What this calculator does
Find out how much you can trust an average worked out from a sample. That's what the standard error of the mean (SEM) tells you. It measures how much a sample's average would change from one random sample to the next. The standard deviation describes the spread of single values. The standard error describes the spread of averages. Statisticians call it the standard deviation of the sampling distribution of the mean. It's the building block of confidence intervals and many hypothesis tests.
You can start from a standard deviation and sample size you already have. Or you can paste the raw data, and the calculator finds the mean and sample standard deviation first. Either way, it shows each step.
Where it's used
Why this math matters outside the classroom:
- Survey precision. A survey of 100 people with SD 15 has a standard error of 15 ÷ √100 = 1.5 for its mean.
- Lab averages. A student averaging 25 measurements with SD 0.5 mm reports the mean ± 0.1 mm, its standard error.
- Bigger samples. Quadrupling a sample from 50 to 200 halves the standard error, because it shrinks with the square root of n.
- Error bars. Scientists draw error bars one standard error long on bar charts to show how precisely each mean is known.
- Class averages. A tutor's 36 students average 78 with SD 12, so the mean's standard error is 12 ÷ 6 = 2 points.
How to use it
- Choose whether you're starting from a standard deviation and sample size, or from raw data.
- For the first option, enter the sample standard deviation s and the number of values n (at least 2).
- For raw data, paste at least two numbers separated by commas, spaces or new lines.
- Read the standard error. Open the working to see the square root of n and the division.
What the inputs mean
- Sample standard deviation (s)
- Usually the sample standard deviation, which divides by n − 1. If you know the true standard deviation of the whole population (σ, sigma), you can enter that instead.
- Sample size (n)
- The number of values that went into the average. It isn't the number of groups or the size of the population.
- Data set
- The calculator uses the sample standard deviation (dividing by n − 1). That's the standard choice when you estimate the standard error from data.
Formula and method
SE = s ÷ √n s = √( Σ (xi − x̄)² ÷ (n − 1) )
SE is the standard error, s is the sample standard deviation and n is the number of values. √ means square root. In the second line, each xi is one value, x̄ ("x-bar") is their average and Σ means "add up".
Averaging n independent values cancels out some of their random ups and downs. The variance (spread, squared) of the average is the variance of a single value divided by n. So its standard deviation, the standard error, is s divided by the square root of n.
Because of the square root, a sample four times as big only halves the standard error. Each extra bit of precision costs more as the sample grows.
Worked example
A sample of 36 delivery times has a standard deviation of 12 minutes.
- Square root of the sample size: √36 = 6.
- Standard error: 12 ÷ 6 = 2.
Result: The standard error of the mean is 2 minutes. Starting from the raw data 2, 4, 4, 4, 5, 5, 7, 9 instead gives s ≈ 2.1381 and SE ≈ 2.1381 ÷ √8 ≈ 0.7559.
Understanding the result
A small standard error means the sample average is a precise estimate of the true population average. A large one means another sample could easily give a clearly different average.
If the averages follow a rough bell curve, the sample average ± about 2 standard errors gives a rough 95% confidence interval. The confidence interval calculator does this exactly, using the t distribution for small samples.
Don't mix up the two measures. Report the standard deviation to show how varied single values are. Report the standard error to show how closely you've pinned down the average.
Assumptions and limitations
- Values are independent and picked at random from the population.
- The population is much larger than the sample. If you sample more than about 5% of a limited population, a finite population correction would make the standard error smaller.
- This is the standard error of an average. Proportions, differences between averages and regression slopes and intercepts have their own standard error formulas.
- Grouped (clustered), paired or otherwise linked data needs a different calculation. This formula would understate the uncertainty.
- Raw data is limited to 10,000 values.
Frequently asked questions
What is the difference between standard deviation and standard error?
Standard deviation measures how spread out single values are around their average. Standard error measures how much the average itself would change across repeated samples. For n of 2 or more, the standard error is always smaller than the standard deviation. It shrinks as the sample grows, while the standard deviation doesn't.
Why does the standard error get smaller with a bigger sample?
Larger samples average out more of the random variation. So their averages bunch more tightly around the true average. The standard error falls in step with 1 ÷ √n.
Should I use n or n − 1?
The standard error divides by √n. The n − 1 belongs inside the sample standard deviation, which is worked out first. In raw-data mode, the calculator does both for you.
Can I use the standard error to build a confidence interval?
Yes. A confidence interval for the average is the sample average plus or minus a critical value times the standard error. Use the confidence interval calculator for the full calculation.
How this calculator is checked
Method: Standard error of the sample mean, SE = s/√n, from sampling theory; sample standard deviation with the n − 1 divisor. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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