18.5 Standard deviation vs. standard error

Even experienced researchers confuse the meaning and the usage of the terms standard deviation and standard error (Ko et al. 2014), so understanding the difference is important.

The standard deviation, in general, quantifies the amount of variation in any variable. Without further qualification, the standard deviation quantifies how much individual observations vary from individual to individual (for quantitative data).

The standard error is a standard deviation that quantifies how much a sample statistic varies from sample to sample.

Crucially, the standard error is a standard deviation, but has a special name to indicate that it is the standard deviation of something very specific.

Any numerical quantity estimated from a sample (a statistic) can vary from sample to sample, and so has sampling variation, a sampling distribution, and hence a standard error:

• the sample mean \(\bar{x}\);
• the sample proportion \(\hat{p}\);
• the sample odds ratio;
• the sample median;
• the sample standard deviation \(s\);
• etc.

The standard error is often abbreviated to ‘SE’ or ‘s.e.’

For example, the ‘standard error of the sample mean’ is written as \(\text{s.e.}(\bar{x})\), and the ‘standard error of the sample proportion’ is written as \(\text{s.e.}(\hat{p})\).