Quick answer: The standard deviation (SD) describes how spread out individual values are around the mean. The standard error (SE) describes how precisely the mean itself is estimated, and it equals SD divided by the square root of the sample size. SD is a property of your data; SE is a property of your estimate. As the sample grows, SD settles toward the population spread while SE keeps shrinking — which is why SE is the right error bar when the point of the figure is the mean.
Part of our statistics series. New to these ideas? Start with data types in statistics and distributions in statistics.
One line of arithmetic separates them
SD = sqrt( Σ(x − mean)² / (n − 1) )
SE = SD / sqrt(n)The only difference is the division by the square root of n. That small factor has a large consequence: with a sample of 100, the standard error is one tenth of the standard deviation. Report SE when the reader needs to judge how well you have pinned down the mean; report SD when they need to know how variable the raw observations are.
What each one is for
| Question the reader asks | Report |
|---|---|
| How variable are the observations? | Standard deviation |
| How confident are we in the mean? | Standard error |
| What range will a new observation fall in? | Standard deviation |
| What range is likely to contain the true mean? | Standard error or a confidence interval |
| How do I show error bars on a bar chart? | Usually standard error, SD for raw spread |
A 95% confidence interval is roughly the mean plus or minus two standard errors, so the two are close cousins. Publishing the interval is usually better than the bare SE because it states the range directly and readers do not have to multiply in their heads. The interpretation is worked through in confidence intervals explained.
Why the confusion matters
Because SE is always smaller than SD, error bars drawn with SE look tighter and the data look more precise than they are. Two papers plotting SE instead of SD can appear to agree when they do not. In a small sample the gap is largest: at n = 4, SE is half of SD. Always read the figure legend to see which was used before comparing studies, and never compare a mean-with-SE against a mean-with-SD.
| Sample size n | SE as a fraction of SD |
|---|---|
| 4 | 0.50 |
| 25 | 0.20 |
| 100 | 0.10 |
| 400 | 0.05 |
# R: both values and an interval in one call
m <- mean(x); s <- sd(x); n <- length(x)
se <- s / sqrt(n)
ci <- m + c(-1, 1) * qt(0.975, df = n - 1) * se
c(mean = m, sd = s, se = se, ci = ci)In jamovi the same figures come from the Descriptives and Exploration menus, with the SE of the mean shown alongside the SD — a quick way to sanity-check your numbers without writing code. The menu path is covered in descriptive statistics in jamovi.
The standard error of other statistics
The idea generalises. Every estimate has a standard error — of a proportion, a difference between means, a regression slope. The standard error of a proportion is the square root of p(1 − p) divided by n, and the standard error of a difference between two means is built from both group variances. These are the ingredients of every t-test and ANOVA, which is why understanding SD and SE early makes the later formulas readable. The tests themselves are set out in hypothesis testing in statistics and hypothesis testing in biology.
Common mistakes
- Labeling error bars SD when they are SE (or the reverse). Readers cannot tell from the length; state it in the caption.
- Reporting SD when the mean is the claim. The precision of a mean is an SE question.
- Comparing intervals across studies without checking which quantity each plotted.
- Assuming a small SE proves a large effect. A precisely estimated tiny difference is still tiny.
- Using SD to describe a skewed variable. When the data are skewed, prefer the median and an interquartile range.
Getting your summary statistics and their error bars right? Ampersand Academy teaches statistics one-to-one, using your own dataset as the curriculum.
Frequently asked questions
What is the difference between standard deviation and standard error?
Standard deviation measures how spread out the individual values are. Standard error measures how precisely the mean is estimated and equals the SD divided by the square root of the sample size.
Which should I plot, standard deviation or standard error?
Plot the standard error when the figure is about the mean and how well it is known. Plot the standard deviation when the figure is about the spread of the raw observations. Always label which one you used.
Why is standard error always smaller than standard deviation?
Because the standard error divides the standard deviation by the square root of the sample size, a number greater than one. The larger the sample, the smaller the standard error relative to the SD.
How do I calculate the standard error of the mean?
Divide the sample standard deviation by the square root of n. Most software reports it directly in its descriptive statistics output alongside the mean and SD.
Does a small standard error mean the result is important?
No. A small standard error means the estimate is precise, not that the effect is large. A tiny difference can be estimated very precisely with a big enough sample.