Standard Error Calculator (SEM, Proportion, Confidence Interval)
Calculate the standard error of the mean (SEM) or the standard error of a proportion in seconds. Choose between entering raw data, summary statistics (sample SD and n), or proportion inputs, add an optional confidence level, and get the result with a full step-by-step worked solution. The result updates as you type.
Formula
Worked example
Ten measurements: 12, 15, 14, 10, 13, 16, 11, 14, 15, 12. Mean = 13.2, sample SD = 1.932, n = 10. SE = 1.932 / sqrt(10) = 0.611. For a 95% CI with df = 9, the t critical value is 2.262, giving a margin of error of 2.262 x 0.611 = 1.382. The 95% CI is (11.818, 14.582).
What is standard error?
The standard error of the mean (SEM) measures how much the sample mean is expected to vary from one random sample to another. If you drew many samples from the same population and computed the mean of each, those means would form a distribution - the standard error is the standard deviation of that sampling distribution. A smaller SE means your sample mean is a more precise estimate of the population mean.
Standard error formula
For the standard error of the mean, the formula is SE = s / sqrt(n), where s is the sample standard deviation (calculated with n-1 in the denominator, known as Bessel's correction) and n is the sample size. For a proportion, the formula is SE = sqrt(p(1-p)/n), where p is the observed sample proportion. Both formulas share the same inverse-square-root relationship with n: doubling n reduces SE by a factor of sqrt(2) (about 29%), and quadrupling n cuts SE in half.
Standard error vs. standard deviation
These two measures are often confused but describe different things. The standard deviation (SD) describes how spread out the individual data points are within your sample. The standard error (SE) describes how spread out the sample means would be across many repeated samples. As sample size grows, the SD stays roughly constant (it estimates a population parameter) while the SE shrinks toward zero (the estimate becomes more precise). A large SD does not imply a large SE if you have a large sample.
Confidence intervals and margin of error
The SE is the building block for a confidence interval. Multiply the SE by the appropriate critical value (a t value for small samples with n under 30, or a z value for larger ones) to get the margin of error. The confidence interval then runs from the sample mean minus the margin of error to the sample mean plus the margin of error. A 95% CI, for example, will contain the true population mean in 95% of repeated experiments - not a statement about any single interval.
How sample size affects the standard error (SD = 1)
| Sample size (n) | SE (SD = 1) | Relative precision |
|---|---|---|
| 5 | 0.447 | Low |
| 10 | 0.316 | Low |
| 20 | 0.224 | Moderate |
| 30 | 0.183 | Moderate |
| 50 | 0.141 | Good |
| 100 | 0.100 | Good |
| 200 | 0.071 | High |
| 500 | 0.045 | High |
For a fixed sample standard deviation of 1, the SE shrinks as n increases. To halve the SE you must quadruple n.
Frequently asked questions
What is the difference between standard error and standard deviation?
Standard deviation measures how spread out individual observations are within one sample. Standard error measures how precisely the sample mean estimates the population mean - it is the standard deviation of the sampling distribution of the mean. SE = SD / sqrt(n), so a larger sample always produces a smaller SE even if the raw data are just as variable.
When should I use a t critical value instead of a z?
Use a t critical value when your sample size is below 30 and you are estimating a mean (not a proportion). The t-distribution has heavier tails than the normal distribution, reflecting the extra uncertainty from estimating the population SD with a small sample. As n grows the t-distribution converges to z, so for n of 30 or more the difference is negligible.
How do I calculate the standard error of a proportion?
Use the formula SE = sqrt(p(1-p)/n), where p is the sample proportion (between 0 and 1) and n is the total sample size. For example, if 42 out of 100 respondents agreed with a statement, p = 0.42 and n = 100, giving SE = sqrt(0.42 x 0.58 / 100) = sqrt(0.002436) = 0.0494. Multiply by 1.96 (the 95% z critical value) for the margin of error: about 0.097, or roughly plus or minus 10 percentage points.
Does a larger sample always reduce the standard error?
Yes, as long as the population SD is fixed. Because SE = SD / sqrt(n), increasing n always decreases SE. However, the returns diminish quickly: you need to quadruple the sample size to halve the SE. Going from n = 25 to n = 100 cuts SE in half; going from 100 to 400 cuts it in half again.
What is a "good" standard error?
There is no universal threshold. The SE is meaningful in context: a good SE is one small enough that your confidence interval is narrow enough to answer your research question. Compare the SE to the effect size or difference you are trying to detect. If the margin of error is larger than the effect you care about, your sample is too small to draw firm conclusions.
Can I compute a standard error from just n and the SD without the raw data?
Yes. The SE of the mean depends only on the sample SD and n, not on the individual values. Use the "Summary stats" mode in this calculator: enter s (the sample standard deviation) and n, and the SE is computed as s / sqrt(n). You do need the raw data only if you want the calculator to compute the SD for you.