Descriptive statistics
Standard Error Calculator
How much a sample statistic would bounce around if you collected the data again.
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How to read this result
Standard deviation describes the data; standard error describes the estimate. The SD says how far individual observations sit from the mean. The SE says how far the sample mean would sit from the true mean if you repeated the study. They answer different questions and are routinely confused.
Only one of them shrinks with more data. The standard deviation is a property of the population: collect ten times as many observations and it stays put. The standard error falls as 1/√n, which is the entire reason larger samples give better estimates.
Quadruple the data to halve the error. That square root is unforgiving. Going from 100 observations to 400 halves the standard error; getting another halving takes 1,600.
It is the building block of everything else. A confidence interval is the estimate plus or minus a multiplier times the standard error. A t statistic is a difference divided by one. Getting the standard error right is most of getting inference right.
The formula
- the sample standard deviation
- the sample proportion
- the sample size
The proportion version needs no separate SD, because for a binary outcome the variance is determined by the proportion itself.
Worked example
How precise is the average of twenty days?
Twenty daily customer counts: 18, 22, 25, 19, 31, 28, 24, 21, 27, 33, 26, 23, 29, 20, 35, 30, 22, 26, 24, 28. The mean is 25.55 and the standard deviation 4.639.
- Take the standard deviation.s = 4.6394
- Divide by the square root of the sample size.√20 = 4.4721
- The standard error of the mean.4.6394 / 4.4721 = 1.0374
- Approximate 95% interval: about two standard errors either side.25.55 ± 2.09 → roughly 23.5 to 27.6
The daily counts vary by about 4.6 customers; the twenty-day average is pinned down to about 1.0. Same data, two very different statements — which is why the SD and the SE must never be swapped in a report.
Checked against R's sd(x)/sqrt(length(x)).
The calculator above is loaded with these numbers by the Load the worked example button.
Assumptions, and when to use something else
A random sample of independent observations. The formula s/√n also assumes the observations are identically distributed — a systematic drift over time, or clustering, makes the true standard error larger than this formula says.
For a proportion, the same normal-approximation caveats apply as elsewhere: with fewer than about 10 successes or 10 failures, the SE is a poor description of the estimate's uncertainty and an exact interval is better.
- You want the intervalConfidence interval calculatorThe estimate ± a multiplier × the standard error.
- You want the spread of the data insteadStandard deviationHow far individual values sit from the mean.
- You are planning the sample sizeSample size calculatorWorks out the n that gives the precision you need.
Questions people ask
What is the difference between standard deviation and standard error?
The standard deviation measures the spread of individual observations and does not change as you collect more data. The standard error measures the precision of a statistic — usually the mean — and shrinks as 1/√n. Error bars on a chart should say which one they show, because they can differ by a factor of ten.
Why does the standard error decrease with sample size?
Because averaging cancels noise. Individual values scatter, but their errors partly offset each other, and the more you average the more they cancel — at a rate of exactly 1/√n.
Should error bars show SD or SE?
Whichever answers the question, and the caption must say which. SD bars describe how variable the data are; SE bars describe how precisely the mean is known. SE bars are always shorter, which is why they are more popular and more often misread.