What the standard error of the mean measures
The standard error of the mean (SEM) tells you how much the mean of your sample is likely to differ from the true mean of the whole population. Every time you draw a fresh sample you get a slightly different sample mean; the SEM is the standard deviation of that hypothetical distribution of sample means. A small SEM means your sample mean is a precise estimate of the population mean; a large SEM means it could be off by quite a bit.
The formula
The standard error of the mean is the sample standard deviation divided by the square root of the sample size:
SEM = s / √n
Here s is the sample standard deviation and n is the number of observations. The sample standard deviation itself uses the Bessel-corrected (n − 1) denominator:
s = √[ Σ(xᵢ − x̄)² / (n − 1) ]
where x̄ is the sample mean and the sum runs over all n values. This calculator uses the n − 1 denominator, which is the standard choice for estimating population spread from a sample.
Why divide by √n and not n?
Variances of independent measurements add, so the variance of a sample mean is the population variance divided by n. Taking the square root to return to standard-deviation units gives √(σ²/n) = σ/√n. That is why doubling your precision (halving the SEM) requires quadrupling the sample size, not merely doubling it — the √n in the denominator makes each additional data point buy progressively less precision.
Standard error vs. standard deviation
These two are constantly confused. The standard deviation describes the spread of the raw data — how far individual observations sit from the mean — and it does not systematically shrink as you collect more data. The standard error describes the precision of the mean and always shrinks as n grows. If you report ± values on a chart, use the standard deviation to show variability in the data and the standard error (or a confidence interval) to show uncertainty in an estimate.
Using the SEM for confidence intervals
For a reasonably large sample, an approximate 95% confidence interval for the population mean is x̄ ± 1.96 × SEM. For small samples you should replace 1.96 with the t-distribution critical value for n − 1 degrees of freedom, which is larger and widens the interval to account for the extra uncertainty in estimating s.
A worked reference example
Suppose you measure 25 items and find a sample standard deviation of s = 10. Then SEM = 10 / √25 = 10 / 5 = 2. If you collected 100 items instead with the same s = 10, the SEM would fall to 10 / √100 = 10 / 10 = 1 — four times the data for half the standard error.