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Standard Error Calculator

Find the standard error of a sample mean.

Result

2.0000

How it is calculated

The standard error of the mean is SEM = SD ÷ √n, where SD is the standard deviation of your sample (in the same unit as the data) and n is the number of observations. The result is in the same unit as SD and is shown to four decimal places. The tool returns nothing if SD is negative or if n is below 1. It does not check that n is a whole number, so enter the real count of observations. It only divides: it does not compute SD from raw data, and it does not apply a t or z multiplier, so it gives no confidence interval by itself.

Worked example

A sample of 36 measurements has a standard deviation of 12. Take the square root of the sample size: √36 = 6. Divide the standard deviation by it: 12 ÷ 6 = 2.0000. The sample mean therefore has a standard error of 2 units. With a sample four times larger (n = 144), the root would be 12 and the SEM would fall to 1.

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Common mistakes

  • Reporting SEM when the goal is to describe how spread out the individual values are. That is the job of the standard deviation.
  • Entering the variance instead of the standard deviation. The variance must be square-rooted first.
  • Entering n − 1 (degrees of freedom) instead of the sample size n.
  • Expecting SEM to halve when the sample doubles. It shrinks by a factor of √2, so you need four times the data to halve it.
  • Treating SEM as a 95% interval. A confidence interval needs the SEM multiplied by a t or z value, roughly 2 for large samples.

Sources

About this calculator

The standard error of the mean is calculated as SEM = SD / √n, where SD is the sample standard deviation and n is the sample size. It's easy to confuse with standard deviation, but the two answer different questions: standard deviation describes how spread out individual data points are, while SEM estimates how much the sample mean itself would vary if you repeated the sampling — and it shrinks as the sample gets larger, since bigger samples give more stable estimates of the true population mean.

Researchers report SEM to build confidence intervals (mean ± t·SEM or ± z·SEM) and to draw error bars on charts in biology, psychology, medicine, and social science papers. Anywhere a study reports "mean ± SEM" instead of "mean ± SD," it's using this formula to convey precision of the estimate rather than variability of the raw data.

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Comments (1)

  • Editorial team

    Turn the formula around to plan a study: the sample size needed for a target SEM is (SD divided by that SEM) squared. With SD 12 and a target of 0.5, you need 576 observations. Take the SD from a pilot sample, since the true value is unknown. For a confidence interval on a small sample, multiply the SEM by a t value, not 2: with n = 10 it is about 2.26. How big is your sample, and what precision are you aiming for?

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