Standard Error of the Mean

STANDARD ERROR OF THE MEAN

The measure of variability of sample means, the standard deviation of the distribution of the sample mean, is called the standard error of the mean (s.e.m.). The s.e.m. is to the distribution of the sample means what the standard deviation is to the pop-ulation distribution. It has the nice property that it decreases in magnitude as the sample size increases, showing that the sample mean becomes a better and better approximation to the population mean as the sample size increases.

Because of the central limit theorem, we can use the normal distribution approximation to assert that the population  mean  μ will be within plus or minus two standard errors of the sample mean with a probability of approximately 95%. This is be-cause slightly over 95% of a standard normal distribution lies between ±2 and the sampling distribution for the mean is centered at μ with a standard deviation equal to one standard error of the mean.

A proof of the central limit theorem is beyond the scope of the course. However, the sampling experiment of Section 7.1 should be convincing to you. If you gener-ate random samples of larger sizes on the computer using an assumed population distribution, you should be able to generate histograms that will have the changing shape illustrated in Figure 7.4 as you increase the sample size.

Suppose we know the population standard deviation σ. Then we can transform the sample mean so that it has an approximate standard normal distribution, as we will show you in the next section.