How to Calculate Standard Error of the Mean
The standard error of the mean (SEM) measures how much the sample mean would vary if you took multiple random samples from the same population. It's the key ingredient in confidence intervals and hypothesis testing.
The Formula
SE = s / √n
Where:
- s = sample standard deviation
- n = sample size
Step-by-Step Example
You measure resting heart rate in 25 patients and find:
- Mean (x̄) = 72 bpm
- Standard deviation (s) = 10 bpm
SE = 10 / √25 = 10 / 5 = 2 bpm
Interpreting Standard Error
The SE of 2 bpm means your sample mean of 72 bpm is likely within ~2 bpm of the true population mean. Specifically, the 95% confidence interval is:
95% CI = x̄ ± 1.96 × SE = 72 ± 1.96 × 2 = 72 ± 3.92
So the 95% CI is [68.1, 75.9] bpm.
Standard Deviation vs. Standard Error
| Metric | Measures | Decreases with larger n? |
|---|---|---|
| Standard Deviation | Spread of individual data points | No |
| Standard Error | Precision of the sample mean | Yes (SE = SD/√n) |
A key implication: doubling sample size reduces SE by a factor of √2 ≈ 1.41. To cut SE in half, you need to quadruple sample size.
When to Report Which
- Report SD when describing the variability of the individuals in your sample
- Report SE (or CI) when describing how precisely you've estimated the population mean
Use our standard error calculator for any dataset or summary statistics.