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Sample mean: M. Population mean: . ispronounced ‘mew,’ like the sound a kitten makes. We used ‘m’ with the symbol font to get . We introduced the concept of a sampling distribution, which is the distribution a statistic will take when calculated from multiple independent samples. One particularly important (or central) sampling distribution is the distribution of the mean across very many, or even all possible, samples of a particular size. The central limit theorem tells us the properties of that sampling distribution. The Central Limit Theorem Part One: The mean of all possible sample means will equal the population mean. (‘All possible’ is roughly the same as ‘a very large number’.) The standard deviation of those sample means will equal the population standard deviation divided by the square root of the sample size. Part Two: If the population is normally distributed, then the distribution of sample means will also be normally distributed. If the population is not normally distributed, but is not so horrible that it doesn’t have a mean or has infinite variability, then the distribution of sample means will become normal as the sample size becomes large. (How large is ‘large’ is a question for which the answer is ‘it depends.’ For some non-normal distributions, sample size as small as 10 might be sufficient, and for others, 200 might not be a large enough sample size.) To avoid confusion, the standard deviation of a sampling distribution is known as the standard error. So we would refer to the standard deviation of the sampling distribution of the mean as ‘the standard error of the mean.’