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Sampling distributions Example • Take random sample of students. • Ask “how many courses did you study for this past weekend?” • Calculate a statistic, say, the sample mean. Sample 1: 1 0 2 Mean = 1.0 Sample 2: 1 1 4 Mean = 2.0 Situation • Different samples produce different results. • Value of a statistic, like mean or proportion, depends on the particular sample obtained. • But some values may be more likely than others. • A “sampling distribution” is a probability distribution of a statistic. It indicates the likelihood of getting certain values. Sampling distribution of mean IF: • data are normally distributed with mean and standard deviation , and • random samples of size n are taken, THEN: The sampling distribution of the sample means is also normally distributed. The mean of all of the sample means is . The standard deviation of the sample means (“standard error of the mean”) is /sqrt(n). Example • Adult nose length is normally distributed with mean 45 mm and standard deviation 6 mm. • Take random samples of n = 4 adults. • Then, sample means are normally distributed with mean 45 mm and standard error 3 mm [from 6/sqrt(4) = 6/2]. Using empirical rule... • 68% of samples of n=4 adults will have an average nose length between 42 and 48 mm. • 95% of samples of n=4 adults will have an average nose length between 39 and 51 mm. • 99% of samples of n=4 adults will have an average nose length between 36 and 54 mm. What happens if we take larger samples? • Adult nose length is normally distributed with mean 45 mm and standard deviation 6 mm. • Take random samples of n = 36 adults. • Then, sample means are normally distributed with mean 45 mm and standard error 1 mm [from 6/sqrt(36) = 6/6]. Again, using empirical rule... • 68% of samples of n=36 adults will have an average nose length between 44 and 46 mm. • 95% of samples of n=36 adults will have an average nose length between 43 and 47 mm. • 99% of samples of n=36 adults will have an average nose length between 42 and 48 mm. • So … the larger the sample, the less the sample averages vary. What happens if data are not normally distributed? The Central Limit Theorem tells us ... Central Limit Theorem • Even if data are not normally distributed, as long as you take “large enough” samples, the sample averages will at least be approximately normally distributed. • Mean of sample averages is still . • Standard error of sample averages is still /sqrt(n). • In general, “large enough” means more than 30 measurements. Big Deal? • Knowing the distribution of sample means allows us to make decisions about the value of a population mean. • Let’s look at an application …

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