# Mean Deviation Formula

The mean deviation is also known as the mean absolute deviation and is defined as the mean of the absolute deviations of the observations from the suitable average which may be the arithmetic mean, the median or the mode.

The formula to calculate Mean deviation is as stated below:

$\large Mean\;Deviation\;from\;Mean=\frac{\sum \left |X-\overline{X}\right|}{N}$

$\large Mean\;Deviation\;from\;Median=\frac{\sum \left |X-M\right|}{N}$

Here,
$\sum$ represents the summation.
X =Observations
$\overline{X}$ = Mean
N = The number of observations

M = Median

For frequency distribution, the mean deviation is given by:

$\large M.D=\frac{\sum f \left | X-\overline{X} \right |}{\sum f}$

When the mean deviation is calculated about the median, the formula becomes:

$\large M.D (about\;median)=\frac{\sum f\left | X-Median \right |}{\sum f}$

The mean deviation about the mode is:

$\large M.D (about\;mode)=\frac{\sum f\left | X-Mode\right |}{\sum f}$

For a population data, the mean deviation about the population mean $\mu$ is:

$\large M.D=\frac{\sum f\left | X-\mu \right |}{\sum f}$

### Solved Example

Question: Anubhav scored 85, 91, 88, 78, 85 for a series of exams. Calculate the mean deviation for his test scores.

Ans:  Given test score; 85, 91, 88, 78, 85

Mean, $$\bar{x}$$ = (85 + 91 + 88 + 78 + 85)/5

= 427/5

= 85.4

Subtracting mean from each score;

 x $$\mathbf{x_i – \bar{x}}$$ $$\mathbf{|x_i – \bar{x}|}$$ 85 -0.4 0.4 91 5.6 5.6 88 2.6 2.6 78 -7.4 7.4 85 -0.4 0.4

Mean deviation = 16.4/5 = 3.28

Thus we can say that on average, the student’s test scores vary by a deviation of 3.28 points from the mean.
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