Measuring the precision of the sample mean as an estimate of the true value

Most practical exercises are based on a limited number of individual data values (a sample) which are used to make inferences about the population from which they were drawn. For example, the lead content might be measured in blood samples from 100 adult females and used as an estimate of the adult female lead content, with the sample mean (Ÿ (Y bar)) and sample standard deviation (s) providing estimates of the true values of the underlying population mean (µ) and the population standard deviation (δ) The reliability of the sample mean as an estimate of the true (population) mean can be assessed by calculating the standard error of the sample mean (often abbreviated to standard error or SE), from:

⇒ Equation [40.3]

SE = s/√n

Strictly, the standard error is an estimate of the standard deviation of the means of n-sized samples from the population. At a practical level, it is clear from eqn [40.3] that the SE is directly affected by sample dispersion and inversely related to sample size. This means that the SE will decrease as the number of data values in the sample increases, giving increased precision.

Summary descriptive statistics for the sample mean are often quoted as Ÿ (Y bar) &plusm; SE (n), with the SE being given to one significant figure more than the mean. You can use such information to carry out a t-test between two samples (Box 41.1); the SE is also useful because it allows calculation of confidence limits for the sample mean.