Short answer
Standard deviation describes how much observations differ from their mean; standard error describes how much an estimate would vary across possible samples. The key question is what is varying: individual measurements or the estimate calculated from them? Standard error is itself a standard deviation, but of a sampling distribution rather than the observations being summarized. 1 2
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At a glance
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| Question | Standard deviation | Standard error |
|---|---|---|
| What varies? | Observations | An estimate across possible samples |
| Main purpose? | Describe data spread | Express sampling uncertainty |
| Effect of a larger sample? | Does not tend to change | For the mean, tends to decrease |
| Applies only to means? | Describes observation variability around a mean | Can describe uncertainty in other estimates too |
These contrasts concern the role of each measure, not simply its numerical size. 2
What each thing is
A sample standard deviation summarizes observed spread and estimates variability in the underlying population. Standard error instead concerns a sampling distribution: the variation an estimate would show across possible samples. For the mean, it expresses the precision of the sample mean as an estimate of the population mean. 2
Key differences
For the sample mean, the supplied relationship is SE = SD/√n, where n is sample size. Thus, with the same SD, a larger sample produces a smaller SE. Increasing sample size does not, by itself, make individual observations less variable; SD does not tend to shrink as SE does. 2
How to tell them apart
Look for an explicit SD or SE label, then ask whether the number describes measurements or uncertainty in an estimate. That rule has a limit: a notation such as “69.4 ± 9.3 kg” does not identify the second number. Without a label or explanation, it could be SD, SE, or something else. 2
Where they overlap
Both measures concern variability, and the mean’s SE is calculated using SD and sample size. Their mathematical connection explains the similar terminology. Nevertheless, they summarize different distributions: the distribution of observations and the sampling distribution of an estimate. One cannot substitute for the other without changing the meaning. 2
Edge cases
Standard error is not restricted to the mean; other sample-based estimates can have standard errors. Accordingly, the displayed SD/√n relationship is for the mean, not a definition for every SE. Also, SD remains a valid variability measure for nonnormal data, although a different summary may be chosen for skewed data. 2
Why the distinction exists
The distinction separates describing the observed data from drawing conclusions about a population. SD answers how scattered measurements are. SE helps express uncertainty in an estimated population quantity and is used in calculating confidence intervals and, in many circumstances, P values. These are different statistical jobs. 2
Common misconceptions
A small SE does not mean individual observations cluster tightly together: it concerns precision of the estimate. Nor is SD limited to normal distributions. The familiar statement that about 95% of observations lie within two SDs of the mean has a normal-distribution context; it should not become a universal coverage rule. 1 2
Examples
Two hypothetical cases apply the distinction. First, a study of 100 measurements has SD = 10 units; using the supplied mean formula, SE = 1 unit. The measurements’ spread is 10, not 1. Second, a study of 400 measurements with the same SD has SE = 0.5 units: less uncertainty in the mean despite unchanged spread. 2