Mathematics · Ch 13 — Statistics
Limitations of Mean Deviation
Limitations of Mean Deviation
The Need for a Better Measure
Mean deviation has two serious weaknesses that make it unreliable as a general-purpose measure of dispersion. First, when data is highly variable — when values are spread far apart — the median itself stops being a good representative of the centre. If the central value is unreliable, any measure of spread calculated around it (like mean deviation about median) is also unreliable.
Second, mean deviation about the mean has a mathematical flaw. The sum of absolute deviations from the mean (ignoring minus signs) is actually larger than the sum of absolute deviations from the median. This means the mean is not the point that minimises absolute deviations — the median is. So using the mean as the reference point for a measure of spread is not scientifically sound; it gives a number that is not the smallest possible for that data set.
Because mean deviation uses absolute values, it cannot be used in further algebraic operations like addition, multiplication, or differentiation. This severely limits its usefulness in advanced statistics.
These limitations force us to look for another measure of dispersion — one that is mathematically tractable and gives consistent, reliable results. That measure is standard deviation.
Why Mean Deviation Fails for Highly Variable Data
Consider a series where values are extremely spread out — for example, incomes in a large population, or marks in a very difficult exam. In such a series, the median may lie far from many observations. The mean deviation about median, which measures the average distance of observations from the median, will then be large and may not reflect the true pattern of spread. The central tendency itself is questionable, so any dispersion measure based on it is questionable too.
This is not just a theoretical point. In practice, when data has outliers or is heavily skewed, the median is preferred as a measure of centre. But even the median can be unrepresentative if the variability is extreme — for instance, if half the data is clustered near zero and the other half is scattered over a huge range.
The Mathematical Flaw: Mean vs. Median as Reference Point
For any set of numbers, the sum of absolute deviations (ignoring signs) is minimised when taken from the median, not from the mean. That is:
This inequality is a known property of absolute deviations. The mean deviation about the mean is therefore not the smallest possible value of its kind. Using the mean as the reference point inflates the measure unnecessarily. A good measure of dispersion should be based on a reference point that gives the most compact description of spread — and that point is the median for absolute deviations, not the mean.
Because the mean does not minimise the sum of absolute deviations, the mean deviation about the mean is not a "natural" or optimal measure. This is one reason statisticians prefer standard deviation, which uses squared deviations — and the mean does minimise the sum of squared deviations.
Algebraic Intractability
The use of absolute values in mean deviation creates a second, equally serious problem. Absolute value functions are not differentiable at zero, and they do not obey simple algebraic rules. You cannot, for example, add two mean deviations and get a meaningful result, nor can you manipulate them in formulas for combined data sets. This makes mean deviation unsuitable for building more advanced statistical theory — for hypothesis testing, analysis of variance, or regression.
| Property | Mean Deviation | Standard Deviation |
|----------|----------------|-------------------|
| Uses absolute values | Yes | No |
| Algebraically tractable | No | Yes |
| Minimised by | Median | Mean |
| Affected by outliers | Less | More | …