Distributions Updated November 2026

Standard error

The standard error is the standard deviation of a sample statistic. For a mean it is s / sqrt(n); for a proportion it is sqrt(p(1 - p) / n). It shrinks as the sample grows and is the building block of margins of error, confidence intervals and test statistics.

Syntax

SE(mean) = s / sqrt(n);   SE(proportion) = sqrt( p * (1 - p) / n )

Part	What it is
`s`	the sample standard deviation
`p`	the sample proportion
`n`	the sample size

Examples

s = 12, n = 36
SE = 12 / sqrt(36) = 12 / 6

Result: 2

Excel: =STDEV.S(A1:A36) / SQRT(COUNT(A1:A36))

Proportion p = 0.5, n = 100
SE = sqrt(0.5 * 0.5 / 100) = sqrt(0.0025)

Result: 0.05

Same s = 12, n = 144
SE = 12 / 12

Result: 1 (four times the sample halves the SE)

Common mistakes

Reporting the standard deviation when the standard error is meant, or the other way round; sd describes the data, SE describes the precision of the estimate.
Dividing by n instead of sqrt(n).
Using the population formula with an estimated s and calling it exact; with s instead of sigma the interval and test use t, not z.

Practice Standard error

These drills check your answer on live data, free, no account needed.

Sampling distribution of the meanOrder values at an online shop have mean $197 and standard deviation $2. A random sample of 16 orders is taken. What is the probability that the sample mean is above $196.10? Round to 4 decimal places.
Margin of error for a mean (t)A random sample of 8 minutes customers waited on hold: 19, 14, 11, 9, 10, 6, 11, 15. What is the margin of error of a 90% confidence interval for the population mean, using the t distribution? Round to 2 decimal places.

Questions

What is the difference between standard error and standard deviation?

Standard deviation measures how spread out the individual values are. Standard error measures how much the sample mean would vary from sample to sample, and it gets smaller with more data while the sd does not.

How do I reduce the standard error?

Increase the sample size. Because SE = s / sqrt(n), halving it requires four times the data.

Is there a STDERR function in Excel?

No. Use =STDEV.S(range) / SQRT(COUNT(range)).

Central limit theoremThe central limit theorem makes sample means approximately normal with sd sigma / sqrt(n). Explanation, worked probability examples and the n = 30 rule.
Margin of errorMargin of error is the critical value times the standard error: z times sqrt(p(1 - p) / n) for a proportion, t times s over root n for a mean. Poll examples.
Confidence interval for a meanHow to calculate a confidence interval for a mean with x-bar plus or minus t times s over root n, a full 95% example, when to use z, and Excel CONFIDENCE.T.
Standard deviationThe standard deviation formula, sample versus population versions, a worked example with exact numbers and the Excel functions STDEV.S and STDEV.P.