Sample Mean Calculator

What the sample mean calculator does

The sample mean is the average of a set of measurements taken from a larger group. This calculator works it out from your data and shows the sum and the count behind it. The arithmetic is the same as any average, but in statistics this particular average has a name and a job, and both are worth knowing.

It is written as x-bar, and it stands in for the average of a whole population you cannot measure in full. Below is how it is worked out and what it is really telling you.

How to use it

1. Enter your numbers in the box, separated by commas, spaces, or line breaks.
2. Press Calculate to see the count, the sum, and the sample mean, or Reset to clear it.

How the sample mean is worked out

It is the plain average. Add up all your values and divide by how many there are:

Sample mean = sum of the values ÷ how many values there are

The calculator shows you that sum and that count next to the answer, so you can check that every value was read and nothing was doubled. On the arithmetic alone, the sample mean and the ordinary mean are identical numbers. What sets the sample mean apart is not the sum, it is what the result is used for.

The sample mean as an estimate

Here is the idea behind the name. Often you cannot measure everyone or everything you care about, the whole population, so you take a sample and measure that instead. The average of your sample, the sample mean, becomes your best estimate of the average of the whole population, which is written as the Greek letter mu and usually stays unknown.

Because it is an estimate, it comes with a little wobble. Take a different sample and you would get a slightly different sample mean, and the value dances around the true population average from sample to sample. The good news is that the larger your sample, the more tightly that dance settles around the truth, which is why a bigger sample gives a more trustworthy estimate. That spread of the sample mean is measured by the standard error, if you want to pin down how precise your estimate is.

A worked example: five measurements

Suppose you measure five items pulled from a much larger batch and get 4, 8, 6, 5, and 7.

They add up to 30, and there are 5 of them, so the sample mean is 30 ÷ 5 = 6. That 6 is your best single guess at the average of the entire batch, not just the five you happened to measure. If you pulled another five items, you would likely land near 6 but not exactly on it, and averaging more items each time would tighten that estimate.

Entering your data, and the rounding

You can separate your numbers with commas, spaces, or line breaks, in any mix, and the order does not matter. The calculator reads the numbers out of whatever you type, so a stray space will not trip it. The sum and the sample mean are shown to three decimal places with trailing zeros trimmed, which is plenty for most work.

Questions people ask

What is the sample mean?

The average of a sample of data, written as x-bar. It is calculated the same way as any average, by adding the values and dividing by the count, and it is used to estimate the average of a larger population.

What is the difference between the sample mean and the population mean?

The sample mean, x-bar, is the average of the data you actually collected. The population mean, mu, is the average of the whole group, which is usually unknown. The sample mean is your best estimate of it.

Is the sample mean the same as the arithmetic mean?

The calculation is identical. The name simply signals that your numbers are a sample standing in for a larger group, which is how the result is meant to be used.

How do I make the estimate more accurate?

Use a larger sample. The more values you measure, the more tightly the sample mean settles around the true population average, so bigger samples give more reliable estimates.

References

A quick note on where the methods here come from. The sample mean, its role as an estimate of the population mean, and the way its accuracy improves with sample size 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 that covers sampling and the sample mean.

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