How to read error bars: SD, SE and 95% CI
An error bar is a line through a plotted mean, and three different statistics routinely hide behind it: the standard deviation, the standard error, and a 95% confidence interval. For the same data they can differ in length by a factor of three or more. If the caption does not say which one you are looking at, you cannot read the chart.
Three bars, three questions
Each kind of bar answers a different question about the data behind the point.
SD bars answer "how spread out are the individual values?" They show the mean ± one standard deviation. They do not shrink as you collect more data, because more data does not make people, plants or parts less variable. It only gives a better estimate of how variable they are.
SE bars answer "how precisely is the mean estimated?" The standard error of the mean is the SD divided by √n, so it shrinks as the sample grows. The difference between describing spread and describing a mean's precision is set out in standard deviation vs mean.
95% CI bars answer "what range of values for the true mean is consistent with this data?" For a small sample the half-width is the SE times a t critical value, which is a little above 2 for moderate n and grows as n falls.
Worked example: n = 10, s = 4
Suppose a group of 10 measurements has a mean of 50 and a sample SD of 4. The three bars are:
SD bar: ± s = ± 4.00 SE bar: ± s / √n = ± 4 / √10 = ± 1.26 95% CI: ± t(0.975, 9) × SE = ± 2.262 × 1.265 = ± 2.86
The critical value 2.262 comes from the t distribution with 9 degrees of freedom. Using the normal value 1.96 instead would give ± 2.48, too narrow for a sample this small. Here is how the widths compare, and what happens with four times as much data:
| Bar type | Half-width, n = 10 | Full length, n = 10 | Half-width, n = 40 |
|---|---|---|---|
| ± 1 SD | 4.00 | 8.00 | 4.00 |
| ± 1 SE | 1.26 | 2.53 | 0.63 |
| 95% CI | 2.86 | 5.72 | 1.28 |
The n = 40 column assumes the SD stays at 4. The SD bar does not move. The SE bar halves, because √40 is twice √10. The CI bar shrinks by a bit more than half, from 2.86 to 1.28, because the t value also falls, from 2.262 to 2.023. You can reproduce these numbers with the standard error calculator and the confidence interval calculator.
The practical consequence: the same experiment drawn with SE bars looks about three times more precise than when drawn with SD bars, and more than twice as precise as with 95% CI bars. Authors sometimes choose SE bars for that reason. As a reader, the first thing to find is the caption.
Try it: standard deviation of the sample mean calculator
The calculator opens with its own defaults; enter an SD of 4 and n = 10 to reproduce the SE bar of ± 1.26, or your own SD and sample size to size your bars.
Applies the finite population correction √((N − n)/(N − 1)).
┄ Individual values: SD σ = 15 ━ Sample means (n = 25): SD σ/√n = 3
| Sample size | σ ÷ √n | Compared with n = 25 |
|---|---|---|
| 25 | 3 | — |
| 100 | 1.5 | 2× smaller |
| 400 | 0.75 | 4× smaller |
| 2500 | 0.3 | 10× smaller |
Show the working, step by step
Divide the standard deviation by the square root of the sample size.
σx̄ = σ ÷ √n = 15 ÷ √25 = 15 ÷ 5 = 3
This quantity is also called the standard error of the mean. With a sample SD s in place of σ it is an estimate, and intervals built from it use the t distribution.
Open the full standard deviation of the sample mean calculator for the finite population correction and raw-data input.
What overlapping bars do and do not mean
The most common use of error bars is eyeballing whether two groups differ. Rules of thumb exist, but they depend entirely on the bar type. Take two independent groups, each with n = 10 and SD 4, as above, and ask: if the bars of the two groups are just touching, how big is the gap between the means, and what p-value would a two-sample t-test give?
| Bars just touching | Gap between means | t (18 df) | Two-sided p |
|---|---|---|---|
| ± 1 SE bars | 2.53 | 1.41 | 0.17 |
| 95% CI bars | 5.72 | 3.20 | 0.005 |
| ± 1 SD bars | 8.00 | 4.47 | 0.0003 |
The t statistic is the gap divided by the standard error of the difference, √(1.265² + 1.265²) = 1.789. The gap needed for p = 0.05 is 2.101 × 1.789 = 3.76.
That gives three rules for independent groups of this size:
Touching SE bars are not evidence of a difference. The means would need to be about 3 SEs apart (3.76 ÷ 1.265 = 2.97), so there has to be a visible gap between the SE bars, here of about 1.2 units.
Overlapping 95% CI bars do not rule out a difference. At the p = 0.05 boundary the two CIs still overlap by 5.72 − 3.76 = 1.96 units, about two-thirds of one arm. Bars that do not overlap at all correspond to p of roughly 0.005 or smaller.
SD bars say almost nothing about whether means differ, because their length does not depend on n. Two groups with heavily overlapping SD bars can have very different means if the samples are large.
When the rules break down
All of the above assumes two independent groups of similar size and SD. Three common situations break it.
Repeated measures. If the same subjects are measured before and after, the relevant spread is that of each subject's change, which is often far smaller than the between-subject SD. The bars on the two group means can overlap completely while the paired difference is highly significant. For paired data, a chart of the differences, with their own CI, is the honest picture.
Very unequal groups. When one group has n = 5 and the other n = 200, the bars have very different lengths and the overlap rules above no longer apply. Compute the test.
Many comparisons on one chart. With a dozen bars, some pairs will look separated by chance. Eyeballing does not adjust for multiple comparisons; a formal method does.
If you are drawing the chart yourself, the mean and standard error calculator gives the numbers for each group. Pick the bar that matches your message, write in the caption "error bars show ± 1 SD" (or SE, or 95% CI), and give n for each group. That single sentence makes the chart readable.
Related calculators
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Standard error calculator
SD ÷ √n from raw data or a known SD.
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Confidence interval calculator
A t- or z-based interval for a mean, with the working shown.
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t-test calculator
The formal test the error bars are often used to guess at.
Common questions
Which error bar should I put on my chart?
Use SD bars when you want to show how much individual values vary. Use 95% confidence interval bars when the reader will compare means, since they can be read directly as a range of plausible values for each mean. SE bars are common but the hardest to read correctly; if you use them, say so in the caption and give n.
If two 95% confidence intervals overlap, is the difference not significant?
Not necessarily. For two independent groups, 95% CIs can overlap by a fair amount and the difference can still be significant at p < 0.05. In the example on this page, with n = 10 per group, the bars can overlap by about two-thirds of one arm and the difference is still right at p = 0.05. Only the absence of overlap is a reliable signal.
Why are the error bars so small on a chart with thousands of data points?
They are probably standard errors or confidence intervals, which shrink as n grows. With n = 10,000 the SE is one hundredth of the SD. Tiny bars in that case say the mean is precisely estimated, not that the individual values are tightly bunched.