By contrast the standard deviation will not tend to change as we increase the size of our sample. The standard error falls as the sample size increases, as the extent of chance variation is reduced-this idea underlies the sample size calculation for a controlled trial, for example. The standard error of the sample mean depends on both the standard deviation and the sample size, by the simple relation SE = SD/√(sample size). Another way of considering the standard error is as a measure of the precision of the sample mean. As the standard error is a type of standard deviation, confusion is understandable.

We can estimate how much sample means will vary from the standard deviation of this sampling distribution, which we call the standard error (SE) of the estimate of the mean. Now the sample mean will vary from sample to sample the way this variation occurs is described by the “sampling distribution” of the mean. We usually collect data in order to generalise from them and so use the sample mean as an estimate of the mean for the whole population. When we calculate the sample mean we are usually interested not in the mean of this particular sample, but in the mean for individuals of this type-in statistical terms, of the population from which the sample comes. We may choose a different summary statistic, however, when data have a skewed distribution. About 95% of observations of any distribution usually fall within the 2 standard deviation limits, though those outside may all be at one end. Contrary to popular misconception, the standard deviation is a valid measure of variability regardless of the distribution. For data with a normal distribution, 2 about 95% of individuals will have values within 2 standard deviations of the mean, the other 5% being equally scattered above and below these limits. When we calculate the standard deviation of a sample, we are using it as an estimate of the variability of the population from which the sample was drawn. The standard deviation (often SD) is a measure of variability. 1 The contrast between these two terms reflects the important distinction between data description and inference, one that all researchers should appreciate. The terms “standard error” and “standard deviation” are often confused.

Standard error of the mean