Mean Deviation about the Mean and Median (Ungrouped Data)

Mean deviation (also called mean absolute deviation) measures the average distance of the observations from a fixed central value — usually the mean or the median — treating every distance as a positive quantity regardless of whether the observation lies above or below that central value.

Why absolute value, and not the plain (signed) deviation? If the signed deviations $x_i-\bar x$ were simply averaged, the result would always be exactly zero for the mean, since by definition $\sum(x_i-\bar x)=\sum x_i - n\bar x = n\bar x - n\bar x = 0$ — the positive deviations (observations above the mean) always exactly cancel the negative deviations (observations below the mean). A similar near-cancellation happens around the median. So a plain average of signed deviations can never measure spread; it is identically zero no matter how scattered the data is. Taking the absolute value $|x_i-\bar x|$ of each deviation before averaging removes the sign, so that a value $10$ units above the mean and a value $10$ units below the mean both contribute the same positive amount ($10$) to the total spread, instead of cancelling to $0$.

Mean deviation about the mean, for ungrouped data. If $x_1,x_2,\ldots,x_n$ is the data with mean $\bar x = \dfrac1n\sum_{i=1}^n x_i$, the mean deviation about the mean is

Mean deviation about the median, for ungrouped data. If $M$ denotes the median of the data (the middle value when the data is arranged in ascending order — the average of the two middle values when $n$ is even), the mean deviation about the median is

Steps to compute mean deviation about the mean (ungrouped).

1. Find the mean $\bar x$ of the data.
2. Find the absolute deviation $|x_i-\bar x|$ of every observation from $\bar x$.
3. Sum these absolute deviations and divide by $n$.

The same three steps apply to the mean deviation about the median, using $M$ in place of $\bar x$ throughout. (When $n$ is odd, the median is the unique middle value of the sorted data; when $n$ is even, it is the average of the two middle values — this second case is illustrated in the exercises.) …