Published on July 6, 2022 by Shaun Turney. Revised on August 17, 2022. The central limit theorem states that if you take sufficiently large samples from a population,
the samples’ means will be normally distributed, even if the population isn’t normally distributed. What is the central limit theorem?The central limit theorem relies on the concept of a sampling distribution, which is the probability distribution of a statistic for a large number of samples taken from a population. Imagining an experiment may help you to understand sampling distributions:
The distribution of the sample means is an example of a sampling distribution. The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal. A normal distribution is a symmetrical, bell-shaped distribution, with increasingly fewer observations the further from the center of the distribution. Central limit theorem formulaFortunately, you don’t need to actually repeatedly sample a population to know the shape of the sampling distribution. The parameters of the sampling distribution of the mean are determined by the parameters of the population:
We can describe the sampling distribution of the mean using this notation:
Where:
What can proofreading do for your paper?Scribbr editors not only correct grammar and spelling mistakes, but also strengthen your writing by making sure your paper is free of vague language, redundant words and awkward phrasing. See editing example Sample size and the central limit theoremThe sample size (n) is the number of observations drawn from the population for each sample. The sample size is the same for all samples. The sample size affects the sampling distribution of the mean in two ways. 1. Sample size and normalityThe larger the sample size, the more closely the sampling distribution will follow a normal distribution. When the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.” By convention, we consider a sample size of 30 to be “sufficiently large.”
2. Sample size and standard deviationsThe sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the variability or spread of the distribution (i.e., how wide or narrow it is).
Conditions of the central limit theoremThe central limit theorem states that the sampling distribution of the mean will always follow a normal distribution under the following conditions:
Importance of the central limit theoremThe central limit theorem is one of the most fundamental statistical theorems. In fact, the “central” in “central limit theorem” refers to the importance of the theorem. NoteParametric tests, such as t tests, ANOVAs, and linear regression, have more statistical power than most non-parametric tests. Their power comes from assumptions about populations’ distributions that are based on the central limit theorem.Central limit theorem examplesApplying the central limit theorem to real distributions may help you to better understand how it works. Continuous distributionSuppose that you’re interested in the age that people retire in the United States. The population is all retired Americans, and the distribution of the population might look something like this: Age at retirement follows a left-skewed distribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years. Imagine that you take a small sample of the population. You randomly select five retirees and ask them what age they retired. Example: Central limit theorem; sample of n = 5
The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5. Example: Central limit theorem; mean of a small samplemean = (68 + 73 + 70 + 62 + 63) / 5mean = 67.2 years Suppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a sampling distribution of the mean.
If you repeat the procedure many more times, a histogram of the sample means will look something like this: Although this sampling distribution is more normally distributed than the population, it still has a bit of a left skew. Notice also that the spread of the sampling distribution is less than the spread of the population. The central limit theorem says that the sampling distribution of the mean will always follow a normal distribution when the sample size is sufficiently large. This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large. Now, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired. Example: Central limit theorem; sample of n = 50
The mean of the sample is an estimate of the population mean. It’s a precise estimate, because the sample size is large. Example: Central limit theorem; mean of a large samplemean = 64.8 yearsAgain, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample: In the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem. The standard deviation of this sampling distribution is 0.85 years, which is less than the spread of the small sample sampling distribution, and much less than the spread of the population. If you were to increase the sample size further, the spread would decrease even more. We can use the central limit theorem formula to describe the sampling distribution: µ = 65 σ = 6 n = 50 Discrete distributionApproximately 10% of people are left-handed. If we assign a value of 1 to left-handedness and a value of 0 to right-handedness, the probability distribution of left-handedness for the population of all humans looks like this: The population mean is the proportion of people who are left-handed (0.1). The population standard deviation is 0.3. Imagine that you take a random sample of five people and ask them whether they’re left-handed. Example: Central limit theorem; sample of n = 5
The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5. mean = 0.2 Imagine you repeat this process 10 times, randomly sampling five people and calculating the mean of the sample. This is a sampling distribution of the mean. Example: Central limit theorem; means of 10 small samples
If you repeat this process many more times, the distribution will look something like this: The sampling distribution isn’t normally distributed because the sample size isn’t sufficiently large for the central limit theorem to apply. As the sample size increases, the sampling distribution looks increasingly similar to a normal distribution, and the spread decreases: The sampling distribution of the mean for samples with n = 30 approaches normality. When the sample size is increased further to n = 100, the sampling distribution follows a normal distribution. We can use the central limit theorem formula to describe the sampling distribution for n = 100. µ = 0.1 σ = 0.3 n = 100 Practice questionsFrequently asked questions about the central limit theoremWhat are the three types of skewness? The three types of skewness are:
Why are samples used in research? Samples are used to make inferences about populations. Samples are easier to collect data from because they are practical, cost-effective, convenient, and manageable. Sources in this articleWe strongly encourage students to use sources in their work. You can cite our article (APA Style) or take a deep dive into the articles below. This Scribbr article
Is this article helpful?You have already voted. Thanks :-) Your vote is saved :-) Processing your vote... How does sample size affect the shape of the sampling distribution of the mean?In other words, as the sample size increases, the variability of sampling distribution decreases. Also, as the sample size increases the shape of the sampling distribution becomes more similar to a normal distribution regardless of the shape of the population.
How does increasing sample size affect?Higher sample size allows the researcher to increase the significance level of the findings, since the confidence of the result are likely to increase with a higher sample size. This is to be expected because larger the sample size, the more accurately it is expected to mirror the behavior of the whole group.
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