Mean absolute deviation: how to find the MAD, step by step
The mean absolute deviation (MAD) is the average distance of the data values from their mean. It measures spread: a small MAD means the values sit close to the mean, a large one means they are spread out. It is quick to work out by hand and easy to interpret.
On this page
The formula
MAD = ∑|x − x̄| / n, where x̄ is the mean and n the number of values. The bars | | mean absolute value: every distance counts as positive.
The absolute value is the key step. Without it, the deviations above and below the mean always add up to zero, so they would tell you nothing about spread.
How to find the MAD in four steps
- Find the mean of the data.
- Subtract the mean from each value.
- Take the absolute value of each difference.
- Find the mean of those absolute values.
Worked examples
Example 1: a simple data set
Find the MAD of 2, 4, 6, 8, 10.
- The mean is 30/5 = 6.
- The distances from 6 are 4, 2, 0, 2, 4.
- Their sum is 12, and 12/5 = 2.4.
- On average, the values are 2.4 away from the mean.
| Value x | x − mean | |x − mean| |
|---|---|---|
| 2 | −4 | 4 |
| 4 | −2 | 2 |
| 6 | 0 | 0 |
| 8 | 2 | 2 |
| 10 | 4 | 4 |
| Total | 0 | 12 |
Example 2: comparing two data sets
Two classes scored 12, 15, 15, 18, 20 and 5, 10, 15, 25, 25. Compare them.
- Both means are 16.
- Class A: distances 4, 1, 1, 2, 4, so MAD = 12/5 = 2.4.
- Class B: distances 11, 6, 1, 9, 9, so MAD = 36/5 = 7.2.
- Same average, but class B's scores are three times as spread out.
Example 3: from a frequency table
Scores of 20 students: score 1 (2 students), 2 (3), 3 (5), 4 (6), 5 (4). Find the MAD.
- Mean = (2 + 6 + 15 + 24 + 20)/20 = 67/20 = 3.35.
- Multiply each distance by its frequency: 2(2.35) + 3(1.35) + 5(0.35) + 6(0.65) + 4(1.65) = 21.
- MAD = 21/20 = 1.05.
Example 4: the effect of an outlier
Find the MAD of 3, 5, 5, 6, 7, 8, then add a value of 30.
- Without 30: mean 5.67, MAD 1.33.
- With 30: mean 9.14, MAD 5.96.
- One outlier more than quadruples the MAD. The median, 6 in the second set, barely moves, which is why the median and IQR are preferred for skewed data.
Interpreting the MAD in context
A number on its own earns little in a statistics answer. Say what it means for the data: "The mean absolute deviation of the delivery times is 2.4 days, so on average a delivery time is 2.4 days away from the mean of 6 days." When comparing two groups, compare both the centre and the spread, in context, as in Example 2.
Example 5: negative values
Morning temperatures over five days were −3, 0, 2, 5 and 6 °C. Find the MAD.
- Mean = (−3 + 0 + 2 + 5 + 6)/5 = 10/5 = 2 °C.
- Distances from 2: 5, 2, 0, 3, 4.
- MAD = 14/5 = 2.8 °C. Distances are always positive, even for negative values.
Mean absolute deviation vs standard deviation
| Mean absolute deviation | Standard deviation | |
|---|---|---|
| What it averages | The distances from the mean | The squared distances, then a square root |
| Formula | ∑|x − x̄| / n | √(∑(x − x̄)² / n), or n − 1 for a sample |
| For 2, 4, 6, 8, 10 | 2.4 | 2.83 (population) or 3.16 (sample) |
| Effect of outliers | Large | Larger, because distances are squared |
| Where it is used | Middle school and introductory statistics | AP Statistics, A level and beyond, because of its link to the normal distribution |
For any data set, the MAD is never larger than the population standard deviation. AP Statistics uses the standard deviation and the IQR to describe spread; see our standard deviation guide.
Mean absolute deviation vs median absolute deviation
Both can be shortened to MAD, which causes confusion. The median absolute deviation takes the median of the distances from the median. It is much less affected by outliers. If a question says "mean absolute deviation", use the method on this page.
Practice questions
Try each one before opening the answer.
Q1. Find the MAD of 1, 3, 5, 7.
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Q2. Find the MAD of 10, 10, 10, 10.
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Q3. Find the MAD of 4, 7, 9, 12, 13.
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Q4. Data set A has MAD 1.5 and data set B has MAD 6. Which is more consistent?
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Q5. Five delivery times in days are 3, 5, 6, 8 and 8. Find the MAD and interpret it.
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Common mistakes
- Forgetting the absolute value, so the deviations add up to zero.
- Dividing by n − 1, which belongs to the sample standard deviation, not the MAD.
- Finding distances from the median instead of the mean.
- In a frequency table, forgetting to multiply each distance by its frequency.
Frequently asked questions
What does the mean absolute deviation tell you?
How far, on average, the values are from the mean. A larger MAD means the data are more spread out.
Can the mean absolute deviation be negative?
No. It is an average of distances, which are never negative. It is zero only when every value is the same.
Is MAD the same as standard deviation?
No. Both measure spread, but standard deviation squares the distances first, so it is usually larger and more affected by outliers.
Is mean absolute deviation on the AP Statistics exam?
AP Statistics uses standard deviation and IQR as its measures of spread. MAD is mainly taught in middle school statistics, but it is a useful way to understand spread.
Key takeaways
- MAD = the mean of the distances from the mean.
- Take absolute values, or the deviations cancel to zero.
- MAD is never larger than the population standard deviation.