How to use this standard error calculator
Paste a sample above and press Calculate. The standard error appears with the sample standard deviation and size it came from. The two ingredients are always shown, because the SE is meaningless without them.
What is the standard error?
The standard deviation describes how much individual values vary. The standard error describes how much a sample mean would wobble if you repeated the study. Collect 100 different samples of heights and their means will not agree exactly. The standard error is the standard deviation of those means, the yardstick of a sample's precision.
Eight measured heights (cm): 168, 172, 165, 180, 175, 170, 169, 174. Sample mean = 171.6 cm, sample SD ≈ 4.69 cm, so SE = 4.69 ÷ √8 ≈ 1.66 cm. Read it as: this sample's mean is probably within a couple of centimetres of the true population mean, and quadrupling the sample (to 32 people) would roughly halve that wobble, because √n grows while s stays put.
Standard error vs standard deviation
The most confused pair in introductory statistics. SD = spread of the data; SE = precision of the estimate. Error bars on a graph of means should be standard errors (or confidence intervals built from them). A shrinking SE with bigger samples is not your data becoming less variable. It is your aim improving.
Frequently asked questions
What is the formula for standard error?
SE = s ÷ √n, where s is the sample standard deviation and n is the sample size.
Why does standard error decrease with sample size?
Averaging over more observations cancels more random noise. The √n in the denominator means quadrupling the sample halves the standard error.
Is standard error the same as margin of error?
They are closely related: a 95% margin of error is roughly 1.96 × SE. The margin of error adds a confidence multiplier to the standard error.