Standard Error of the Mean Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/6/2026
The standard error of the mean (SEM) is calculated with standard deviation ÷ square root of the sample size. For a standard deviation of 10 over a sample of 25 observations, the SEM is 2.
Explanation
The standard error of the mean (often abbreviated SEM) measures how precisely a sample's mean estimates the true mean of the population it was drawn from. It shouldn't be confused with the standard deviation, which measures the spread of individual values around the mean: the SEM instead measures the variability that would be observed in the mean itself if the sampling were repeated several times. The larger the sample, the lower the SEM (it decreases with the square root of the sample size): this reflects the fact that a larger sample gives a more precise, more stable estimate of the population mean, even though the standard deviation of the individual data stays the same. This is why the SEM is commonly used to build error bars on scientific charts and confidence intervals, while the standard deviation remains the reference indicator for describing the spread of the raw data itself.
Example: a standard deviation of 10, a sample of 25 observations
Inputs
Standard deviation: 10. Sample size: 25.
Calculation
Standard error = 10 ÷ √25 = 10 ÷ 5 = 2.
Result
The SEM of this sample is 2.
Frequently asked questions
What is the difference between standard deviation and standard error of the mean?
The standard deviation describes the spread of individual values around the mean of a given sample, and doesn't necessarily decrease as the sample size grows. The SEM, on the other hand, measures the precision of the estimate of the population mean and systematically decreases as the sample size increases, since a larger sample gives a more reliable estimate of the true mean.
Why does the SEM decrease with the square root of n, rather than directly with n?
This is a mathematical consequence of the central limit theorem, which shows that the variance of the sampling mean equals the population variance divided by n. Since the standard deviation is the square root of the variance, the standard error of the mean is therefore the standard deviation divided by the square root of n, not by n directly. In practice, quadrupling the sample size only divides the SEM by 2, not by 4.
When should I use the SEM rather than the standard deviation on a chart?
Use the standard deviation to show the actual spread of individual data points (how much observations vary from each other). Use the SEM to show the precision of an estimated mean, for example to compare whether two group means are statistically distinct. Confusing the two is a common mistake that can make a chart misleading about what it actually represents.