The Central Limit Theorem
The central limit theorem is one of the most powerful ideas in statistics. This chapter explains that when samples are large enough, the distribution of sample means, and of sample sums, is approximately normal regardless of the shape of the original population, and shows how this result is used to calculate probabilities.
The Central Limit Theorem for Means
The central limit theorem states that if you draw large enough random samples from any population and calculate each sample’s mean, the distribution of those sample means is approximately normal, whatever the shape of the original population. The mean of this sampling distribution equals the population mean, and its standard deviation, called the standard error, equals the population standard deviation divided by the square root of the sample size. Larger samples give a tighter distribution of means.
The Central Limit Theorem for Sums
The theorem applies to sample sums as well as means. If large samples are drawn from a population, the distribution of the sample sums is also approximately normal. The mean of the sum distribution is the sample size times the population mean, and its standard deviation is the square root of the sample size times the population standard deviation. This lets questions about totals, not just averages, be answered using the normal distribution.
Using the Central Limit Theorem
The theorem matters because it allows normal-distribution methods to be used even when the underlying population is not normal, provided the sample is large enough, commonly taken as at least thirty. To find the probability that a sample mean or sum falls in a given range, the value is converted to a z-score using the standard error, and the area under the standard normal curve is read. This underpins much of the inference that follows, including confidence intervals and hypothesis tests.