What Is Mean and MAD?
MAD is an abbreviation used for two different measures of spread. The mean absolute deviation averages the distances of values from the mean, while the median absolute deviation takes the median of the distances from the median, which makes it resistant to outliers.
Enter your values and the calculator returns the mean, both versions of MAD, the mean absolute deviation about the median, the population standard deviation for comparison, the relative MAD and the scaled median absolute deviation that estimates σ for normal data. A working table shows each absolute deviation.
Below are the formulas, a variables table, a worked example and answers to common questions about which MAD to use.
How to Use the Mean and MAD Calculator
- Enter your data values.
- Read the mean and both versions of MAD.
- Compare with the standard deviation.
- Use the median absolute deviation if outliers are present.
Mean and MAD Formula
Mean absolute deviation
Median absolute deviation
Mean and MAD Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x̄ | Mean | |
| MAD (mean) | Mean absolute deviation | Σ|x − x̄| ÷ n |
| MAD (median) | Median absolute deviation | median of |x − median| |
| 1.4826 | Scale factor | makes median MAD estimate σ for normal data |
Mean and MAD Worked Example
For 2, 4, 6, 8: the mean is 5. The absolute deviations are 3, 1, 1, 3, so the mean absolute deviation is 8 ÷ 4 = 2. The median is 5 and the absolute deviations from it are the same, so the median absolute deviation is the median of 1, 1, 3, 3, which is 2.
With the example values loaded in the calculator above, the steps are:
- Mean x̄ = Σx ÷ n = 6.5.
- Absolute deviations |x − x̄| are averaged: MAD = Σ|x − x̄| ÷ n = 16 ÷ 8 = 2.
- Median = 6.5; the median of |x − median| is 2.
- Compare: population SD = 2.2912878, which is at least as large as the mean absolute deviation.
Result: Mean x̄ = 6.5; Mean absolute deviation (about the mean) = 2; Median absolute deviation (about the median) = 2.
How to Read Mean and MAD
The mean absolute deviation is the average distance from the mean. For 2, 4, 6, 8 the mean is 5, the distances are 3, 1, 1 and 3, and the MAD is 2. The median absolute deviation uses the median as the centre and then takes the median of the distances, which resists outliers.
The scaled median absolute deviation multiplies by 1.4826. For normally distributed data this puts it on the same scale as the standard deviation, so it can be compared with it directly.
Where Mean and MAD Is Used
Spread without squaring
Describe variability in the original units with a measure that is easy to explain.
Outlier-resistant spread
Use the median absolute deviation when a few values are extreme.
Comparing to the SD
See the standard deviation next to the MAD for the same data.
Relative spread
Use MAD divided by the mean to compare data on different scales.
Common Mean and MAD Mistakes to Avoid
- "MAD" can mean mean absolute deviation or median absolute deviation. Say which one you use.
- The mean version is affected by outliers, the median version much less.
- The scaled median absolute deviation matches the SD only for normal data.
- For the same data, the mean absolute deviation is never larger than the standard deviation.
Mean and MAD Terms and Definitions
- Absolute deviation
- The distance of a value from a centre, ignoring sign.
- Robust
- Not strongly affected by outliers.
Mean and MAD FAQs
What is the mean absolute deviation?
It is the average of the absolute differences between each value and the mean. For 2, 4, 6, 8 it is (3 + 1 + 1 + 3) ÷ 4 = 2.
What is the difference between mean and median absolute deviation?
The mean version averages the distances from the mean. The median version takes the median of the distances from the median. The second changes much less when an outlier is added.
How do I calculate MAD by hand?
Find the centre, subtract it from each value, drop the signs, then average the results, or take their median for the median version.
Is MAD smaller than the standard deviation?
For the mean absolute deviation, yes, it is never larger than the standard deviation for the same data. Equal values are possible only when all deviations have the same size.