Standard Error Calculator

Standard error vs. standard deviation

The standard deviation (SD) measures how much individual values vary from the mean of the sample. The standard error (SE) measures how much the sample mean itself would vary if you took many samples. SE = SD / √n, so the standard error shrinks as the sample grows larger.

If you measure 10 people's heights: the SD tells you how spread out their heights are (e.g. SD = 6 cm). The SE tells you how reliable the sample mean is as an estimate of the population mean (e.g. SE = 6 / √10 ≈ 1.9 cm).

Standard error and confidence intervals

The standard error is used to construct confidence intervals for the population mean. For large samples (n > 30), the 95% CI ≈ x̄ ± 1.96 × SE. For small samples, a t-distribution with (n−1) degrees of freedom is used instead of the z = 1.96 multiplier.

Error bars on scientific charts and graphs almost always show either ±1 SD (spread of data) or ±1 SE (precision of the mean). These have very different interpretations — ±1 SE bars from two groups can overlap while still having a statistically significant difference.

Central limit theorem

The central limit theorem explains why the standard error matters: regardless of the population distribution, the distribution of sample means approaches a normal distribution as n grows. The mean of this sampling distribution equals the population mean, and the standard deviation of the sampling distribution is exactly the standard error SD / √n.