Loading the player...

What is the 'Central Limit Theorem - CLT'

The central limit theorem (CLT) is a statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population. Furthermore, all of the samples will follow an approximate normal distribution pattern, with all variances being approximately equal to the variance of the population divided by each sample's size.

BREAKING DOWN '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, and whether it is normal or non-normal. As a general rule, sample sizes equal to or greater than 30 are considered sufficient for the central limit theorem to hold, meaning the distribution of the sample means is fairly normally distributed.

The Central Limit Theorem in Finance

The central limit theorem is very useful when examining returns for a given stock or index because it simplifies many analysis procedures. An appropriate sample size depends on the data available, but generally speaking, having a sample size of at least 50 observations is sufficient. Due to the relative ease of generating financial data, it is often easy to produce much larger sample sizes. The central limit theorem is the basis for sampling in statistics, so it holds the foundation for sampling and statistical analysis in finance as well. Investors of all types rely on the central limit theorem to analyze stock returns, construct portfolios and manage risk.

Example of Central Limit Theorem

If an investor is looking to analyze the overall return for a stock index made up of 1,000 stocks, he can take random samples of stocks from the index to get an estimate for the return of the total index. The samples must be random, and at least 30 stocks must be evaluated in each sample for the central limit theorem to hold. Random samples ensure a broad range of stock across industries and sectors is represented in the sample. Stocks previously selected must also be replaced for selection in other samples to avoid bias. The average returns from these samples approximates the return for the whole index and are approximately normally distributed. The approximation holds even if the actual returns for the whole index are not normally distributed.

RELATED TERMS
  1. Sampling

    A process used in statistical analysis in which a predetermined ...
  2. Sampling Distribution

    A probability distribution of a statistic obtained through a ...
  3. Sample

    A subset containing the characteristics of a larger population. ...
  4. Representative Sample

    A subset of a statistical population that accurately reflects ...
  5. Systematic Sampling

    A type of probability sampling method in which sample members ...
  6. Simple Random Sample

    A subset of a statistical population in which each member of ...
Related Articles
  1. Investing

    Explaining the Central Limit Theorem

    Central limit theorem is a fundamental concept in probability theory.
  2. Investing

    How Does Sampling Work?

    Sampling is a term used in statistics that describes methods of selecting a pre-defined representative number of data from a larger data population.
  3. Investing

    What is Systematic Sampling?

    Systematic sampling is similar to random sampling, but it uses a pattern for the selection of the sample.
  4. Investing

    What is a Representative Sample?

    In statistics, a representative sample accurately represents the make-up of various subgroups in an entire data pool.
  5. Investing

    Explaining Standard Error

    Standard error is a statistical term that measures the accuracy with which a sample represents a population.
  6. Investing

    Understanding the Simple Random Sample

    A simple random sample is a subset of a statistical population in which each member of the subset has an equal probability of being chosen.
  7. Investing

    How to Use Stratified Random Sampling

    Stratified random sampling is a technique best used with a sample population easily broken into distinct subgroups. Samples are then taken from each subgroup based on the ratio of the subgroup’s ...
  8. Personal Finance

    Birch Box Review: Is It Worth It?

    Learn more about the convenience of the subscription beauty box industry, and discover why the Birchbox company in particular has become so popular.
  9. Investing

    Using Historical Volatility To Gauge Future Risk

    Use these calculations to uncover the risk involved in your investments.
  10. Investing

    Understanding the Modigliani-Miller Theorem

    The Modigliani-Miller (M&M) theorem is used in financial and economic studies to analyze the value of a firm, such as a business or a corporation.
RELATED FAQS
  1. How can a representative sample lead to sampling bias?

    Learn how using representative samples alone is not enough to make sampling bias negligible and why elements such as randomization ... Read Answer >>
  2. What is the difference between systematic sampling and cluster sampling?

    Learn about the differences between systematic sampling and cluster sampling, including how the samples are created for each ... Read Answer >>
  3. What's the difference between a representative sample and a convenience sample?

    Learn the difference between convenience sampling and representative sampling and the advantages and disadvantages of each ... Read Answer >>
  4. What percentage of the population do you need in a representative sample?

    Learn about representative samples and how they are used in conjunction with other strategies to create useful data with ... Read Answer >>
  5. What's the difference between a representative sample and a random sample?

    Explore the differences between representative samples and random samples, and discover how they are often used in tandem ... Read Answer >>
  6. When is it better to use systematic over simple random sampling?

    Learn when systematic sampling is better than simple random sampling, such as in the absence of data patterns and when there ... Read Answer >>
Hot Definitions
  1. 403(b) Plan

    A retirement plan for certain employees of public schools, tax-exempt organizations and certain ministers. Generally, retirement ...
  2. Master Of Business Administration - MBA

    A graduate degree achieved at a university or college that provides theoretical and practical training to help graduates ...
  3. Liquidity Event

    An event that allows initial investors in a company to cash out some or all of their ownership shares and is considered an ...
  4. Job Market

    A market in which employers search for employees and employees search for jobs. The job market is not a physical place as ...
  5. Yuppie

    Yuppie is a slang term denoting the market segment of young urban professionals. A yuppie is often characterized by youth, ...
  6. SEC Form 13F

    A filing with the Securities and Exchange Commission (SEC), also known as the Information Required of Institutional Investment ...
Trading Center