 ## Measure of dispersion #### Overview

##### Measure of dispersion

The measures that describe the degree of scatteredness (spread) of a data set are called the measure of dispersion. A measure of central tendency do not explain the nature of distribution of the data, it indicates the location of the central position of the given data.

Properties of a good measure of dispersion:

1. It should be rigidly defined.
2. It should be easy to calculate and understand.
3. It should be based on all the observations.
4. It should be suitable for the further mathematical treatment.
5. It should be least affected by the fluctuation in sampling.
6. It should not be affected by extreme observations.

Absolute and Relative Measure of Dispersion:

The measure of dispersion whose unit is same as the unit of the given data is called the absolute measure of dispersion. Range, quartile deviation, mean deviation and standard deviation are the absolute measure of dispersion. Similarly, The relative measure of dispersion is obtained as the ratio of an absolute measure of dispersion to suitable average.

i.e Relative measure of dispersion = $\frac{Absolute measure of dispersion}{Average}$

Therefore, we can say that relative measure is independent of units. It is also called as the coefficient of dispersion. So, a coefficient of range, a coefficient of quartile deviation, a coefficient of mean deviation and coefficient of variation are the relative measure of dispersion.

Methods of measuring Dispersion

The following are the commonly used measures of dispersion,

1. #### Range

The range is the easiest method of measure of dispersion. It is the difference between the largest item and smallest item of the series. It is denoted by R. Mathematically,
Range () = L – S
Where,
L = largest item
S = smallest item
Coefficient of Range = $\frac{L - S}{L + S}$
2. #### Quartile Deviation

The half of the interquartile range is known as the quartile deviation. The interquartile range is the difference between the 1st and 3rd quartiles and is defined as the interquartile range. i.e Q3 – Q1. This interval contains the middles 50% of the distribution.
Quartile deviation (Q.D) =$\frac{Q_3 - Q_1}{2}$
Coefficient of quartile deviation = $\frac{Quartile deviation}{Median}$
$\frac{Q_3 – Q_1}{Q_3 + Q_1}$
3. #### Mean Deviation

It is also known as the Average deviation. It can be defined as the arithmetic mean of the absolute deviation of various items from an average either from mean or median or mode. It is denoted by M.D.
Calculation of mean deviation
a) For individual series:

Mean deviation from mean = $\frac{∑â”‚x- \overline{x}â”‚}{n}$
Mean deviation from median = $\frac{∑â”‚x- M_d|}{n}$
Mean deviation from mode = $\frac{∑â”‚x- M_o|}{n}$
Where n = total number of items.
b) For discrete or continuous series
Mean deviation from mean = $\frac{∑â”‚x- \overline{x}â”‚}{N}$/
Mean deviation from median = $\frac{∑x- M_d}{N}$
Mean deviation from mode = $\frac{∑â”‚x- M_o}{N}$
Where, N = ∑f
The relative measure of mean deviation is defined as follows:
Coefficient of M.D from mean = $\frac{M.D from mean}{Mean}$
Coefficient of M.D from median = $\frac{M.D from median}{Median}$
Coefficient of M.D from mode = $\frac{M.D from mode}{Mode}$
4. #### Standard deviation

The positive square root of the arithmetic mean of the square of the deviation of the given observation taken from their arithmetic mean is called standard deviation. It is denoted by the Greek letter $\sigma$ (Sigma).

References: