Technical Mathematics, Sixth Edition

21.6. Standard Errors

When we draw a random sample from a population, it is usually to infer something about that population. Typically, from our sample we compute a statistic(such as the sample mean ) and use it to infer a population parameter(such as the population mean μ).

Example 55: A researcher measured the heights of a randomly drawn sample of students at Tech College and calculated the mean height of that sample. It was mean height = 67.50 in. From this he inferred that the mean height of the entire population of students at Tech College was mean height μ = 67.50 in.

In general,

population parameter ≈ sample statistic

How accurate is the population parameter that we find in this way? We know that all measurements are approximate, and we usually give some indication of the accuracy of a measurement. In fact, a population parameter such as the mean height is often given in the form

mean height μ = 67.50 ± 0.24 in.

where ±0.24 is called the standard error of the mean. In this section we show how to compute the standard error of the mean and the standard error of the standard deviation.

Further, when we give a range of values for a statistic, we know that it is possible that another sample can have a mean that lies outside the given range. In fact, the true mean itself can ...