# 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 *x*_{1,} *x*_{2, }*x*_{3, }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 |

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

- Mean deviation takes its minimum value when the deviations are taken from the median.
- 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*+*bx*,*a*and*b*being constants, then MD of*y*= |*b*|*x*MD of*x*.

# Merits

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

# Demerits

- In mean deviation the signs of all deviations are taken as positive and, therefore, it is not suitable for further algebraic treatment.
- It is rarely used in social sciences.
- 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.
- It is often not useful for statistical inferences.