Mean Deviation

Why the simple mean of deviations fails

When we talk about the spread of a set of observations, a natural first thought is to look at how far each observation lies from some central value $a$. The deviation of an observation $x$ from $a$ is simply $x - a$. If we take the mean of all these deviations, we get:

But here is the problem. Any sensible measure of central tendency — mean, median, mode — lies somewhere between the smallest and largest observation. This means some deviations will be positive (observations above $a$) and some will be negative (observations below $a$). When you add them up, the positives and negatives cancel each other out. In fact, for the special case where $a = \bar{x}$ (the arithmetic mean), the sum of deviations is exactly zero:

So the mean of deviations is always zero when taken about the mean. That tells us nothing about how spread out the data is. A measure of dispersion that is always zero is useless.

The solution: absolute deviations

The problem is that negative and positive deviations cancel. What we actually care about is the distance of each observation from $a$, not the signed difference. On a number line, the distance between two numbers is given by the absolute value of their difference. So instead of taking the mean of the signed deviations, we take the mean of the absolute deviations.

This gives us a genuine measure of dispersion. It is called the mean deviation about $a$, denoted as M.D.$(a)$.

The mean deviation about a central value $a$ is the mean of the absolute values of the deviations of the observations from $a$.

Which central value to use?

Mean deviation can be computed about any measure of central tendency — mean, median, or mode. However, in statistical studies, mean deviation about the mean and mean deviation about the median are the ones you will encounter most often. The choice depends on the nature of the data and what you want to emphasise.

The next step is to learn how to calculate mean deviation about the mean and about the median for different types of data — ungrouped data, discrete frequency distributions, and continuous frequency distributions. Each case has its own formula and procedure, which we will now develop systematically.

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Key formulas recap …