Mean Absolute Deviation Calculator

How to use the mean absolute deviation calculator

1. Enter the data. Separate values with commas, spaces or new lines, or paste a column from a spreadsheet. Negative numbers and decimals are fine.
2. Choose the center. “Mean” gives the mean absolute deviation, the version taught in middle school and most statistics courses. “Median” gives the median absolute deviation, which robust statistics uses.
3. Check the table. Each row shows a value, its signed distance from the center and the absolute distance. The bottom line shows the sum (for the mean version) or the median (for the median version) of those distances, so you can follow every step of a homework answer.

The results also show the mean, the median, the other kind of absolute deviation and the population standard deviation, so you can see how the measures compare on the same data.

The formula, step by step

The NIST/SEMATECH e-Handbook of Statistical Methods calls this measure the average absolute deviation and writes it as:

In words: find the mean, find how far each value is from it (ignoring the sign), and average those distances. Using the default data 2, 4, 4, 4, 5, 5, 7, 9:

1. The mean is 40 ÷ 8 = 5.
2. The distances from 5 are 3, 1, 1, 1, 0, 0, 2 and 4.
3. They add up to 12.
4. 12 ÷ 8 = 1.5. On average, a value is 1.5 away from the mean.

The absolute value matters. Without it, the deviations from the mean cancel out and add up to zero for every data set, which tells you nothing about spread. Taking the absolute value is one fix; squaring the deviations, as the standard deviation does, is the other.

Median absolute deviation

The median absolute deviation replaces both averages with medians:

For the same eight values, the median is 4.5. The distances from 4.5 are 2.5, 0.5, 0.5, 0.5, 0.5, 0.5, 2.5 and 4.5. Sorted, the middle two are 0.5 and 0.5, so the median absolute deviation is 0.5.

Both are abbreviated MAD, which causes confusion. In a middle school or introductory class, MAD usually means the mean version. In data analysis software it often means the median version, and some programs multiply it by about 1.4826 so it lines up with the standard deviation for bell-shaped data. This calculator reports the unscaled value. Check your assignment or software documentation if the numbers don’t match.

MAD compared with standard deviation

All three measures describe spread in the same units as the data, but they react differently to extreme values. NIST notes that the average absolute deviation is less affected by extremes than the variance and standard deviation, because it doesn’t square the distances, and that the median absolute deviation is even less affected by extremes in the tails.

A quick example shows the difference. Take 10, 20, 30 and 1,000:

The single outlier drags the mean far from the other three values, and both the mean absolute deviation and the standard deviation balloon. The median absolute deviation stays at 10, describing the tight cluster of the first three numbers. Which one is “right” depends on the question: if the 1,000 is a real value you care about, the mean-based measures show it; if it is a typo or a rare event, the median-based measure ignores it.

For the default data the population standard deviation is 2, compared with a mean absolute deviation of 1.5. The standard deviation is at least as large as the mean absolute deviation for any data set, because squaring gives extra weight to the larger distances. For a full standard deviation workup, including the sample (n − 1) version and quartiles, use the standard deviation calculator.

Reading the result

Because the mean absolute deviation is an average distance, it is easy to explain: “test scores were 76 on average, and a typical score was about 8 points from that.” Two classes with the same mean but different MADs had different spreads: the class with the larger MAD had more students far above and far below average.

A MAD of 0 means every value is identical. There is no fixed “large” or “small” MAD; compare it with the mean (a MAD of 5 is large for values around 10 and tiny for values around 10,000), or with the MAD of another group measured the same way.

Forecasters use a close cousin, the mean absolute error, which averages the distances between forecasts and actual results. The arithmetic is the same, with the forecast taking the place of the mean.

Example: comparing two groups

Two basketball players each average 15 points over five games. Player A scored 14, 15, 15, 16 and 15. Player B scored 5, 25, 10, 20 and 15. The means are equal, but the spreads aren’t:

• Player A: distances from 15 are 1, 0, 0, 1 and 0. The sum is 2, so the MAD is 2 ÷ 5 = 0.4 points.
• Player B: distances from 15 are 10, 10, 5, 5 and 0. The sum is 30, so the MAD is 30 ÷ 5 = 6 points.

Player A is consistent; Player B is streaky. Paste each list into the calculator to confirm the numbers. This kind of comparison, where two data sets have the same center but different variability, is the standard middle school exercise on MAD.

Common mistakes

• Dropping the absolute value. If some deviations are negative and you add them as they are, the sum is zero.
• Dividing by the wrong count. Divide by the number of values, including any with a deviation of 0.
• Rounding the mean too early. If the mean is 5.333…, use the full value. Rounding it to 5.3 before finding each distance can shift the answer in the second decimal place, which matters when a worksheet checks two decimals.
• Mixing the two MADs. Distances from the mean averaged with the mean, or distances from the median taken at their median. Mixing a median center with a mean of distances gives a third measure that neither textbook defines.

Sources & assumptions

How this calculator gets its numbers. Full method: our methodology.

• Mean absolute deviation = Σ|x − mean| ÷ n, which NIST calls the average absolute deviation.
• Median absolute deviation = the median of |x − median|, without the 1.4826 scale factor some software applies.
• Population standard deviation (divide by n) is shown for comparison because it is the closest like-for-like measure.

Sources

1. National Institute of Standards and Technology, NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.6 Measures of Scale. Average absolute deviation, AAD = Σ|Yᵢ − Ȳ| ÷ N, less affected by extreme values than the variance and standard deviation; median absolute deviation, MAD = median(|Yᵢ − Ỹ|).
2. OpenStax (Rice University), Introductory Statistics 2e, 2.7 Measures of the Spread of the Data. Sample (n − 1) and population (N) standard deviation formulas; Example 2.32 (mean 10.525, s² = 0.5125, s = 0.715891) and Example 2.33 (mean 73.5, s = 17.9, median 73, Q1 61, Q3 90).

Last reviewed November 2, 2026. Some statistics packages use “MAD” for the median version and scale it; check which one your class or software expects.