Standard Error of the Mean

The standard error of the mean (or standard error, for short) is a measure of how representative a sample is of the population from which it was drawn. Using information about how people vary within a sample and how big that sample is, we can estimate how much variation we would expect to see if we drew multiple samples from a given population.

The concept of the standard error of the mean is similar to the standard deviation. Just as the standard deviation allows us to see how much individuals vary within a sample, so the standard error allows us to estimate how much samples will vary within a population. This is important because in research, it is seldom possible to observe the whole population we are interested in. We nearly always observe a smaller sample and make inferences from it to the population. For example, if we were interested in finding out whether Americans, on average, have a higher body mass index (BMI) than Europeans do, then we would take a sample of Americans and a sample of Europeans and find out the average BMI of both samples. It would be unfeasible to get this information for all members of both populations, but using the information we get from our sample, we can build up a picture about what the distribution of all possible samples would look like.

Take a look at Figure 1a. This shows a normal distribution of BMI scores in a sample of 50 Americans using some hypothetical data. The mean BMI score for this sample is 24.35. Based on the nature of normal distributions, the standard deviation of 3.31 tells us that 68% of all individuals in this sample have a [Page 942]score between 21.04 and 27.66 (±1 standard deviation) and that 95% have a score between 17.73 and 30.97 (±2 standard deviations).

Figure 1a A Normal Distribution of Body Mass Index Scores From a Hypothetical Sample of 50 Americans

Now take a look at Figure 1b. This also shows a normal distribution, but this time it is for all possible sample means from the population of Americans (again using hypothetical data). Undoubtedly there will be some variation between the different possible samples we could have selected. We may have selected a particularly overweight group of Americans, for instance. Imagine we could take all possible samples. Each one would have a different makeup of people and thus would have a slightly different average BMI. The standard error tells us the degree to which sample means drawn from the same population are likely to differ.

Notice that the mean score of samples is the same as the mean score of the sample of 50 individuals. Notice also that the distribution of all sample means is narrower. This makes sense when we realize that we are less likely to observe extreme means than we are to observe extreme individuals. What is really important, though, is that we are now able to estimate the upper and lower boundaries of the mean that we expect to see if the sample has been drawn from this population. Using the standard error of .47, we see that 68% of all sample means will be between 23.88 and 24.82, and 95% will be between 23.41 and 25.29. If we find a sample that has a mean score outside those limits, then we can assume that it is not from the same population (using 95% confidence intervals).

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