Average Deviation Calculator

What the average deviation calculator does

The average deviation tells you how far, on average, your numbers sit from their centre. It is a plain, readable measure of spread: the typical distance from the middle, in the same units as your data. This calculator works it out two ways, measured from the mean and measured from the median, and shows both centres alongside.

Giving you both is the useful part, because the centre you measure from changes the answer, and one of the two is always the smaller. Below is how it works and how to read the pair.

How to use it

1. Enter your numbers in the box, separated by spaces or commas.
2. Press Calculate to see the mean, the median, and the average deviation from each, or Reset to clear it.

How the average deviation is worked out

Pick a centre, either the mean or the median. Then, for each number, measure how far it is from that centre, ignoring whether it is above or below, and average those distances:

Average deviation = the average of the distances from the centre, each taken as positive

Taking each distance as positive, its absolute value, is the key step. Without it the distances above the centre would cancel the ones below and the total would always come out to zero. Taking them all as positive lets them add up into a real measure of spread.

Measuring from the mean or the median

The calculator measures from both centres because they answer slightly different questions. The average deviation from the mean pairs naturally with the mean itself, the balance point of the data. The average deviation from the median has a special property worth knowing: of every possible centre you could choose, the median gives the smallest average deviation. Nothing beats it.

That makes the median version the more robust of the two. When your data has a few extreme values pulling the mean around, the distances measured from the median stay steadier, so the median-based average deviation is often the truer description of a typical gap.

Absolute distances, not squared

The average deviation is a close cousin of the standard deviation, with one difference that shapes everything. The standard deviation squares each distance before averaging, which makes large gaps count for a great deal more than small ones. The average deviation just takes each distance as it is.

That makes the average deviation easier to interpret, since it is literally the average distance from the centre, and steadier when a lone outlier is present, because it does not blow that outlier up by squaring it. The standard deviation is still the more common measure and the one that feeds into most other statistics, but for a plain, robust sense of spread, the average deviation is hard to beat.

A worked example: a set with an outlier

Take the five numbers 1, 2, 3, 4, 100. The mean is 22, dragged high by that 100, while the median is 3, sitting calmly in the middle.

Measured from the mean of 22, the distances average to 31.2. Measured from the median of 3, they average to just 20.2. The median-based figure is smaller, as it always is, and here it is the more honest one, because four of the five numbers really are clustered down near 3. The gap between the two averages is the outlier making itself felt.

Entering your data, and the rounding

You can separate your numbers with spaces or commas, and the order does not matter, since the average deviation depends only on distances from the centre. All four figures, the two centres and the two average deviations, are shown to three decimal places.

Questions people ask

What is the average deviation?

The average distance of your values from a centre, with each distance taken as positive. It is a straightforward measure of how spread out the data is, in the same units as the data.

Should I measure from the mean or the median?

The median version is always the smaller and is steadier against outliers, so it is often the better description of a typical gap. The mean version pairs naturally with the mean itself.

How is it different from the standard deviation?

The standard deviation squares each distance before averaging, so big gaps count for much more. The average deviation takes distances as they are, which makes it easier to read and less swayed by outliers.

Why not just average the distances without taking them as positive?

Because distances above and below the centre would cancel out and the total would always be zero. Taking each as positive is what turns them into a usable measure of spread.

References

A quick note on where the methods here come from. The average, or mean absolute, deviation and its place among the measures of spread 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 measures of spread.

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