How to calculate the standard error
The standard error of the mean estimates how much a sample mean would vary from sample to sample. Divide the sample standard deviation (using n − 1) by the square root of the sample size. For 2, 4, 4, 4, 5, 5 the sample standard deviation is about 1.095 and n is 6, so the standard error is 1.095 ÷ √6 ≈ 0.45. It is the standard deviation divided by the square root of the sample size, so start there.
How to use this calculator
Enter at least two sample values and leave unused fields blank. The badge shows the sample standard deviation and sample size used. A larger sample shrinks the standard error. For relative spread, see the coefficient of variation calculator.
The standard error carries the same units as your data. Because n sits under a square root, gains come slowly: quadrupling the sample size only halves the standard error.
Standard error vs. standard deviation
The standard deviation describes spread among individual values, while the standard error describes the precision of the sample mean. As the sample grows, the standard error falls even though the standard deviation stays roughly the same. For percentiles of the data, see the percentile calculator.
Quoting one when you mean the other is a common reporting error, and it makes results look far more or far less precise than they are. Also remember that the standard error only captures random sampling variation — it says nothing about bias from a badly chosen sample, which no amount of extra data will fix.
Frequently asked questions
- How do I calculate the standard error?
- Divide the sample standard deviation by the square root of the sample size. For 2, 4, 4, 4, 5, 5 the standard error is about 0.45.
- Does this use sample or population standard deviation?
- It uses the sample standard deviation, dividing the sum of squared deviations by n − 1 (Bessel's correction).
- Why does a bigger sample give a smaller standard error?
- Dividing by the square root of a larger n shrinks the result, so larger samples estimate the mean more precisely.
- How is standard error different from standard deviation?
- Standard deviation measures spread of the data; standard error measures the uncertainty of the sample mean.
- How do I turn the standard error into a confidence interval?
- Multiply it by the appropriate critical value and add and subtract that from the mean. For a rough 95% interval, use about 2 standard errors either side of the sample mean.