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.

How do I find mean absolute deviation with this calculator?

Enter the complete list under Your numbers and choose Mean about the mean. The result updates as you type, showing the mean, each absolute deviation, their sum and the division by n. The same division by n is used for a full population or a supplied sample list; there is no n minus 1 adjustment. Copy puts the labelled result, values and deviations on your clipboard as tab-separated text.

How does median absolute deviation respond to an extreme value?

It uses the median of the absolute deviations about the median, so the size of one extreme value may have little effect. The effect depends on the list size and distribution. For 1, 2, 100 the median is 2, the absolute deviations are 1, 0 and 98, and their median is 1. The mean absolute deviation about the mean is 43.777778 on the same list. These are descriptive statistics; the page does not choose a statistic or an outlier threshold for you.

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?

Separate values with commas, spaces, tabs, semicolons or new lines. Negative numbers, decimals, exponent notation such as 1e3, currency symbols and parentheses for a negative are accepted. Commas always separate values: type 1234 for one value, because 1,234 is read as 1 and 234 with a note. Up to 10,000 values and 200,000 characters are accepted. Invalid entries are named in an error and stop the calculation. Nonzero inputs that round to zero below the numeric range are refused; scale all values up by the same factor and interpret the result in those scaled units.

Can I enter grouped data or compare actual values with forecasts?

The input accepts a raw list, not a frequency table or paired actual and forecast columns. A forecasting MAD is the mean of absolute actual-minus-forecast errors, which differs from measuring each value from the list mean. This page does not calculate those paired forecast errors.

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.

Mean Absolute Deviation Calculator

Paste a list to calculate mean absolute deviation with steps: the centre, each absolute deviation, their total and the final average. Choose mean about the mean, mean about the median, or median absolute deviation. Each result uses its full name, and Copy includes tab-separated values and deviations. Calculations use double precision and accept up to 10,000 values.

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.

Your numbers

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. Median-based spread is less sensitive to extremes in many lists; the effect depends on the list size and distribution. This page computes whichever you pick and labels it by its full name rather than by the abbreviation.

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.
How do I find mean absolute deviation with this calculator?: Enter the complete list under Your numbers and choose Mean about the mean. The result updates as you type, showing the mean, each absolute deviation, their sum and the division by n. The same division by n is used for a full population or a supplied sample list; there is no n minus 1 adjustment. Copy puts the labelled result, values and deviations on your clipboard as tab-separated text.
How does median absolute deviation respond to an extreme value?: It uses the median of the absolute deviations about the median, so the size of one extreme value may have little effect. The effect depends on the list size and distribution. For 1, 2, 100 the median is 2, the absolute deviations are 1, 0 and 98, and their median is 1. The mean absolute deviation about the mean is 43.777778 on the same list. These are descriptive statistics; the page does not choose a statistic or an outlier threshold for you.
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?: Separate values with commas, spaces, tabs, semicolons or new lines. Negative numbers, decimals, exponent notation such as 1e3, currency symbols and parentheses for a negative are accepted. Commas always separate values: type 1234 for one value, because 1,234 is read as 1 and 234 with a note. Up to 10,000 values and 200,000 characters are accepted. Invalid entries are named in an error and stop the calculation. Nonzero inputs that round to zero below the numeric range are refused; scale all values up by the same factor and interpret the result in those scaled units.
Can I enter grouped data or compare actual values with forecasts?: The input accepts a raw list, not a frequency table or paired actual and forecast columns. A forecasting MAD is the mean of absolute actual-minus-forecast errors, which differs from measuring each value from the list mean. This page does not calculate those paired forecast errors.
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 formulas are evaluated with double-precision numbers; displayed values are rounded to six decimal places, or shown in scientific notation for very small or large values. Mean absolute deviation and median absolute deviation are labelled separately, with every deviation shown. The 1.4826 normal-consistency scaling is an approximation shown apart from the raw median absolute deviation.