What Is Average Deviation?
Average deviation (also called mean deviation) is the mean of the absolute distances between each value and a measure of centre. When the centre is the mean you get the same number as the mean absolute deviation; using the median as the centre gives a smaller or equal value, because the median minimises the sum of absolute deviations.
The calculator returns both versions, the sum of absolute deviations, the coefficient of average deviation (relative to the centre) and the sample standard deviation for comparison. If you need the median absolute deviation instead, see the MAD calculator linked below.
The sections that follow give the formulas, a variables table, a worked example and answers to common questions such as average deviation versus standard deviation.
How to Use the Average Deviation Calculator
- Enter your numbers.
- Choose whether to measure deviation from the mean or from the median, matching what your textbook or instructor asks for.
- Read the deviation table to see each absolute difference, and compare the mean and median versions in the results.
Average Deviation Formula
Average deviation about the mean
Average deviation about the median
Coefficient of average deviation
A unit-free way to compare spread between data sets.
Average Deviation Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| AD | Average deviation | Same units as the data |
| xᵢ | Each data value | |
| c | Centre | Mean x̄ or median |
| n | Number of values | |
| Coefficient | AD divided by the centre | Unit-free, for comparing data sets |
Average Deviation Worked Example
With the example values loaded in the calculator above, the steps are:
- Centre: mean = 12.7143.
- Absolute deviations |x − mean| are listed in the table; their sum is 33.7143.
- Average deviation = 33.7143 ÷ 7 = 4.81633.
Result: Average deviation about the mean = 4.81633; Mean (x̄) = 12.7143.
How to Read an Average Deviation
The average deviation is the mean of the distances between each value and the centre, with the signs ignored. You can measure from the mean or from the median, and the median version is never larger than the mean version.
Compare it with the sample standard deviation shown next to it. The standard deviation squares the distances before averaging, so it gives large deviations more weight and is normally the bigger of the two. The coefficient of average deviation divides by the centre to compare data of different sizes. For example, 5, 8, 12, 13 and 20 have a mean of 11.6 and absolute distances 6.6, 3.6, 0.4, 1.4 and 8.4, so the average deviation about the mean is 20.4 ÷ 5 = 4.08. About the median of 12 the distances are 7, 4, 0, 1 and 8, giving 4. The sample standard deviation is 5.68.
Average deviation vs standard deviation
Standard deviation squares the distances before averaging, which gives extra weight to values far from the centre. Average deviation treats every unit of distance equally, so it is less affected by outliers and easier to interpret. For normally distributed data the standard deviation is about 1.25 times the mean absolute deviation about the mean, which you can check against the comparison row in the results.
| Feature | Average deviation | Standard deviation |
|---|---|---|
| Distances | Absolute values | Squared |
| Sensitive to outliers | Less | More |
| Algebra and theory | Awkward | Convenient, basis of most tests |
| Ease of explanation | Very easy | Harder |
| Normal data ratio | 1 | about 1.25 |
Mean or median as the centre?
Most textbook problems say which one to use. If they do not, the mean is the default for symmetric data. For skewed data the median is a more representative centre and gives a smaller, more robust average deviation. The sum of absolute deviations is minimised at the median, which is why the median version is never larger.
Average deviation for grouped data
For frequency tables, replace each class by its midpoint x and weight by frequency f: AD = Σ f|x − centre| / Σ f. If you have grouped data and need a standard deviation instead, use the grouped standard deviation calculator.
Where Average Deviation Is Used
Describing spread
Report how far values typically sit from the centre.
Mean or median centre
Choose which centre to measure from.
Comparing scales
Use the coefficient to compare data on different scales.
Checking working
Use the deviation table to follow each step.
Common Average Deviation Mistakes to Avoid
- Take absolute values. Without them the deviations about the mean sum to zero.
- State whether the centre is the mean or the median.
- The result is in the units of the data.
- Average deviation is not the same as standard deviation.
Average Deviation Terms and Definitions
- Deviation
- The signed difference between a value and the centre.
- Absolute deviation
- A deviation with its sign removed.
Average Deviation FAQs
What is average deviation?
It is the mean of the absolute differences between each data value and a central value, usually the mean or median.
Is average deviation the same as mean absolute deviation?
In most textbooks yes. Mean absolute deviation (MAD) often means the average absolute distance from the mean. See our MAD calculator for the variants.
Why do we take absolute values?
Deviations from the mean add up to zero, so you need to remove the signs to measure spread.
Can the average deviation be zero?
Only if every value is identical.
Which is smaller, deviation about the mean or the median?
The deviation about the median is always less than or equal to the deviation about the mean.
How is it different from variance?
Variance averages squared distances, so it is in squared units and weights large deviations more heavily.
Can average deviation be negative?
No. It averages absolute distances, so it is zero when every value is identical and positive otherwise.
How do I find the average deviation in Excel?
Use =AVEDEV(range). It returns the average of the absolute deviations from the mean.