What this calculator does
The standard error is the standard deviation divided by the square root of the sample size. With a standard deviation of 12 and 36 observations, the standard error is 2.
The square root is what governs everything about sampling. Quadrupling the sample halves the standard error: going from 9 observations to 36 takes it from 4 to 2, and from 100 to 400 takes it from 1.2 to 0.6. Precision improves, but at a steadily worsening price, which is why large surveys reach a point where more respondents stop being worth the cost.
The formula
The sample standard deviation is divided by the square root of the sample size. Where a finite population size is supplied, a correction factor of √((N−n)/(N−1)) is applied, which reduces the standard error to reflect that sampling a large share of a small population leaves less uncertainty. With the population left at zero, no correction is applied.
| Term | Meaning |
|---|---|
| Standard error | The expected variability of a sample mean around the true population mean. |
| Standard deviation | The variability of individual observations, which is a different thing. |
| Finite population correction | A reduction applied when the sample is a meaningful share of the whole population. |
| Square root law | Precision improves with the square root of sample size, not in proportion to it. |
The inputs explained
| Field | What to enter |
|---|---|
| Sample standard deviation | Sample standard deviation. |
| Sample size | Sample size. |
| Population size (0 = treat as infinite) | Population size. Leave at zero to treat the population as effectively infinite, which is right unless your sample is more than about 5% of it. |
When to use it
Building a confidence interval
The standard error is the unit a confidence interval is measured in, typically 1.96 of them either side for 95%.
Planning a sample size
Working backwards from a target standard error gives the sample size needed, and shows how quickly the cost rises.
Sampling a small population
Auditing 36 of 100 records is very different from 36 of a million, and the correction accounts for that.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the finite population correction change things?
The same sample drawn from populations of different sizes.
| Population size | Standard error | Without finite-population correction | Finite population correction factor |
|---|---|---|---|
| N = 100 | 1.608 | 2.000 | 0.8040 |
| N = 500 | 1.929 | 2.000 | 0.9643 |
| N = 1,000 | 1.965 | 2.000 | 0.9823 |
Questions
What is the difference between standard deviation and standard error?
The standard deviation describes how spread out individual observations are. The standard error describes how much a sample mean would vary between repeated samples. The standard deviation does not shrink with more data; the standard error does.
Why does the standard error use the square root of n?
Because the variance of a mean is the variance of an observation divided by n, and the standard error is the square root of that. The practical effect is that halving the standard error takes four times the data, which sets a hard economic limit on survey precision.
When do I need the finite population correction?
When your sample is more than about 5% of the population. Below that the factor is close to 1 and makes almost no difference. Above it the correction matters, and at the extreme of sampling the entire population the standard error correctly falls to zero.
Can I calculate standard error without the standard deviation?
Not directly, but it can be estimated. If only a range is available, the range rule of thumb gives an approximate standard deviation to work from. For a proportion rather than a mean, the standard error has its own formula using p and n instead.
For the margin of error on a survey proportion, see the margin of error calculator. For the interval it feeds, see the confidence interval calculator.