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Mean Absolute Deviation Calculator

Paste or type a list and this shows the whole calculation rather than just the answer: the centre, every absolute deviation, the total, and the step that turns those deviations into the statistic. That matters more here than on most pages, because MAD is written for two different statistics. The mean absolute deviation is the mean of the absolute deviations, usually measured from the arithmetic mean; the median absolute deviation is the median of them, measured from the median. On 1, 2, 100 the first is 43.777778 and the second is 1. This page treats them as three separate choices, mean about the mean, mean about the median and median about the median, and labels every result with its full name instead of the abbreviation. Up to 10,000 values at a time, nothing uploaded, and the copied output carries every deviation as tab separated columns.

8 valuesCentre: meanMean absolute deviation
values           2  4  4  4  5  5  7  9        n = 8
mean             40 / 8  =  5

|x - 5|          3  1  1  1  0  0  2  4        sum = 12

mean absolute deviation about the mean   12 / 8  =  1.5
n
8
Centre (mean)
5
Sum of |x - 5|
12
Mean absolute deviation
1.5
Which statistic

Mean absolute deviation about the mean. Add every |x - mean|, then divide by n. The statistic most pages mean by mean absolute deviation, and the one taught alongside the standard deviation.

Mean absolute deviation about the mean of 8 values is 1.5. That is 12 / 8.

Commas, spaces, tabs, semicolons and new lines all separate values, so a column copied from a spreadsheet pastes straight in, and up to 10,000 values are read at a time. Figures on screen are rounded to six decimal places; the arithmetic itself keeps full double precision. Nothing you type leaves this browser.

Worked examples

  • 1, 2, 3centre is the mean, 2
    Mean absolute deviation 0.666667, the sum 2 over n 3.
  • A single valuen = 1
    Mean absolute deviation 0: the only value is the centre, so every deviation is zero.
  • 1, 2, 100 about the medianmedian absolute deviation
    Median absolute deviation 1, scaled 1.4826, where the mean absolute deviation about the mean is 43.777778.
Two statistics, one abbreviation. MAD is written for both the mean of the absolute deviations and the median of the absolute deviations, and they are different numbers: on 1, 2, 100 the mean absolute deviation about the mean is 43.777778 and the median absolute deviation about the median is 1. One wild value drags the first a long way and moves the second hardly at all. This page computes whichever you pick and labels it by its full name rather than by the abbreviation.
Accuracy. Exact arithmetic: the mean of the absolute deviations from the centre you choose, with every deviation shown so you can add them up yourself. Mean deviation about the mean and median absolute deviation about the median are two different statistics and are labelled separately here, never merged. The 1.4826 normal-consistency scaling is an approximation and is shown apart from the raw value.

Common questions

What is the mean absolute deviation?
It is the average distance between the values and their centre. Take every value, measure how far it sits from the centre, drop the minus signs, and average those distances. On 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the absolute deviations are 3, 1, 1, 1, 0, 0, 2 and 4, they total 12, and 12 divided by 8 is 1.5. Nothing is squared along the way, so the result is in the same units as the data and one far-off value pulls it less than it pulls a standard deviation.
Does MAD mean the mean absolute deviation or the median absolute deviation?
Both, depending on the textbook, which is why this page never prints the abbreviation on its own. The mean absolute deviation is the mean of the absolute deviations, normally measured from the arithmetic mean. The median absolute deviation is the median of the absolute deviations, measured from the median. They are different numbers on the same data: for 1, 2, 100 the mean absolute deviation about the mean is 43.777778 and the median absolute deviation about the median is 1. Pick the one you need with the control under the readout: the working and the copied text carry its full name, and the chip and the readout label say whether the answer is a mean or a median of the deviations.
How do I work out the mean absolute deviation by hand?
Four steps, and the page prints all four so you can follow them. Add the values and divide by n to get the mean. Subtract the mean from each value and drop the minus signs. Add those absolute deviations up. Divide that total by n. For 1, 2, 3 the mean is 2, the deviations are 1, 0 and 1, the total is 2, and 2 divided by 3 is 0.666667.
When is the median absolute deviation the better choice?
When one or two extreme values would otherwise dominate. The median absolute deviation about the median hardly moves when a value is far away: on 1, 2, 100 it is 1, because the deviations are 0, 1 and 98 and the middle one of those is 1. That is why robust outlier rules are built on it. The mean absolute deviation weighs every deviation equally, so the same list gives 43.777778 instead.
What is the 1.4826 factor for?
It scales a median absolute deviation so that, for data that follows a normal distribution, it lands near the standard deviation. The factor is 1 divided by the 0.75 quantile of the standard normal, 1.482602218505602, rounded to 1.4826, so it is an approximation and it assumes that normal shape. The scaled figure sits in its own cell beside the raw value, and neither is ever folded into the other.
Can I use the median as the centre and still take the mean of the deviations?
Yes, that is the middle of the three choices, and it is labelled mean absolute deviation about the median. It is worth knowing that no other centre makes that total smaller: the sum of absolute deviations is at its minimum about the median. On 1, 2, 100 it gives 33, against 43.777778 about the mean.
What can I paste, and how many values?
Commas, spaces, tabs, semicolons and new lines all separate values, so a column copied from a spreadsheet works as it is. Negative numbers, decimals, exponent notation such as 1e3, currency symbols and a value in brackets for a negative are all read. Up to 10,000 values at a time. Anything that is not a number is named in a message rather than dropped quietly, so the answer always covers the whole list.
Is the mean absolute deviation the same as the standard deviation?
No. The standard deviation squares each deviation, averages the squares and takes the square root, which gives distant values more weight. The mean absolute deviation simply averages the distances, so it is easier to read off and less sensitive to one odd value. For a normal distribution the two are related by a fixed factor, but that is a property of that shape rather than a conversion for any list, and this page does not convert between them.
Does anything I paste leave my browser?
No. The whole calculation runs on this page, and the only copy kept anywhere is the draft this browser remembers so your list is still there when you come back. Start over, above the tool, forgets it.

Exact arithmetic: the mean of the absolute deviations from the centre you choose, with every deviation shown so you can add them up yourself. Mean deviation about the mean and median absolute deviation about the median are two different statistics and are labelled separately here, never merged. The 1.4826 normal-consistency scaling is an approximation and is shown apart from the raw value.