What is the standard error?

The standard error (SE) measures how much your sample statistic (such as the mean or a proportion) is likely to vary from sample to sample. It equals the standard deviation divided by the square root of the sample size for means, or the square root of p(1-p)/n for proportions. A smaller SE means the estimate is more precise.

How is standard error different from standard deviation?

Standard deviation describes the spread of individual data points around the sample mean. Standard error describes how precisely the sample mean estimates the true population mean. SE shrinks as sample size grows (SE = SD divided by sqrt(n)), while SD is a property of the data set itself and does not depend on n.

How do I use SE to build a confidence interval?

A 95% confidence interval for a mean is approximately the sample mean plus or minus 1.96 times the standard error. This means that if you repeated the sampling process many times, about 95% of the resulting intervals would contain the true population mean.

Standard Error Calculator

The standard error (SE) tells you how much your sample estimate is likely to differ from the true population value if you were to repeat the sampling process. When you calculate a mean or a proportion from a random sample, that number will be slightly different each time you take a new sample due to natural variation. The standard error quantifies how large that expected variation is. For the mean, SE equals the sample standard deviation divided by the square root of the sample size. This means that as you collect more data, the SE shrinks, and your estimate becomes more precise. Quadrupling the sample size halves the standard error. For a proportion (such as a poll result), SE equals the square root of p times (1 minus p) divided by n, where p is the proportion and n is the sample size. The SE is also the building block for confidence intervals: a 95% confidence interval for a mean is approximately the estimate plus or minus 1.96 times the SE. The calculator offers two modes. In mean mode you enter the standard deviation and sample size to get the SE and the 95% margin of error. In proportion mode you enter a percentage and a sample size. Both modes return the standard error, the 95% margin of error (SE times 1.96), and the half-width of the 95% confidence interval. Useful for students, researchers, and analysts checking the precision of a sample-based estimate before reporting results or making decisions.

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Mode

Standard deviation (SD)

Proportion (%)

Sample size (n)

1.0000

standard error

95% margin of error1.9600

95% CI half-width+/-1.96

Sample size (n)100

How it works

In mean mode, SE = SD / sqrt(n). In proportion mode, SE = sqrt(p x (1-p) / n) where p is entered as a decimal. The 95% margin of error is 1.96 times SE (using the z-score for 95% confidence). The confidence interval half-width is the same value expressed as plus or minus SE times 1.96. If n is zero or SD is negative, the calculator returns N/A.

Worked example

In mean mode with SD = 10 and n = 100: SE = 10 / sqrt(100) = 10 / 10 = 1.0000. The 95% margin of error is 1.96 x 1 = 1.96. In proportion mode with p = 50% and n = 100: SE = sqrt(0.5 x 0.5 / 100) = sqrt(0.0025) = 0.0500 (5.00%). These match the default values pre-filled above.

Related calculators

Margin of Error Calculator: calculate the full survey margin of error with confidence level selection.
P-value Calculator: convert a z-score or t-score to a p-value for hypothesis testing.
Correlation Coefficient Calculator: measure the linear association between two variables.
Percent Error Calculator: measure deviation from an accepted reference value.

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