Find the standard error of the mean from your data or from s and n, and understand how it differs from the standard deviation.
Separate numbers with commas, spaces or line breaks. You can paste a column from Excel or Sheets. 10 values read.
| Mean | 12.08 |
|---|---|
| Standard deviation, s | 0.2781 |
| n | 10 |
| Rough 95% interval for the mean (±1.96 SE) | ±0.1724 |
Working: SE = s ÷ √n = 0.2781 ÷ √10 = 0.2781 ÷ 3.1623 = 0.08794. For small samples use a t critical value; see the confidence interval calculator.
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Each column is one imaginary study that averaged n values from the same population (SD = 10). With n = 1, the “averages” are just single values and spread widely. Slide n up and the averages crowd toward the true mean; their spread is the standard error, 10 ÷ √n.
SE = 10 ÷ √10 = 3.162. Shaded band = ±1.96 SE, where about 95% of study averages land.
SE = s ÷ √n
The standard error is the standard deviation of the sampling distribution of the mean. It is the ingredient in every confidence interval and t-test for means. If you only have raw data and need s first, the standard deviation calculator shows that step.
Standard deviation describes how spread out individual values are. Standard error describes how much the sample mean would vary if you repeated the study, so it measures the precision of the mean. SE = SD ÷ √n, so it is always smaller than the SD for n > 1.
Report SD to describe the data (how variable the subjects were). Report SE, or better a confidence interval, when the point is how precisely you estimated a mean. Error bars should always be labelled; many journals now prefer 95% confidence intervals.
SE shrinks with the square root of n. Quadrupling the sample size halves the standard error; to cut it to a tenth you need 100 times the data.
There is no built-in function. Use =STDEV.S(range)/SQRT(COUNT(range)).