Standard Error Calculator
Calculate the Standard Error of the Mean (SEM) from standard deviation and sample size. Includes margin of error for 90%, 95%, and 99% confidence levels.
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How to use this calculator
Standard Error of the Mean measures how precisely the sample mean estimates the population mean. Larger samples (bigger n) give smaller SE and more precise estimates.
- 1
Enter the standard deviation of your sample (or known population standard deviation).
- 2
Enter the sample size n.
- 3
The Standard Error and the margin of error at 90%, 95%, and 99% confidence are calculated.
Frequently asked questions
What is calculated, and how are the margins of error produced?
SE = SD ÷ √n, with n rounded down to a whole number. The three margin rows multiply SE by fixed critical values: 1.645 (90%), 1.960 (95%) and 2.576 (99%). For SD = 15 and n = 30, SE is 2.738613 and the 95% margin is ±5.367681.
Which standard deviation should I enter, s or σ?
The calculator treats whatever you enter the same way. If σ is a known population value, SE is exact. If you have only a sample standard deviation s, computed with the n − 1 divisor, the result is an estimated standard error. The variance calculator’s sample setting gives s.
Are the margins of error right for small samples?
They use normal-curve multipliers, not t values, so they are too narrow when n is small and the SD is estimated from the sample. For SD = 12.5 and n = 8, the tool gives a 95% margin of ±8.662058, whereas the t multiplier for 7 degrees of freedom is 2.365 and would give about ±10.45. The discrepancy fades as n grows.
How does the standard error change when I collect more data?
It falls with the square root of n, so you need four times the data to halve it. With SD = 15, SE is 2.738613 at n = 30, 1.369306 at n = 120 and 0.684653 at n = 480. Rearranging to n = (SD ÷ SE)² gives the sample needed for a target: SD 10 and a target SE of 1 need n = 100.
What are the input limits?
The standard deviation must be zero or positive (zero gives SE = 0), and n must be at least 1. Non-integer n is rounded down. With n = 1 the SE equals the SD, but a sample standard deviation cannot exist for a single observation, so only use n = 1 with a known σ.
Is standard error the same as standard deviation?
No. SD describes how far individual observations spread around their mean. SE describes how far a sample mean is likely to fall from the true mean. Collecting more data does not shrink the SD, which reflects the real variability, but it does shrink the SE. Error bars labelled SE are therefore narrower than SD bars and say nothing about the spread of the data.
What does the calculator not cover?
Only the standard error of a mean. It does not do the standard error of a proportion, a median, a regression slope or a difference between two means, and it applies no finite-population correction. It also assumes independent observations; clustered or serially correlated data have a larger true SE than SD ÷ √n suggests.
Standard error of the mean: SD ÷ √n with margin-of-error rows
Precision of an average, not spread of the data
The standard error of the mean measures how much a sample average would bounce around if you repeated the study with fresh samples of the same size. It equals the standard deviation divided by the square root of the sample size. The calculator shows SE, the square root of n, and margins of error at 90%, 95% and 99% found by multiplying the SE by 1.645, 1.960 and 2.576. Add and subtract a margin from your sample mean to obtain an interval.
| n | Standard error | 95% margin |
|---|---|---|
| 30 | 2.738613 | ±5.367681 |
| 120 | 1.369306 | ±2.683841 |
| 480 | 0.684653 | ±1.34192 |
Worked example: quadrupling the sample
With SD = 15, a sample of 30 yields SE = 2.738613. Multiplying n by four to 120 halves the SE to 1.369306, and four times again to 480 halves it again to 0.684653. Each halving costs four times the data, so precision gets progressively more expensive. The 95% margin shrinks from ±5.367681 to ±2.683841 to ±1.34192 along the way.
Worked example: planning for a target precision
Suppose past data suggest SD ≈ 10 and you want an SE of 1. Solving n = (SD ÷ SE)² gives 100, and the calculator confirms SE = 1 and a 95% margin of ±1.96 for n = 100. Cutting the SE to 0.5 would need n = 400. With n = 25 the SE is 2 and the 95% margin is ±3.92. Planning this way works for means only, and the SD must come from a pilot or earlier study.
SE versus SD in reports
Charts sometimes show error bars without saying what they represent. SD bars display the scatter of individual values; SE bars display uncertainty about the mean and are narrower by a factor of √n. A reader seeing SE bars can wrongly infer that the data are tightly clustered. State which one you are plotting, or show a confidence interval instead.
Small samples and other caveats
The margin rows use normal multipliers. When the SD is estimated from a small sample, the true t-based multiplier is larger: for n = 8 and SD = 12.5, the 95% margin printed is ±8.662058, while a t multiplier of 2.365 would give about ±10.45. The SE formula also assumes independent observations, and it does not apply to proportions or medians.
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Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →