The central limit theorem states that if we take repeated random samples from a population and calculate the mean value of each sample, then the distribution of the sample means will be approximately normally distributed, even if the population the samples came from is not normal. Show
The central limit theorem also states that the mean of the sampling distribution will be equal to the mean of the population distribution: x = μ The central limit theorem is useful because it allows us to use a sample mean to draw conclusions about a larger population mean. The following examples show how the central limit theorem is used in different real-life situations. Example 1: EconomicsEconomists often use the central limit theorem when using sample data to draw conclusions about a population. For example, an economist may collect a simple random sample of 50 individuals in a town and use the average annual income of the individuals in the sample to estimate the average annual income of individuals in the entire town. If the economist finds that the average annual income of the individuals in the sample is $58,000, then her best guess for the true average annual income of individuals in the entire town will be $58,000. Example 2: BiologyBiologists use the central limit theorem whenever they use data from a sample of organisms to draw conclusions about the overall population of organisms. For example, a biologist may measure the height of 30 randomly selected plants and then use the sample mean height to estimate the population mean height. If the biologist finds that the sample mean height of the 30 plants is 10.3 inches, then her best guess for the population mean height will also be 10.3 inches. Example 3: ManufacturingManufacturing plants often use the central limit theorem to estimate how many products produced by the plant are defective. For example, the manager of the plant may randomly select 60 products produced by the plant in a given day and count how many of the products are defective. He can use the proportion of defective products in the sample to estimate the proportion of all products that are defective that are produced by the entire plant. If he finds that 2% of products are defective in the sample, then his best guess for the proportion of defective products produced by the entire plant is also 2%. Example 4: SurveysHuman Resources departments often use the central limit theorem when using surveys to draw conclusions about overall employee satisfaction at companies. For example, the HR department of some company may randomly select 50 employees to take a survey that assesses their overall satisfaction on a scale of 1 to 10. If it’s found that the average satisfaction among employees in the survey is 8.5 then the best guess for the average satisfaction rating of all employees at the company is also 8.5. Example 5: AgricultureAgricultural scientists use the central limit theorem whenever they use data from samples to draw conclusions about a larger population. For example, an agricultural scientist may test a new fertilizer on 15 different fields and measure the average crop yield of each field. If it’s found that the average field produces 400 pounds of wheat, then the best guess for the average crop yield for all fields will also be 400 pounds. Additional ResourcesThe following tutorials provide additional information about the central limit theorem: Introduction to the Central Limit Theorem The central limit theorem states that the sampling distribution of the mean approaches a normal distribution, as the sample size increases. This fact holds especially true for sample sizes over 30. Therefore, as a sample size increases, the sample mean and standard deviation will be closer in value to the population mean μ and standard deviation σ . The central limit theorem tells us that no matter what the distribution of the population is, the shape of the sampling distribution will approach normality as the sample size (N) increases. This is useful, as the research never knows which mean in the sampling distribution is the same as the population mean, but by selecting many random samples from a population the sample means will cluster together, allowing the research to make a very good estimate of the population mean. Thus, as the sample size (N) increases the sampling error will decrease. • As the sample size increases, the distribution of frequencies approximates a bell-shaped curved (i.e. normal distribution curve). • Sample size equal to or greater than 30 are required for the central limit theorem to hold true. • A sufficiently large sample can predict the parameters of a population such as the mean and standard deviation. How to reference this article:How to reference this article:McLeod, S. A. (2019, Nov 25). What is central limit theorem in statistics? Simply psychology: https://www.simplypsychology.org/central-limit-theorem.html How to reference this article:How to reference this article:McLeod, S. A. (2019, November 25). What is central limit theorem in statistics? Simply Psychology. www.simplypsychology.org/central-limit-theorem.html What is the usefulness of the central limit theorem in solving problems involving sampling?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.
How is the central limit theorem useful?The Central Limit Theorem is important for statistics because it allows us to safely assume that the sampling distribution of the mean will be normal in most cases. This means that we can take advantage of statistical techniques that assume a normal distribution, as we will see in the next section.
Why is the central limit theorem important to the study of sampling distributions?The central limit theorem is known to be important to the study of sampling distributions because it enables us to disregard the shape of the population when the value of n is relatively large.
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