Mean absolute deviation calculator
Paste your numbers, get the MAD with every step.
How do you find the mean absolute deviation?
Find the mean, subtract it from each value and drop the minus signs, then average those distances: MAD = Σ|x − mean| ÷ n. It tells you how far a typical value is from the mean.
Example
For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 40 ÷ 8 = 5. The distances from 5 are 3, 1, 1, 1, 0, 0, 2, 4, which add up to 12, so the MAD is 12 ÷ 8 = 1.5.
What is the mean absolute deviation about the median?
The same average of distances, measured from the median instead of the mean. No other center gives a smaller average distance, so it is never larger than the MAD about the mean. For 2, 4, 4, 4, 5, 5, 7, 9 the median is 4.5 and this MAD is also 1.5.
Is that the same as the median absolute deviation?
No — the median absolute deviation takes the median of the distances from the median, not their average. It ignores extreme values even more. For the example above the distances from 4.5 are 2.5, 0.5, 0.5, 0.5, 0.5, 0.5, 2.5, 4.5 and their median is 0.5. It is shown here unscaled, as NIST defines it.
How is MAD different from the standard deviation?
The standard deviation squares the distances before averaging, so large deviations weigh more. The MAD uses plain distances, so it is easier to compute by hand; the Common Core grade 6 standard 6.SP.B.5c lists it, with the interquartile range, as the measure of variability students should report.
Where do the formulas come from?
The average absolute deviation and the median absolute deviation are defined in the NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.6 (Measures of Scale); the version about the median follows NIST Dataplot’s “average absolute deviation from the median”. Mean absolute deviation is named in Common Core standard 6.SP.B.5c.
Do I divide by n or n − 1?
By n. The mean absolute deviation is defined as the average of the distances, so the sum of |x − mean| is divided by the number of values. The n − 1 divisor belongs to the sample variance and standard deviation, not to this measure.
Why not just average x − mean without the absolute value?
Because the deviations from the mean always add up to exactly 0: the values below the mean cancel the values above it. Taking the distance (dropping the sign) first is what makes the average a measure of spread.
What does a MAD of 0 mean?
Every value is the same. The MAD is never negative, and it grows as the values spread out. If you multiply every value by 10 the MAD is 10 times larger; adding the same amount to every value leaves it unchanged.
Worked example with an outlier
Take 12, 15, 11, 14, 13, 40. The mean is 105 ÷ 6 = 17.5, and the distances from it add up to 45, so the MAD about the mean is 45 ÷ 6 = 7.5. The median is 13.5; the distances from the median add up to 33, so the average distance about the median is 5.5. The median absolute deviation, the middle of those distances, is only 1.5.
The one large value, 40, drags the mean up to 17.5 and inflates the first figure. Without it, 12, 15, 11, 14, 13 has a mean of 13 and a MAD of 1.2. The median absolute deviation is far less sensitive (1.5 with the outlier, 1 without it, against 7.5 and 1.2 for the mean-based MAD), which is why it is the choice when a data set may contain errors. For comparison, the population standard deviation of the six values is 10.14, larger than the MAD about the mean (7.5) because squaring weights the outlier more.
Which center to use
The mean-centered MAD is the version taught in grade 6 statistics and matches the Common Core wording. The median-centered average is never larger, because the median is the center that minimizes the average distance. The median absolute deviation is a different statistic, a median of distances rather than an average; it is shown unscaled, so do not compare it directly with a standard deviation.
Common mistakes
Leaving the minus signs in is the classic slip: the deviations then sum to zero and the “MAD” comes out as 0. Another is mixing up the three similarly named statistics (mean absolute deviation about the mean, about the median, and median absolute deviation) when comparing with a textbook answer — check which center and which average the book means. Finally, make sure you divide by the count of values, including repeated values, not by the number of distinct ones.
Sources: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.6 (Measures of Scale); NIST Dataplot reference for the average absolute deviation from the median; Common Core State Standards for Mathematics, 6.SP.B.5c. Related: standard deviation calculator, quartile calculator, average calculator.