# Mean Deviation (Average Deviation)

Mean deviation is the arithmetic mean of the absolute values of deviations of observations from a measure of central tendency. The measure of central tendency is generally taken as the A.M. or the median.

Let x assume n values x1, x2, x3, â€¦ xn and  be its mean. Then, mean deviation about when the observations are a set of discrete data, is expressed as

For a grouped frequency distribution, mean deviation about  is given by

Example
Compute the mean deviation and the coefficient of mean deviation about the arithmetic mean for the following data:

 X 1 3 5 7 9 f 5 8 9 2 1
Solution

 X f 1 5 2.88 14.40 3 8 0.88 7.04 5 9 1.12 10.08 7 2 3.12 6.24 9 1 5.12 5.12 Total 25 42.88

MD about A.M. is given by

We can compute the mean deviation about the median also.
The mean deviation about median for discrete data
The mean deviation about median for grouped data

Example
Compute the mean deviation and coefficient of mean deviation of weight about the median for the following data:

 Weight (lb) 131â€“140 141â€“150 151â€“160 161â€“170 171â€“180 181â€“190 No. of Persons 3 8 13 15 6 5
Solution
 Class Boundary Frequency Less Than Cumulative Frequency 130.5â€“140.5 3 3 140.5â€“150.5 8 11 150.5â€“160.5 13 24 160.5â€“170.5 15 39 170.5â€“180.5 6 45 180.5â€“190.5 5 50

24 < 25 <39, hence 1 = 160.5, C f = 24, f = 15, C = 10

 xi | xi â€“ Me | fi| xi â€“ Me | 135.5 25.66 76.98 145.5 15.66 125.28 155.5 5.66 73.58 165.5 4.34 65.1 175.5 14.34 86.04 185.5 24.34 121.7 Total 548.68

# Properties of Mean Deviation

1. Mean deviation takes its minimum value when the deviations are taken from the median.
2. Mean deviation remains unchanged due to a change of origin but it changes in the same ratio due to a change in scale, i.e., if two variables x and y are related as y = a + bxa and b being constants, then MD of y = | b | x MD of x.

# Merits

1. It is easy to understand and compute.
2. Mean deviation is less affected by the extreme values as compared to standard deviation.
3. Mean deviation about an arbitrary point is least when the point is median.

# Demerits

1. In mean deviation the signs of all deviations are taken as positive and, therefore, it is not suitable for further algebraic treatment.
2. It is rarely used in social sciences.
3. It does not give accurate results because the mean deviation from the median is least but median itself is not considered a satisfactory average when the variation in the series is large.
4. It is often not useful for statistical inferences.