Mean Absolute Deviation Calculator

What the mean absolute deviation calculator does

The mean absolute deviation, or MAD, is the average distance of your numbers from their mean. It answers a plain question in a plain way: on average, how far does a value fall from the middle? This calculator works it out from your data and shows the mean and the count it used.

It is one of the most readable measures of spread, because the answer is a real distance in your own units, not a squared or scaled figure. Below is how it works and how it compares to the standard deviation.

How to use it

1. Enter your numbers in the box, separated by commas, spaces, or new lines.
2. Press Calculate for the MAD, the mean, and the count, or Reset to clear it.

How the mean absolute deviation is worked out

Three steps. Find the mean of your numbers. For each number, measure how far it is from the mean, taking that distance as positive. Then average those distances:

MAD = the average of the distances from the mean, each taken as positive

Taking each distance as positive, its absolute value, is what makes the sum add up instead of cancelling to zero. The result is simply the typical distance between a value and the mean, which is about as direct as a measure of spread can be.

MAD versus standard deviation

The mean absolute deviation and the standard deviation are measuring the same thing, the spread of data around its mean, and they differ in one step. The standard deviation squares each distance before averaging and then takes a square root at the end. The MAD skips the squaring and just averages the distances.

That one difference has two effects. It makes the MAD easier to interpret, since it is a plain average distance rather than a squared-then-unsquared quantity. And it makes the MAD steadier when your data has outliers, because squaring is what gives a far-flung value its outsized pull on the standard deviation. The trade-off is that the standard deviation is the established measure and the one that plugs into most other statistics, so the MAD tends to be chosen when clarity and robustness matter more than convention.

A worked example: eight numbers

Take the eight numbers 2, 4, 4, 4, 5, 5, 7, 9. Their mean is 40 ÷ 8 = 5. The distances from 5, all taken as positive, are 3, 1, 1, 1, 0, 0, 2, and 4, which add up to 12.

Averaging those gives a MAD of 12 ÷ 8 = 1.5. So a typical value in this set sits one and a half units from the mean. For comparison, the standard deviation of the same numbers is 2, a little larger, because its squaring gives the 9 at the top end more weight than the MAD does.

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 MAD depends only on distances from the mean. The MAD and the mean are shown to four decimal places with trailing zeros trimmed.

Questions people ask

What is the mean absolute deviation?

The average distance of your values from the mean, with each distance taken as positive. It is a direct, easy-to-read measure of how spread out the data is.

How is MAD different from standard deviation?

The standard deviation squares each distance before averaging, so outliers count for more. The MAD averages the plain distances, which makes it easier to interpret and less swayed by extreme values.

Why is the MAD usually smaller than the standard deviation?

Because squaring in the standard deviation gives large gaps extra weight, which pulls its value up. Without that squaring, the MAD comes out a little lower for most data.

When should I use the MAD?

When you want a spread figure that is easy to explain and steady against outliers. The standard deviation remains the more common choice where convention or further calculation calls for it.

References

A quick note on where the methods here come from. The 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