Standard Error Calculator
This standard error calculator works from a list of measurements or directly from the standard deviation s and the sample size n. Example: the values 12, 15, 11, 18, 14, 16, 13 and 17 have s ≈ 2.449, so the standard error is ≈ 0.866.
Source: Wikipedia – Standard error. Updated: .
How it is calculated
The standard error of the mean tells you how precisely a sample mean estimates the true population mean. It is often written SE or SEM and is computed from the standard deviation s and the sample size n.
Formula and steps
The formula is SE = s / √n. First find the mean x̄, then the sum of squared deviations Σ(x − x̄)². Dividing that sum by n − 1 gives the variance s², and its square root is the standard deviation s. Finally divide s by the square root of n. The sample standard deviation (denominator n − 1) is the right one here, not the population version.
Worked example
For 12, 15, 11, 18, 14, 16, 13, 17 we have n = 8 and x̄ = 14.5. The squared deviations add up to 42, so s² = 42 / 7 = 6 and s ≈ 2.449. That gives SE = √(6 / 8) ≈ 0.866. With known summary values s = 10 and n = 25, SE = 10 / 5 = 2.
- More data shrinks the standard error, but only with the square root: four times the data halves it.
- Standard deviation and standard error are different: one describes the spread of the data, the other the uncertainty of the mean.
- A confidence interval follows from it, for example x̄ ± 1.96 · SE for 95 %.
Frequently asked questions
What is the difference between standard deviation and standard error?
The standard deviation measures how far individual values scatter around the mean. The standard error measures how far the mean itself would scatter if you drew new samples again and again. It is always smaller than s once n is above 1.
How do I calculate the standard error from s and n?
Divide s by the square root of n. With s = 10 and n = 25, SE = 10 / √25 = 2.
How many values do I need for a meaningful standard error?
Two values are enough to compute it, but it only becomes informative with a larger sample, because s changes a lot when you have just a few values.
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