Standard Deviation vs Standard Error: What's the Difference?

They look alike, they are often printed side by side, and journals have spent decades correcting the confusion. Yet standard deviation (SD) and standard error (SE) answer different questions. One describes your data. The other describes how well you know the average.

Two questions, two numbers

The SD measures the spread of individual values. For roughly normal data, about 95% of individuals fall within two SDs of the mean. The SE describes how much a sample mean would vary from sample to sample, and equals the SD divided by the square root of the sample size. Altman and Bland framed this as the distinction between description and inference (BMJ, 2005).

Same SD, growing sample

Suppose adult heights have a mean of 170 cm and an SD of 10 cm. Individuals then range from roughly 150 to 190 cm. With 25 people, SE = 10 / √25 = 2 cm, and the 95% interval for the mean is about 165.9 to 174.1 cm. With 100 people, SE = 1 cm and the interval is 168.0 to 172.0. With 400 people, SE = 0.5 cm and it is 169.0 to 171.0.

The SD stays near 10 the whole time, because people did not become less varied. Only our knowledge of the mean sharpened. The standard deviation calculator and the confidence interval calculator for unknown variance compute the two views.

The SD describes people. The SE describes our certainty about the average.

Why error bars deceive

A bar chart with mean ± SE error bars looks reassuringly tight, but those bars are shorter than SD bars by a factor of √n. In our n = 100 example, the SE bar is ten times shorter than the SD bar. Bars of ±1 SE correspond to roughly a 68% confidence interval, not 95% (Carter, 2013). Using SE bars to describe the spread of a sample therefore makes the data look less variable than they are.

Say which one you used

The notation 69.4 ± 9.3 kg does not say whether 9.3 is an SD or an SE. A review of 88 articles found that 14% failed to identify which measure they reported, and the BMJ removes ± signs and asks authors to specify (Altman and Bland). Write "mean 69.4 kg (SD 9.3)" or "mean 69.4 kg (SE 0.9)," never a bare ±.

Which one to report

Use the SD to describe your sample: how variable are people? Use a confidence interval, built from the SE, to make a claim about the population mean or compare groups. Descriptive tables want SDs, and inferential results want intervals. The descriptive statistics calculator summarizes the mean and spread of a dataset.

Because the SE shrinks with √n, halving it takes four times as many observations, which is why precise estimates get expensive. Our article on poll margins of error shows the same square-root rule at work in surveys. With very small samples there is a further catch. The SE itself is an estimate, so the multiplier for a 95% interval is larger than 1.96: with five observations it is about 2.78, which widens the interval noticeably.

In short, report the SD to show how people differ, and an interval to show how sure you are about their average. Plot the raw points when you can, since they show what neither number can.