Standard Error of the Mean

The standard error of the mean is a number that represents the variability within a sampling [Page 1431]distribution. In other words, it is how measurements collected from different samples are distributed around the mean of the total population from which the sample measurements could be taken. The mean of each sample of measurements, therefore, is an estimate of the population mean. Although the means from each sample provide an estimate of the mean of the entire population, measurements collected from numerous samples are quite unlikely to be equivalent. This raises the question of how well each sample mean represents the true population mean. That accuracy can be estimated by calculating the standard error of the mean, which provides an indication of how close the sample mean is likely to be to the population mean.

Figure 1 Formula for Calculating Standard Error of the Mean

In random samples of the population, the standard error of the mean is the standard deviation of those sample means from the population mean. So strictly speaking, the functional difference between the two measures is that the standard deviation is used to calculate the variability of individual measurements around the mean of a sample, and the standard error of the mean is used to calculate an estimate of the variability of the means of multiple samples around the mean of the population. The formula for calculating the standard error of the mean (see Figure 1) is closely related to that of the standard deviation (for calculation, see Figure 3 in Standard Deviation entry) in that it is the standard deviation divided by the square root of the sample size. And, like the standard deviation (see Figure 2 in Standard Deviation entry), the means of 68.3%, or about two thirds, of all the sample means will be within ±1 standard errors of the mean (rather than ±1 standard deviation), 95.4% will be within ±2 standard errors of the population mean, and 99.7%, or almost all the sample means will be within ±3 standard errors from the population mean (only 0.15% will be more than three standard errors either above or below the population mean).

For example, assuming the height of all individuals in a total population of 330 people was measured, the mean of those measurements would be [Page 1432]the true mean of that entire population. Because it is rarely feasible to conduct an experiment that tests an entire population (and entire populations are typically much larger than this fictitious example), experimenters collect data from smaller samples that are randomly selected from the population. The size of the sample will influence the accuracy of the estimate of the true population mean. Looking at Figure 2, the graph on the left shows data for 6 samples of 5 individuals each and the graph on the right shows data for 6 samples of 50 individuals each. For both graphs, each vertical line (with the cross bars on the ends) shows the spread of the data in that sample (asterisks and circles represent extreme values within each sample), and the heavy horizontal line is the mean height of the individuals in each sample. The horizontal dotted line on each graph illustrates the location of the true mean of the total population of 330 individuals. The means of each of the small samples (left side, n = 5) deviate from the true population mean much more than the means of each of the larger samples (right, n = 50). The means of the smaller sample sizes also show much more variability in the distance from the population mean. On both graphs, the calculated estimates of the deviation of those sample means from the population mean (i.e., the standard error of the mean) are shown just above the x-axis. Comparing the standard error calculations on each graph, it is easy to observe that the larger samples provide a more consistent and truer estimate of the population mean than the smaller samples. The larger samples deviate from about one half to two thirds of a standard error from the mean, whereas the smaller sample sizes vary from one half to more than two and a half standard errors from the mean. Clearly, when feasible, testing a larger sample will provide a better estimate of the population, although there are practical limits.

Figure 2 The Effect of Sample Size on Accuracy of Estimating the Population Mean

Note: The population mean = 67.7 inches tall (dashed line).

Figure 3 The Function of Sample Size on the Standard Error of the Mean

Note: Holding the SD constant at 20.

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