Mean Deviation: The Intuition
You have a set of numbers — say, the marks of five students in a test: 40, 50, 60, 70, 80. The average (mean) is 60. Now, not every student scored 60. Some are above, some below. The question is: on average, how far away from the centre are these marks?
That is exactly what Mean Deviation measures. It answers: "If I pick any one mark at random, how many units away from the mean should I expect it to be?"
The Problem with Signs
If you simply add up the differences (40 − 60 = −20, 50 − 60 = −10, 60 − 60 = 0, 70 − 60 = +10, 80 − 60 = +20), the positives and negatives cancel: (−20) + (−10) + 0 + 10 + 20 = 0. That tells you nothing about spread — it always gives zero for any symmetric data.
So we need to get rid of the sign. The simplest way: take the absolute value of each deviation. That is the core idea.
The Definition
For a set of n observations x1,x2,…,xn with mean xˉ, the Mean Deviation about the mean is:
MD(xˉ)=n1∑i=1n∣xi−xˉ∣
You can also compute it about the median instead of the mean — the formula is the same, just replace xˉ with the median M.
MD(mean)=n∑∣xi−xˉ∣
Worked Example
Take the marks: 40, 50, 60, 70, 80.
- Mean xˉ=60.
- Deviations: |40−60|=20, |50−60|=10, |60−60|=0, |70−60|=10, |80−60|=20.
- Sum of absolute deviations = 20 + 10 + 0 + 10 + 20 = 60.
- Mean Deviation = 60 / 5 = 12.
Interpretation: On average, a student's mark is 12 points away from the class average of 60.
Why Not Just Use Standard Deviation?
Mean Deviation is simpler to explain and compute — no squaring, no square roots. But squaring (as in standard deviation) gives more weight to extreme values, which is often desirable in statistics. Mean Deviation treats every deviation equally, which can be both a strength (robustness to outliers) and a weakness (less sensitive to large errors).
For Grouped Data
When data is grouped into classes with frequencies fi and midpoints xi:
MD(xˉ)=∑fi∑fi∣xi−xˉ∣
The logic is identical — you just weight each absolute deviation by how many observations fall in that class.
For discrete data, always list all values explicitly before summing. For continuous grouped data, use the class midpoints as xi.
A Common Mistake
Do not take the absolute value of the mean itself. The absolute value goes inside the sum, applied to each individual deviation. The mean deviation is always non-negative, and equals zero only if every observation is identical.
The Big Picture
Mean Deviation is your first tool for measuring dispersion — how spread out the data is. It is intuitive, easy to calculate by hand, and gives a direct answer in the original units of the data. Later you will meet variance and standard deviation, which are more mathematically convenient, but the idea remains the same: how far, on average, are the data points from the centre?