Range, Quartile Deviation, and Standard Deviation || Measures of Dispersion || Bcis Notes

Range, Quartile Deviation, and Standard Deviation

1. Range

The Range is defined as the difference between the largest item and the smallest item in a set of observations. If L = Largest item and S = Smallest item and their range are denoted by ‘R’ and given by:

R = L – S

Coefficient of Range

The range is of absolute measure. The relative measures corresponding to the range is known as the coefficient of range and is given by:

 

2. Quartile Deviation or Semi-interquartile range

The measures of dispersion based on upper quartile and lower quartile is called quartile deviation.

The difference between the upper quartile and lower quartile is called the inter-quartile range.

Half of the inter-quartile range is called the quartile deviation.

 

Coefficient of Quartile Deviation

The relative measure of dispersion based on the upper quartile and lower quartile is called the coefficient of quartile deviation.

 

 

3. Standard Deviation

It is defined as a positive square root of the mean of the square of deviation of items taken from their arithmetic. It is denoted by:

Individual Series

Direct Method

Shortcut Method

where,

d = X – a

a = Assumed Mean

Discrete Series

Direct Method

Shortcut Method

Continuous Series

Step-deviation Method

Coefficient of Standard Deviation

The relative measures of dispersion based on the standard deviation is called the coefficient of standard deviation.

Property of Standard Deviation

1. Standard Deviation is independent of the change of origin but not of the scale.

Change of origin = (+/-)

Change of scale=

2. Let n1 and n2 be the sizes and   be the mean and σ1 and σ2 be the Standard Deviation of two series and their combined                 Standard Deviation is denoted by:

Here,

 

4. Variance

The square of standard deviation is called variance. It is denoted by σ² or µ.

Coefficient of Variance (C.V)

The relative measure of Standard Deviation is multiplied by 100 is called Coefficient of Variance.

Two series can be compared with the help of the Coefficient of Variance for their variability. Less the value of Coefficient of Variance more will be consistent or uniform and vice versa.

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