizes get bigger, the distribution of the means from the repeated samples tends to normalize and resemble a normal distribution. The result remains the same regardless of what the original shape of the distribution was. It can be illustrated in the figure below:\n\n![Central Limit Theorem (CLT) - How It Works]()\n\nFrom the figure above, we can deduce that despite the fact that the original shape of the distribution was uniform, it tends towards a normal distribution as the value of n (sample size) increases.\n\nApart from showing the shape that the sample means will take, the central limit theorem also gives an overview of the mean and variance of the distribution. The sample mean of the distribution is the actual population mean from which the samples were taken.\n\nThe variance of the sample distribution, on the other hand, is the variance of the population divided by *n*. Therefore, the larger the sample size of the distribution, the smaller the variance of the sample mean.\n\n### Example of Central Limit Theorem\nAn investor is interested in estimating the return of ABC stock market index that is comprised of 100,000 stocks. Due to the large size of the [index](https:\/\/corporatefinanceinstitute.com\/resources\/equities\/dow-jones-industrial-average-djia\/), the investor is unable to analyze each stock independently and instead chooses to use random sampling to get an estimate of the overall return of the index.\n\nThe investor picks random samples of the stocks, with each sample comprising at least 30 stocks. The samples must be random, and any previously selected samples must be replaced in subsequent samples to avoid bias.\n\nIf the first sample produces an average return of 7.5%, the next sample may produce an average return of 7.8%. With the nature of randomized sampling, each sample will produce a different result. As you increase the size of the sample size with each sample you pick, the sample means will start forming their own distributions.\n\nThe distribution of the sample means will move toward normal as the value of n increases. The average return of the stocks in the sample index estimates the return of the whole index of 100,000 stocks, and the average return is normally distributed.\n\n### History of the Central Limit Theorem\nThe initial version of the central limit theorem was coined by Abraham De Moivre, a French-born mathematician. In an [article](https:\/\/www.jstor.org\/stable\/2334818?seq=1#page_scan_tab_contents) published in 1733, De Moivre used the normal distribution to find the number of heads resulting from multiple tosses of a coin. The concept was unpopular at the time, and it was forgotten quickly.\n\nHowever, in 1812, the concept was reintroduced by Pierre-Simon Laplace, another famous French mathematician. Laplace re-introduced the normal distribution concept in his work titled “Théorie Analytique des Probabilités,” where he attempted to approximate binomial distribution with the normal distribution.\n\nThe mathematician found that the average of independent random variables, when increased in number, tends to follow a normal distribution. At that time, Laplace’s findings on the central limit theorem attracted attention from other theorists and academicians.\n\nLater in 1901, the central limit theorem was expanded by Aleksandr Lyapunov, a Russian mathematician. Lyapunov went a step ahead to define the concept in general terms and prove how the concept worked mathematically. The characteristic functions that he used to provide the theorem were adopted in modern probability theory.\n\n### Related Readings\nThank you for reading CFI’s guide to Central Limit Theorem. 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[Log in](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/central-limit-theorem\/)\n\n×\n\n### Access Exclusive Templates\nGain unlimited access to more than 250 productivity Templates, CFI's full course catalog and accredited Certification Programs, hundreds of resources, expert reviews and support, the chance to work with real-world finance and research tools, and more.\n\n[Discover Full-Immersion Membership](https:\/\/corporatefinanceinstitute.com\/pricing\/)\n\nAlready have a Full-Immersion membership? [Log in](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/central-limit-theorem\/)","attrs_readable_markdown":"\"The sample mean of a random variable will assume a near normal or normal distribution if the sample size is large enough\"\n\nPublished May 14, 2020\n\nRead Time  4 minutes\n\nOver 2.8 million + professionals use CFI to learn accounting, financial analysis, modeling and more. Unlock the essentials of corporate finance with our free resources and get an exclusive sneak peek at the first module of each course.    [Start Free](https:\/\/learn.corporatefinanceinstitute.com\/auth\/registration)\n## What is the Central Limit Theorem (CLT)?\nThe Central Limit Theorem (CLT) is a statistical concept that states that the sample mean distribution of a random variable will assume a near-normal or normal distribution if the sample size is large enough. In simple terms, the theorem states that the sampling distribution of the [mean](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/mean\/) approaches a normal distribution as the size of the sample increases, regardless of the shape of the original population distribution.\n\n![Central Limit Theorem (CLT) Diagram Showing Convergence to Normal Distribution](https:\/\/cdn.corporatefinanceinstitute.com\/assets\/Central-Limit-Theorem-CLT-Diagram-1024x626.png)\n\nAs the user increases the number of samples to 30, 40, 50, etc., the graph of the sample means will move towards a normal distribution. The sample size must be 30 or higher for the central limit theorem to hold.\n\nOne of the most important components of the theorem is that the mean of the sample will be the mean of the entire population. If you calculate the mean of multiple samples of the population, add them up, and find their average, the result will be the estimate of the population mean.\n\nThe same applies when using [standard deviation](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/standard-deviation\/). If you calculate the standard deviation of all the samples in the population, add them up, and find the average, the result will be the standard deviation of the entire population.\n\n### How Does the Central Limit Theorem Work?\nThe central limit theorem forms the basis of the probability distribution. It makes it easy to understand how population estimates behave when subjected to repeated [sampling](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/type-ii-error\/). When plotted on a graph, the theorem shows the shape of the distribution formed by means of repeated population samples.\n\nAs the sample sizes get bigger, the distribution of the means from the repeated samples tends to normalize and resemble a normal distribution. The result remains the same regardless of what the original shape of the distribution was. It can be illustrated in the figure below:\n\n![Central Limit Theorem (CLT) - How It Works](https:\/\/cdn.corporatefinanceinstitute.com\/assets\/Central-Limit-Theorem-CLT-How-it-works-and-how-it-arises-1024x734.png)\n\nFrom the figure above, we can deduce that despite the fact that the original shape of the distribution was uniform, it tends towards a normal distribution as the value of n (sample size) increases.\n\nApart from showing the shape that the sample means will take, the central limit theorem also gives an overview of the mean and variance of the distribution. The sample mean of the distribution is the actual population mean from which the samples were taken.\n\nThe variance of the sample distribution, on the other hand, is the variance of the population divided by *n*. Therefore, the larger the sample size of the distribution, the smaller the variance of the sample mean.\n\n### Example of Central Limit Theorem\nAn investor is interested in estimating the return of ABC stock market index that is comprised of 100,000 stocks. Due to the large size of the [index](https:\/\/corporatefinanceinstitute.com\/resources\/equities\/dow-jones-industrial-average-djia\/), the investor is unable to analyze each stock independently and instead chooses to use random sampling to get an estimate of the overall return of the index.\n\nThe investor picks random samples of the stocks, with each sample comprising at least 30 stocks. The samples must be random, and any previously selected samples must be replaced in subsequent samples to avoid bias.\n\nIf the first sample produces an average return of 7.5%, the next sample may produce an average return of 7.8%. With the nature of randomized sampling, each sample will produce a different result. As you increase the size of the sample size with each sample you pick, the sample means will start forming their own distributions.\n\nThe distribution of the sample means will move toward normal as the value of n increases. The average return of the stocks in the sample index estimates the return of the whole index of 100,000 stocks, and the average return is normally distributed.\n\n### History of the Central Limit Theorem\nThe initial version of the central limit theorem was coined by Abraham De Moivre, a French-born mathematician. In an [article](https:\/\/www.jstor.org\/stable\/2334818?seq=1#page_scan_tab_contents) published in 1733, De Moivre used the normal distribution to find the number of heads resulting from multiple tosses of a coin. The concept was unpopular at the time, and it was forgotten quickly.\n\nHowever, in 1812, the concept was reintroduced by Pierre-Simon Laplace, another famous French mathematician. Laplace re-introduced the normal distribution concept in his work titled “Théorie Analytique des Probabilités,” where he attempted to approximate binomial distribution with the normal distribution.\n\nThe mathematician found that the average of independent random variables, when increased in number, tends to follow a normal distribution. At that time, Laplace’s findings on the central limit theorem attracted attention from other theorists and academicians.\n\nLater in 1901, the central limit theorem was expanded by Aleksandr Lyapunov, a Russian mathematician. Lyapunov went a step ahead to define the concept in general terms and prove how the concept worked mathematically. The characteristic functions that he used to provide the theorem were adopted in modern probability theory.\n\n### Related Readings\nThank you for reading CFI’s guide to Central Limit Theorem. To keep learning and advancing your career, the additional CFI resources below will be useful:\n\n- [Bayes’ Theorem](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/bayes-theorem\/)\n- [Central Tendency](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/central-tendency\/)\n- [Law of Large Numbers](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/law-of-large-numbers\/)\n- [Total Probability Rule](https:\/\/corporatefinanceinstitute.com\/resources\/data-science\/total-probability-rule\/)\n- **[See all data science resources](https:\/\/corporatefinanceinstitute.com\/topic\/data-science\/)**","meta_canonical":null}

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Boilerpipe Text	"The sample mean of a random variable will assume a near normal or normal distribution if the sample size is large enough" Published May 14, 2020 Read Time 4 minutes Over 2.8 million + professionals use CFI to learn accounting, financial analysis, modeling and more. Unlock the essentials of corporate finance with our free resources and get an exclusive sneak peek at the first module of each course. Start Free What is the Central Limit Theorem (CLT)? The Central Limit Theorem (CLT) is a statistical concept that states that the sample mean distribution of a random variable will assume a near-normal or normal distribution if the sample size is large enough. In simple terms, the theorem states that the sampling distribution of the mean approaches a normal distribution as the size of the sample increases, regardless of the shape of the original population distribution. As the user increases the number of samples to 30, 40, 50, etc., the graph of the sample means will move towards a normal distribution. The sample size must be 30 or higher for the central limit theorem to hold. One of the most important components of the theorem is that the mean of the sample will be the mean of the entire population. If you calculate the mean of multiple samples of the population, add them up, and find their average, the result will be the estimate of the population mean. The same applies when using standard deviation . If you calculate the standard deviation of all the samples in the population, add them up, and find the average, the result will be the standard deviation of the entire population. How Does the Central Limit Theorem Work? The central limit theorem forms the basis of the probability distribution. It makes it easy to understand how population estimates behave when subjected to repeated sampling . When plotted on a graph, the theorem shows the shape of the distribution formed by means of repeated population samples. As the sample sizes get bigger, the distribution of the means from the repeated samples tends to normalize and resemble a normal distribution. The result remains the same regardless of what the original shape of the distribution was. It can be illustrated in the figure below: From the figure above, we can deduce that despite the fact that the original shape of the distribution was uniform, it tends towards a normal distribution as the value of n (sample size) increases. Apart from showing the shape that the sample means will take, the central limit theorem also gives an overview of the mean and variance of the distribution. The sample mean of the distribution is the actual population mean from which the samples were taken. The variance of the sample distribution, on the other hand, is the variance of the population divided by n . Therefore, the larger the sample size of the distribution, the smaller the variance of the sample mean. Example of Central Limit Theorem An investor is interested in estimating the return of ABC stock market index that is comprised of 100,000 stocks. Due to the large size of the index , the investor is unable to analyze each stock independently and instead chooses to use random sampling to get an estimate of the overall return of the index. The investor picks random samples of the stocks, with each sample comprising at least 30 stocks. The samples must be random, and any previously selected samples must be replaced in subsequent samples to avoid bias. If the first sample produces an average return of 7.5%, the next sample may produce an average return of 7.8%. With the nature of randomized sampling, each sample will produce a different result. As you increase the size of the sample size with each sample you pick, the sample means will start forming their own distributions. The distribution of the sample means will move toward normal as the value of n increases. The average return of the stocks in the sample index estimates the return of the whole index of 100,000 stocks, and the average return is normally distributed. History of the Central Limit Theorem The initial version of the central limit theorem was coined by Abraham De Moivre, a French-born mathematician. In an article published in 1733, De Moivre used the normal distribution to find the number of heads resulting from multiple tosses of a coin. The concept was unpopular at the time, and it was forgotten quickly. However, in 1812, the concept was reintroduced by Pierre-Simon Laplace, another famous French mathematician. Laplace re-introduced the normal distribution concept in his work titled “Théorie Analytique des Probabilités,” where he attempted to approximate binomial distribution with the normal distribution. The mathematician found that the average of independent random variables, when increased in number, tends to follow a normal distribution. At that time, Laplace’s findings on the central limit theorem attracted attention from other theorists and academicians. Later in 1901, the central limit theorem was expanded by Aleksandr Lyapunov, a Russian mathematician. Lyapunov went a step ahead to define the concept in general terms and prove how the concept worked mathematically. The characteristic functions that he used to provide the theorem were adopted in modern probability theory. Related Readings Thank you for reading CFI’s guide to Central Limit Theorem. To keep learning and advancing your career, the additional CFI resources below will be useful: Bayes’ Theorem Central Tendency Law of Large Numbers Total Probability Rule See all data science resources
Markdown	[Corporate Finance Institute](https://corporatefinanceinstitute.com/) Menu - [Certifications](https://corporatefinanceinstitute.com/certifications/) - [Courses](https://corporatefinanceinstitute.com/collections/) - [Resources](https://corporatefinanceinstitute.com/resources/) - [About](https://corporatefinanceinstitute.com/about-cfi/) - [Pricing](https://corporatefinanceinstitute.com/pricing/) - [Teams](https://corporatefinanceinstitute.com/cfi-for-teams/) [My Dashboard](https://learn.corporatefinanceinstitute.com/dashboard) [Log In](https://learn.corporatefinanceinstitute.com/auth/login) [Start Free](https://learn.corporatefinanceinstitute.com/auth/registration) - [Certifications](https://corporatefinanceinstitute.com/certifications/) - [Specializations](https://corporatefinanceinstitute.com/resources/data-science/central-limit-theorem/) - [Courses](https://corporatefinanceinstitute.com/collections/) - [About CFI](https://corporatefinanceinstitute.com/about-cfi/) - [Pricing](https://corporatefinanceinstitute.com/pricing/) - [Resources](https://corporatefinanceinstitute.com/resources/) - [Pricing](https://corporatefinanceinstitute.com/?page_id=207142) - [Why CFI Certifications?](https://corporatefinanceinstitute.com/cfi-for-teams/certified-professionals/) [My Dashboard](https://learn.corporatefinanceinstitute.com/dashboard) [Log In](https://learn.corporatefinanceinstitute.com/auth/login) [Start Free](https://learn.corporatefinanceinstitute.com/auth/registration) [Home](https://corporatefinanceinstitute.com/ "Go to Corporate Finance Institute.") › [Resources](https://corporatefinanceinstitute.com/resources/ "Go to Resources.") › [Data Science](https://corporatefinanceinstitute.com/topic/data-science/ "Go to the Data Science Resource Type archives.") › Central Limit Theorem Table of Contents - [What is the Central Limit Theorem (CLT)?](https://corporatefinanceinstitute.com/resources/data-science/central-limit-theorem/#what-is-the-central-limit-theorem-clt) - [How Does the Central Limit Theorem Work?](https://corporatefinanceinstitute.com/resources/data-science/central-limit-theorem/#how-does-the-central-limit-theorem-work) - [Example of Central Limit Theorem](https://corporatefinanceinstitute.com/resources/data-science/central-limit-theorem/#example-of-central-limit-theorem) - [History of the Central Limit Theorem](https://corporatefinanceinstitute.com/resources/data-science/central-limit-theorem/#history-of-the-central-limit-theorem) - [Related Readings](https://corporatefinanceinstitute.com/resources/data-science/central-limit-theorem/#related-readings) # Central Limit Theorem "The sample mean of a random variable will assume a near normal or normal distribution if the sample size is large enough" Written by [CFI Team](https://corporatefinanceinstitute.com/about-cfi/) Published May 14, 2020 Read Time 4 minutes Over 2.8 million + professionals use CFI to learn accounting, financial analysis, modeling and more. Unlock the essentials of corporate finance with our free resources and get an exclusive sneak peek at the first module of each course. [Start Free](https://learn.corporatefinanceinstitute.com/auth/registration) ## What is the Central Limit Theorem (CLT)? The Central Limit Theorem (CLT) is a statistical concept that states that the sample mean distribution of a random variable will assume a near-normal or normal distribution if the sample size is large enough. In simple terms, the theorem states that the sampling distribution of the [mean](https://corporatefinanceinstitute.com/resources/data-science/mean/) approaches a normal distribution as the size of the sample increases, regardless of the shape of the original population distribution. ![Central Limit Theorem (CLT) Diagram Showing Convergence to Normal Distribution]() As the user increases the number of samples to 30, 40, 50, etc., the graph of the sample means will move towards a normal distribution. The sample size must be 30 or higher for the central limit theorem to hold. One of the most important components of the theorem is that the mean of the sample will be the mean of the entire population. If you calculate the mean of multiple samples of the population, add them up, and find their average, the result will be the estimate of the population mean. The same applies when using [standard deviation](https://corporatefinanceinstitute.com/resources/data-science/standard-deviation/). If you calculate the standard deviation of all the samples in the population, add them up, and find the average, the result will be the standard deviation of the entire population. ### How Does the Central Limit Theorem Work? The central limit theorem forms the basis of the probability distribution. It makes it easy to understand how population estimates behave when subjected to repeated [sampling](https://corporatefinanceinstitute.com/resources/data-science/type-ii-error/). When plotted on a graph, the theorem shows the shape of the distribution formed by means of repeated population samples. As the sample sizes get bigger, the distribution of the means from the repeated samples tends to normalize and resemble a normal distribution. The result remains the same regardless of what the original shape of the distribution was. It can be illustrated in the figure below: ![Central Limit Theorem (CLT) - How It Works]() From the figure above, we can deduce that despite the fact that the original shape of the distribution was uniform, it tends towards a normal distribution as the value of n (sample size) increases. Apart from showing the shape that the sample means will take, the central limit theorem also gives an overview of the mean and variance of the distribution. The sample mean of the distribution is the actual population mean from which the samples were taken. The variance of the sample distribution, on the other hand, is the variance of the population divided by n. Therefore, the larger the sample size of the distribution, the smaller the variance of the sample mean. ### Example of Central Limit Theorem An investor is interested in estimating the return of ABC stock market index that is comprised of 100,000 stocks. Due to the large size of the [index](https://corporatefinanceinstitute.com/resources/equities/dow-jones-industrial-average-djia/), the investor is unable to analyze each stock independently and instead chooses to use random sampling to get an estimate of the overall return of the index. The investor picks random samples of the stocks, with each sample comprising at least 30 stocks. The samples must be random, and any previously selected samples must be replaced in subsequent samples to avoid bias. If the first sample produces an average return of 7.5%, the next sample may produce an average return of 7.8%. With the nature of randomized sampling, each sample will produce a different result. As you increase the size of the sample size with each sample you pick, the sample means will start forming their own distributions. The distribution of the sample means will move toward normal as the value of n increases. The average return of the stocks in the sample index estimates the return of the whole index of 100,000 stocks, and the average return is normally distributed. ### History of the Central Limit Theorem The initial version of the central limit theorem was coined by Abraham De Moivre, a French-born mathematician. In an [article](https://www.jstor.org/stable/2334818?seq=1#page_scan_tab_contents) published in 1733, De Moivre used the normal distribution to find the number of heads resulting from multiple tosses of a coin. The concept was unpopular at the time, and it was forgotten quickly. However, in 1812, the concept was reintroduced by Pierre-Simon Laplace, another famous French mathematician. Laplace re-introduced the normal distribution concept in his work titled “Théorie Analytique des Probabilités,” where he attempted to approximate binomial distribution with the normal distribution. The mathematician found that the average of independent random variables, when increased in number, tends to follow a normal distribution. At that time, Laplace’s findings on the central limit theorem attracted attention from other theorists and academicians. Later in 1901, the central limit theorem was expanded by Aleksandr Lyapunov, a Russian mathematician. Lyapunov went a step ahead to define the concept in general terms and prove how the concept worked mathematically. The characteristic functions that he used to provide the theorem were adopted in modern probability theory. ### Related Readings Thank you for reading CFI’s guide to Central Limit Theorem. To keep learning and advancing your career, the additional CFI resources below will be useful: - [Bayes’ Theorem](https://corporatefinanceinstitute.com/resources/data-science/bayes-theorem/) - [Central Tendency](https://corporatefinanceinstitute.com/resources/data-science/central-tendency/) - [Law of Large Numbers](https://corporatefinanceinstitute.com/resources/data-science/law-of-large-numbers/) - [Total Probability Rule](https://corporatefinanceinstitute.com/resources/data-science/total-probability-rule/) - [See all data science resources](https://corporatefinanceinstitute.com/topic/data-science/) Get Certified for Business Intelligence (BIDA®) Develop analytical superpowers by learning how to use programming and data analytics tools such as VBA, Python, Tableau, Power BI, Power Query, and more. 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Readable Markdown	"The sample mean of a random variable will assume a near normal or normal distribution if the sample size is large enough" Published May 14, 2020 Read Time 4 minutes Over 2.8 million + professionals use CFI to learn accounting, financial analysis, modeling and more. Unlock the essentials of corporate finance with our free resources and get an exclusive sneak peek at the first module of each course. [Start Free](https://learn.corporatefinanceinstitute.com/auth/registration) ## What is the Central Limit Theorem (CLT)? The Central Limit Theorem (CLT) is a statistical concept that states that the sample mean distribution of a random variable will assume a near-normal or normal distribution if the sample size is large enough. In simple terms, the theorem states that the sampling distribution of the [mean](https://corporatefinanceinstitute.com/resources/data-science/mean/) approaches a normal distribution as the size of the sample increases, regardless of the shape of the original population distribution. ![Central Limit Theorem (CLT) Diagram Showing Convergence to Normal Distribution](https://cdn.corporatefinanceinstitute.com/assets/Central-Limit-Theorem-CLT-Diagram-1024x626.png) As the user increases the number of samples to 30, 40, 50, etc., the graph of the sample means will move towards a normal distribution. The sample size must be 30 or higher for the central limit theorem to hold. One of the most important components of the theorem is that the mean of the sample will be the mean of the entire population. If you calculate the mean of multiple samples of the population, add them up, and find their average, the result will be the estimate of the population mean. The same applies when using [standard deviation](https://corporatefinanceinstitute.com/resources/data-science/standard-deviation/). If you calculate the standard deviation of all the samples in the population, add them up, and find the average, the result will be the standard deviation of the entire population. ### How Does the Central Limit Theorem Work? The central limit theorem forms the basis of the probability distribution. It makes it easy to understand how population estimates behave when subjected to repeated [sampling](https://corporatefinanceinstitute.com/resources/data-science/type-ii-error/). When plotted on a graph, the theorem shows the shape of the distribution formed by means of repeated population samples. As the sample sizes get bigger, the distribution of the means from the repeated samples tends to normalize and resemble a normal distribution. The result remains the same regardless of what the original shape of the distribution was. It can be illustrated in the figure below: ![Central Limit Theorem (CLT) - How It Works](https://cdn.corporatefinanceinstitute.com/assets/Central-Limit-Theorem-CLT-How-it-works-and-how-it-arises-1024x734.png) From the figure above, we can deduce that despite the fact that the original shape of the distribution was uniform, it tends towards a normal distribution as the value of n (sample size) increases. Apart from showing the shape that the sample means will take, the central limit theorem also gives an overview of the mean and variance of the distribution. The sample mean of the distribution is the actual population mean from which the samples were taken. The variance of the sample distribution, on the other hand, is the variance of the population divided by n. Therefore, the larger the sample size of the distribution, the smaller the variance of the sample mean. ### Example of Central Limit Theorem An investor is interested in estimating the return of ABC stock market index that is comprised of 100,000 stocks. Due to the large size of the [index](https://corporatefinanceinstitute.com/resources/equities/dow-jones-industrial-average-djia/), the investor is unable to analyze each stock independently and instead chooses to use random sampling to get an estimate of the overall return of the index. The investor picks random samples of the stocks, with each sample comprising at least 30 stocks. The samples must be random, and any previously selected samples must be replaced in subsequent samples to avoid bias. If the first sample produces an average return of 7.5%, the next sample may produce an average return of 7.8%. With the nature of randomized sampling, each sample will produce a different result. As you increase the size of the sample size with each sample you pick, the sample means will start forming their own distributions. The distribution of the sample means will move toward normal as the value of n increases. The average return of the stocks in the sample index estimates the return of the whole index of 100,000 stocks, and the average return is normally distributed. ### History of the Central Limit Theorem The initial version of the central limit theorem was coined by Abraham De Moivre, a French-born mathematician. In an [article](https://www.jstor.org/stable/2334818?seq=1#page_scan_tab_contents) published in 1733, De Moivre used the normal distribution to find the number of heads resulting from multiple tosses of a coin. The concept was unpopular at the time, and it was forgotten quickly. However, in 1812, the concept was reintroduced by Pierre-Simon Laplace, another famous French mathematician. Laplace re-introduced the normal distribution concept in his work titled “Théorie Analytique des Probabilités,” where he attempted to approximate binomial distribution with the normal distribution. The mathematician found that the average of independent random variables, when increased in number, tends to follow a normal distribution. At that time, Laplace’s findings on the central limit theorem attracted attention from other theorists and academicians. Later in 1901, the central limit theorem was expanded by Aleksandr Lyapunov, a Russian mathematician. Lyapunov went a step ahead to define the concept in general terms and prove how the concept worked mathematically. The characteristic functions that he used to provide the theorem were adopted in modern probability theory. ### Related Readings Thank you for reading CFI’s guide to Central Limit Theorem. To keep learning and advancing your career, the additional CFI resources below will be useful: - [Bayes’ Theorem](https://corporatefinanceinstitute.com/resources/data-science/bayes-theorem/) - [Central Tendency](https://corporatefinanceinstitute.com/resources/data-science/central-tendency/) - [Law of Large Numbers](https://corporatefinanceinstitute.com/resources/data-science/law-of-large-numbers/) - [Total Probability Rule](https://corporatefinanceinstitute.com/resources/data-science/total-probability-rule/) - [See all data science resources](https://corporatefinanceinstitute.com/topic/data-science/)
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