Math · Statistics
Standard error calculator.
Standard error of a sample mean or a sample proportion. Choose a mode, enter the standard deviation or proportion and the sample size, and add a population size if you sampled a large share of it.
A sample of 36 with a standard deviation of 12 has a standard error of the mean of 12 / √36 = 2.00. For a proportion of 50% from 400 respondents the standard error is √(0.5 × 0.5 / 400) = 2.5 percentage points.
Your data
Your result
Fill the form and press Calculate.
The formulas
The standard error of a statistic is the standard deviation of its sampling distribution. For the sample mean it equals the standard deviation divided by the square root of the sample size: SE = s / √n. For a sample proportion the corresponding value is SE = √( p(1 − p) / n ), which is the denominator of the z statistic in NIST's proportion test.
Finite population correction. The formulas above assume an infinite population. When you sample without replacement a large share of a known, fixed population of size N, multiply by √( (N − n) / (N − 1) ). Wikipedia gives approximately 5% or more of the population as the point where the correction matters for an enumerative study.
Worked example
s = 12, n = 36: SE = 12 / 6 = 2. Quadruple the sample to n = 144 and SE = 12 / 12 = 1: halving the standard error takes four times the data. For a proportion of 50% with n = 400, SE = √(0.25 / 400) = 0.025, or 2.5 percentage points. With s = 10 and n = 100, SE = 1.00; if those 100 came from a population of 500, the correction is √(400 / 499) = 0.8953, giving an adjusted SE of 0.8953.
Standard error versus standard deviation
The standard deviation describes how spread out individual values are. The standard error describes how far a sample statistic, such as the mean, is likely to be from the population value; it shrinks as the sample grows, while the standard deviation does not. A confidence interval is the estimate plus or minus a critical value times the standard error (see the confidence interval calculator).
Sources, opened 8 October 2026: Wikipedia, Standard error (definition, standard error of the mean as standard deviation over the square root of n, finite population correction and the 5% guide); NIST/SEMATECH e-Handbook, 7.2.4 Does the proportion of defectives meet requirements? (the denominator √(p0(1−p0)/N) of the proportion test statistic).
Frequently asked questions
What is the standard error?
The standard error of a statistic is the standard deviation of its sampling distribution. For the sample mean it is the standard deviation divided by the square root of the sample size.
What is the difference between standard error and standard deviation?
Standard deviation measures the spread of individual values. Standard error measures how precisely a sample statistic estimates the population value, and it falls as the sample size grows.
How do I calculate the standard error of a proportion?
Take the square root of p times (1 - p) divided by n. For 50% from 400 respondents that is the square root of 0.25 / 400, which is 0.025 or 2.5 percentage points.
How do I make the standard error smaller?
Collect more data. Because it scales with the square root of n, you need four times the sample to halve it.
When do I use the finite population correction?
When you sample about 5% or more of a known, fixed population without replacement and want to describe that population. The correction factor is the square root of (N - n) / (N - 1).
How does standard error relate to margin of error?
A margin of error is a critical value times the standard error: 1.96 times the standard error for a 95% interval with the normal distribution.
Planning and learning aid
Educational calculator for independent random samples. Standard errors for other statistics, such as medians or differences between groups, use different formulas.