Standard Error Calculator

What the standard error calculator does

The standard error tells you how precisely your data has pinned down the average, rather than how spread out the data itself is. This calculator works it out from your numbers, and shows the sample standard deviation and the sample size it used along the way.

You paste in a data set, and it does the rest, finding the standard deviation first and then the standard error from it. Below is how that works, and the difference between the two, which is easy to mix up.

How to use it

1. Enter your numbers in the box, separated by commas, spaces, or new lines.
2. Press Calculate for the sample size, standard deviation, and standard error, or Reset to clear it.

How the standard error is worked out

It is two stages. First the calculator finds the sample standard deviation of your data, the spread of the numbers around their mean, dividing by one less than the count. Then it divides that by the square root of the sample size:

Standard error = sample standard deviation ÷ √n

The square root of n in the bottom is the important part. It means that as your sample grows, the standard error shrinks, but slowly. To halve it you need four times the data, which is why chasing a very precise estimate gets expensive.

Standard deviation versus standard error

These two get confused constantly, so here is the clean split. The standard deviation describes your data: how far the individual values spread around their mean. It does not shrink when you collect more data, because more data does not make the values themselves any less varied.

The standard error describes your estimate of the mean: how close the average of your sample is likely to be to the true average of the whole population. It does shrink as you collect more data, because a bigger sample locates the average more tightly. So if you are describing how varied your data is, quote the standard deviation. If you are saying how sure you are about the average, quote the standard error. The standard error is also the engine behind confidence intervals and many hypothesis tests, which is where it earns most of its keep.

A worked example: eight numbers

Take the eight numbers 2, 4, 4, 4, 5, 5, 7, 9. Their sample standard deviation, dividing by one less than the count, works out to about 2.14.

The standard error is that divided by the square root of the sample size: 2.14 ÷ √8 = 2.14 ÷ 2.83 = about 0.76. So while the data spreads about 2.14 units around its mean, the mean itself is pinned down to within roughly 0.76, and gathering more numbers would tighten that further.

Entering your data, and the rounding

You can separate your numbers with commas, spaces, or new lines, in any mix, and the order does not matter. Since the standard error is built on the sample standard deviation, you need at least two numbers for it to be defined. The standard deviation and standard error are shown to several decimal places with trailing zeros trimmed.

Questions people ask

What is the standard error?

The standard error of the mean measures how precisely your sample has estimated the true average, found by dividing the sample standard deviation by the square root of the sample size.

What is the difference between standard error and standard deviation?

The standard deviation measures how spread out your data is. The standard error measures how close your sample mean is likely to be to the true mean. The first stays put as data grows, the second shrinks.

Why does the standard error get smaller with more data?

Because a larger sample pins down the average more tightly. It falls with the square root of the sample size, so four times the data halves the standard error.

What is the standard error used for?

It is the basis of confidence intervals and many hypothesis tests, where it sets how wide the margin around an estimate should be.

References

A quick note on where the methods here come from. The standard error of the mean, and its role in confidence intervals and testing, are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering the standard error and how it differs from the standard deviation.

1. NIST/SEMATECH e-Handbook of Statistical Methods (standard error and sampling). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (sampling and the standard error). https://openstax.org/details/books/introductory-statistics-2e