Standard error calculator

What is the standard error?

The standard deviation of a statistic from one sample to the next. The standard error of the mean, s ÷ √n, tells you how far a sample mean typically lands from the population mean. With a standard deviation of 15 and samples of 25, it is 15 ÷ √25 = 3 (OpenStax Introductory Statistics 2e, Example 7.1).

What is the difference between standard deviation and standard error?

The standard deviation describes how spread out the individual values are; it does not shrink as you collect more data. The standard error describes how precise the mean is, and it falls with the square root of the sample size — four times the data halves it.

How do I calculate the standard error of a proportion?

√(p′(1 − p′) ÷ n). If 421 of 500 people say yes, p′ = 0.842 and the standard error is √(0.842 × 0.158 ÷ 500) = 0.0163. Multiplied by 1.96 it gives the 0.032 error bound of the 95% interval in OpenStax Example 8.10.

What is the standard error of the difference between two means?

√(s₁²/n₁ + s₂²/n₂) for two independent samples. In OpenStax Example 10.1, girls (s = 0.866, n = 9) and boys (s = 1, n = 16): √(0.0833 + 0.0625) = 0.382. If you can assume equal variances, the pooled version uses one combined standard deviation instead.

Pooled or unpooled standard error for two proportions?

Unpooled, √(p₁q₁/n₁ + p₂q₂/n₂), for a confidence interval. For a z-test of equal proportions, use the pooled proportion: with 20 of 200 and 12 of 200, it is 0.08, the pooled standard error √(0.08 × 0.92 × (1/200 + 1/200)) = 0.0271, and z = 0.04 ÷ 0.0271 = 1.47 (OpenStax Example 10.8).

Where do the formulas come from?

From OpenStax Introductory Statistics 2e: section 7.1 (the central limit theorem for sample means), 8.3 (a population proportion), 10.1 (two population means) and 10.3 (two proportions). Their worked examples are part of this calculator’s automated tests.

How many observations do I need to cut the standard error in half?

Four times as many, because the standard error falls with √n. With a standard deviation of 15, the standard error of the mean is 3.00 for n = 25, 1.50 for n = 100 and 0.75 for n = 400. Halving the error again needs four times more data again, so the gains slow down quickly.

Is the standard error the same as the margin of error?

No. The margin of error is the standard error multiplied by a critical value, for example about 1.96 for a 95% interval from a normal distribution; for small samples, a t value is used instead and is larger. See the confidence interval calculator for the multiplication.

What the standard error measures

If you drew many samples of the same size from a population and computed the mean of each, those means would vary. The standard error is the standard deviation of that variation. For the mean it is s ÷ √n, where s is the sample standard deviation (with an n − 1 divisor) and n is the sample size. Other statistics have their own formulas, which the other tabs of the calculator provide for proportions and for differences. In every case a smaller standard error means a more precise estimate. The standard error says nothing about whether the sample was collected without bias; a biased sample can have a tiny standard error and still be wrong.

Worked example: eight measurements

Eight timings, in seconds: 12.1, 11.4, 13, 12.6, 11.8, 12.9, 12.2, 13.4. The mean is 12.425, and the sample standard deviation is 0.669. With n = 8, the standard error of the mean is 0.669 ÷ √8 = 0.237 seconds. A rough 95% interval of the mean is the mean ± 1.96 standard errors, 11.96 to 12.89 seconds; for a sample this small, a t critical value (2.365 for 7 degrees of freedom) makes it slightly wider, which the confidence interval calculator handles.

Common mistakes

Reporting the standard deviation when the question is about the precision of an average, or the reverse, is the most common one: the standard deviation does not shrink with more data, the standard error does. The proportion formula √(p(1 − p) ÷ n) needs p as a fraction between 0 and 1, not a percent, and is only a good basis for a normal-approximation interval when the expected counts of successes and failures are both reasonably large (a common rule of thumb is at least 5). For two groups, use the difference formulas rather than adding the two standard errors together, because variances add, not standard deviations. Finally, a standard error assumes random sampling; for a whole population, there is no sampling error.

Sources: OpenStax Introductory Statistics 2e, sections 7.1, 8.3, 10.1 and 10.3 as listed above; NIST/SEMATECH e-Handbook of Statistical Methods on the standard error of the mean. Related: standard deviation calculator, confidence interval calculator and sample size calculator.