We have examined in detail three components of the central limit theorem -- successive sampling, increasing sample size, and different populations. Let's review what we have learned from each and put them together into a final statement. Remember that the central limit theorem applies only to the mean and not to other statistics.
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Sampling requires that we draw successive samples from a defined population. The samples must be randomly selected and of the same size. |
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Calculate the mean for each sample and plot the sample means. This produces a distribution of sample means. A plot of an "infinite" number of sample means is called the sampling distribution of the mean. |
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Frequency distributions of sample means quickly approach the shape of a normal distribution, even if we are taking relatively few, small samples from a population that is not normally distributed. |
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As we randomly select more and more samples from the population, the distribution of sample means becomes more normally distributed and looks looks smoother. |
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With "infinite" numbers of successive random samples, the sampling distributions all have a normal distribution with a mean that is equal to the population mean (µ). |
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As sample sizes increase, the sampling distributions approach a normal distribution. With "infinite" numbers of successive random samples, the mean of the sampling distribution is equal to the population mean (µ). |
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As the sample sizes increase, the variability of each sampling distribution decreases so that they become increasingly more leptokurtic. The range of the sampling distribution is smaller than the range of the original population. The standard deviation of each sampling distribution is equal to s/ÖN (where N is the size of the sample drawn from the population). |
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Taken together, these distributions suggest that the sample mean provides a good estimate of µ and that errors in our estimates (indicated by the variability of scores in the distribution) decrease as the size of the samples we draw from the population increase. |
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The principles of successive sampling and increasing sample size work for all distributions. |
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We can count on the sampling distribution of the mean being approximately normally distributed, no matter what the original population distribution looks like as long as the sample size is relatively large. |
The central limit theorem states that when an infinite number of successive random samples are taken from a population, the distribution of sample means calculated for each sample will become approximately normally distributed with mean µ and standard deviation s/ÖN ( ~N(µ,s/ÖN)) as the sample size (N) becomes larger, irrespective of the shape of the population distribution.
How does the central limit theorem help us when we are testing hypotheses about sample means? Even if we do not know the distribution of scores in the original population, we know that the sampling distribution of the means will be approximately normally distributed with mean µ and standard deviation s/ÖN, if the sample is relatively large. Knowing the properties of the sampling distribution allows us to continue with the test, even if we don't know what the population distribution looks like.
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Now that you have reviewed all three components of the central limit theorem, test your knowledge with practice exercises. |
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You also might want to check out a very cool website that puts all of the components together into one graphic. |
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Click here to go to practice exercises:
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