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# clt - Sampling distribution of a sample mean The mean and...

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Sampling distribution of a sample mean The mean and standard deviation of For normally distributed populations The central limit theorem x

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Reminder: What is a sampling distribution? The sampling distribution of a statistic is the distribution of all possible values taken by the statistic when all possible samples of a fixed size n are taken from the population. It is a theoretical idea — we do not actually build it. The sampling distribution of a statistic is the probability distribution of that statistic.
Sampling distribution of the sample mean We take many random samples of a given size n from a population with mean μ and standard deviation σ . Some sample means will be above the population mean μ and some will be below, making up the sampling distribution. Sampling distribution of “ x bar” Histogram of some sample averages

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Sampling distribution of x bar μ σ / n For any population with mean μ and standard deviation σ : The mean , or center of the sampling distribution of , is equal to x
Mean of a sampling distribution of There is no tendency for a sample mean to fall systematically above or below μ , even if the distribution of the raw data is skewed. Thus, the mean of the sampling distribution is an unbiased estimate of the population mean μ — it will be “correct on average” in many samples. Standard deviation of a sampling distribution of The standard deviation of the sampling distribution measures how much the sample statistic varies from sample to sample. It is smaller than the standard deviation of the population by a factor of √ n . Averages are less variable than individual observations .

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