Standard error and standard deviation
Stat talk: What’s a standard error and when should I report it rather than the standard deviation of my data?
Perhaps the most popular question I get asked is whether it is better to report a standard deviation or a standard error. This question is always built upon a fundamental misunderstanding of these two very different concepts. I suspect that students have missed the point because their supervisors tell them, for example, to use a standard error because it’s smaller. Some students may find this to be a satisfactory answer for a little while, but the more inquisitive student may begin to feel unsatisfied with this response.
Their intuitions are absolutely right. Standard deviations and standard errors are very different concepts and the correct one to use depends upon the context.
A standard deviation is a descriptive statistic describing the spread of a distribution. It is a very good description when the data are normally distributed. It is less useful when data are highly skewed or bimodal because it doesn’t describe very well the shape of the distribution. One generally uses standard deviation when reporting the characteristics of the sample, because one is describing how variable the data are around the mean. Other useful statistics for describing the spread of the data are interquartile range, the 25th and 75th percentiles, and the range of the data.
Figure 1. The standard deviation is a measure of the spread of the data. When data are a sample from a normally distributed distribution, then one expects two-thirds of the data to lie within 1 standard deviation of the mean.
Variance is a descriptive statistic also, and it is defined as the square of the standard deviation. It is not usually reported when describing results, but it is a more mathematically tractable formula (aka the sum of squared deviations) and plays a role in the computation of statistics. For example, if I have two statistics X & Y with known variances var(X) & var(Y), then the variance of the sum X+Y is equal to the sum of the variances: var(X) +var(Y). So you can see why statisticians like to talk about variances. But standard deviations carry an important meaning for spread, particularly when the data are normally distributed: The interval mean +- 1 SD can be expected to capture 2/3 of the sample, and the interval mean +- 2 SD can be expected to capture 95% of the sample.
A standard error is an inferential statistic that is used when comparing sample means (averages) across populations. It is a measure of precision of the sample mean. The sample mean is a statistic derived from data that has an underlying distribution. We can’t visualize it in the same way as the data, since we have performed a single experiment and have only a single value. Statistical theory tells us that the sample mean (for a large “enough” sample and under a few regularity conditions) is approximately normally distributed. The standard deviation of this normal distribution is what we call the standard error.
Figure 2. The distribution at the bottom represents the distribution of the data, whereas the distribution at the top is the theoretical distribution of the sample mean. The SD of 20 is a measure of the spread of the data, whereas the SE of 5 is a measure of uncertainty around the sample mean.
When we want to compare the means of outcomes from a two-sample experiment of Treatment A vs Treatment B, then we need to estimate how precisely we’ve measured the means. Actually, we are interested in how precisely we’ve measured the difference between the two means. We call this measure the standard error of the difference. You may not be surprised to learn that the standard error of the difference in the sample means is a function of the standard errors of the means:
Now that you’ve understood that the standard error of the mean (SE) and the standard deviation of the distribution (SD) are two different beasts, you may be wondering how they got confused in the first place. Whilst they differ conceptually, they have a simple relationship mathematically:
, where n is the number of data points.
Notice that the standard error depends upon two components: the standard deviation of the sample, and the size of the sample n. This makes intuitive sense: the larger the standard deviation of the sample, the less precise we can be about our estimate of the true mean. Also, the large the sample size, the more information we have about the population and the more precisely we can estimate the true mean.
Finally, to answer the question: when should I report SD? Whenever you want to describe the characteristics of the distribution of your sample, then use SD or other measures of spread. If you are reporting on a group of patients, you may want to report their mean age, and variation around the mean (reported as SD).
When should I report SE? Whenever you are comparing means and you wish to infer that the two means are different, then report SE. This applies to tables and also to graphs.
One of my students sent me a few good references on this subject that you may find interesting. More questions? Come visit the SCU for some more stat talk.
Additional References
Biau, D. J. (2011) Standard deviation and standard error, Clin Orthop Relat Res, 469, 2661 - 2664.