An Introduction to Statistics

A similar process can be used when dealing with sample means. If a suYcient number of samples have been taken, and their means plotted, then they begin to take up a normal distribution. It can be shown mathematically that the mean of this distribution (μx) is the same as the population mean (μ). Furthermore, the standard deviation of the distribution is equal to ó/'n, where n is the number of cases in the sample. This is known as the standard error of the mean (SEM). To estimate the probability of getting a value greater than or equal to a particular sample mean (x), in a population with a known mean (μ) and standard deviation (ó), we again calculate the z statistic. However, as we are dealing with the distribution of the means, we use the SEM rather than the population’s standard deviation: