Mean Absolute Deviation Calculator
Paste your numbers and get the mean absolute deviation — the average distance of the values from their mean — with a full deviation table and the working. You also get the MAD around the median and, clearly kept apart, the median absolute deviation: a different, outlier-resistant statistic that shares the abbreviation.
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How to Use the Mean Absolute Deviation Calculator
- 1. Paste your data. Numbers separated by commas, spaces or new lines — at least two.
- 2. Choose the centre. Around the mean is the definition in school maths and in Excel's AVEDEV; around the median is the variant some courses and robust methods use.
- 3. Read the table. Each row shows a value, its deviation from the mean and the absolute deviations from both centres, with the sums at the foot.
- 4. Compare the spread measures. The population SD sits alongside for comparison, and the median absolute deviation appears in its own section so the two MADs are never confused.
How to Find the Mean Absolute Deviation
The mean absolute deviation answers a plain question: on average, how far is a value from the mean? Take eight students who read 3, 6, 6, 7, 8, 11, 15 and 16 books over a summer:
- Find the mean. x̄ = (3 + 6 + 6 + 7 + 8 + 11 + 15 + 16) ÷ 8 = 72 ÷ 8 = 9.
- Subtract the mean from each value. −6, −3, −3, −2, −1, 2, 6, 7. These always add up to zero, so their plain average is useless.
- Drop the signs. 6, 3, 3, 2, 1, 2, 6, 7 — the distances from the mean.
- Average the distances. MAD = Σ|x − x̄| ÷ n = 30 ÷ 8 = 3.75.
So a typical student's total is 3.75 books away from the average of 9. Measured from the median, 7.5, the absolute deviations sum to 28 and the MAD is 3.5 — the median is the point that makes this sum as small as possible, so the MAD around the median can never exceed the MAD around the mean.
Mean Absolute Deviation vs Standard Deviation
Both measure spread around the mean, but the standard deviation squares the deviations before averaging and then takes a square root. Squaring gives large deviations extra weight, so the SD is never smaller than the MAD: for the book data the population SD is 4.30 against a MAD of 3.75. For normally distributed data the MAD is about 0.8 × σ (exactly σ√(2 ÷ π)).
The MAD is easier to explain — it is literally the average distance — which is why middle-school curricula teach it first. The SD has better mathematical properties: variances add, and it is the scale parameter of the normal distribution, so nearly all inference is built on it. Report the SD in research unless your method calls for an absolute-deviation measure.
Median Absolute Deviation: Same Abbreviation, Different Statistic
The median absolute deviation takes the median of |x − median| instead of the mean of |x − x̄|. For the book data the sorted absolute deviations from 7.5 are 0.5, 0.5, 1.5, 1.5, 3.5, 4.5, 7.5, 8.5, and their median is 2.5. Multiplied by 1.4826 — that is, 1 ÷ Φ⁻¹(0.75) — it becomes 3.71, an estimate of the standard deviation that holds for normal data.
Its strength is resistance to outliers. Change the 15 to 60 and the mean absolute deviation leaps from 3.75 to 11.69 and the SD from 4.30 to 17.54, but the median absolute deviation stays at 2.5. Watch the defaults when checking software: R's mad() reports the scaled version, while SciPy's median_abs_deviation reports the raw one unless you pass scale="normal".
Frequently Asked Questions
- What does the mean absolute deviation tell you?
- It tells you how far, on average, the values sit from the mean, in the data's own units. A MAD of 3.75 books means a typical value is about 3.75 books above or below the average. A larger MAD means the data are more spread out.
- How do you calculate the mean absolute deviation?
- Find the mean, subtract it from every value, drop the minus signs, and average the results: MAD = Σ|x − x̄| ÷ n. The deviation table above shows each step for your own numbers.
- Why do we take the absolute value of the deviations?
- Because the signed deviations from the mean always add up to exactly zero — the values above the mean balance those below it. Taking absolute values turns each deviation into a distance, so they can be averaged. The standard deviation solves the same problem by squaring instead.
- Is the mean absolute deviation the same as the standard deviation?
- No. Both measure spread, but the standard deviation squares the deviations, which gives more weight to values far from the mean, so it is always at least as large as the MAD. For normally distributed data the MAD is about 80% of the standard deviation.
- What Excel function calculates the mean absolute deviation?
- AVEDEV. It returns the mean absolute deviation around the mean, dividing by n, so it matches the default result here. Excel has no built-in median absolute deviation, but
=MEDIAN(ABS(A2:A9-MEDIAN(A2:A9)))computes it (in older versions of Excel, confirm it with Ctrl+Shift+Enter as an array formula). - When should I use the median absolute deviation?
- Use it when the data may contain outliers or are heavily skewed, as a robust replacement for the standard deviation. A common outlier rule flags values more than 2.5 to 3 scaled MADs from the median — Leys and colleagues (2013) recommend 2.5 — and it keeps working when a few extreme values would inflate the SD.