# Introduction

Range and quartile deviations are positional measures of dispersion, wherein all the observations are not taken into account in the calculation. Now we consider a measure of dispersion called mean deviation based on all observations.

Consider the observation 3, 5, 6, 7, 9

A.M. = `(3+5+6+7+9)/(5)` = 6

The sum of the deviations of the items from the mean is zero. Consider the A.M. of theabsolute deviations of those observations from their mean.

i.e., = `(3+1+0+1+3)/(5)` = 8/5 = 1.6

This is called the mean deviation about the mean. This tells that on the average the observations are deviated away from the mean by 1.6 on either side.

Definition of Mean Deviation for Grouped Data:

Let x_{1}, x_{2}, x_{3},…x_{n} be n values. The mean deviation about the mean of these values is given by

M.D. = `(sum I X i – barxI)/(n)`

In a frequency distribution with frequencies f_{1},f_{2,}f_{3},…f_{n} against the values x_{1}, x_{2}, x_{3},…x_{n},

M.D. about the mean = `(sum fi I X i – barx I)/(N)`

where N is the total frequency

Relative measure:

Co efficient of mean deviation about the mean = `(M.D.)/(Mean)`

Co efficient of mean deviation about the median = `(M.D.)/(Median)`

**Finding Mean Deviation for Grouped Data:**

Find the mean deviation for the grouped data.Classfrequency0-535-10510-151215-20620-254Total30

Solution:

Class | Mid X | f | d | fd | X – `barx` | f IX -`barx` I |

0-5 | 2.5 | 3 | -2 | -6 | 10.5 | 31.5 |

5-10 | 7.5 | 5 | -1 | -5 | 5.5 | 27.5 |

10-15 | 12.5 | 12 | 0 | 0 | 0.5 | 6.0 |

15-20 | 17.5 | 6 | 1 | 6 | 4.5 | 27.0 |

20-25 | 22.5 | 4 | 2 | 8 | 9.5 | 38.0 |

30 | 3 | 130.0 |

A = 12.5 d = `(x – 7.5)/(5)`

`barx` = A + `(sum fd )/(N)` x c

= 12.5 + `(3)/(30)` = 13

Mean deviation about the mean = `(sum f I X – barx I)/(N)` x c

= `(130 X 5)/(30)` = 21.67

**Standard Deviation of Grouped Data**

The **standard deviation** measures the spread of the data about the **mean value**. It is useful in comparing sets of data which may have the same mean but a different range. For example, the mean of the following two is the same: 15, 15, 15, 14, 16 and 2, 7, 14, 22, 30. However, the second is clearly more spread out. If a set has a low standard deviation, the values are not spread out too much…

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