# Measures of Dispersion, Properties of Variance and Standard Deviation, Analysis of Frequency Distribution with Equal Mean

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## Properties of Variance and Standard Deviation:

Property I:

The variance is independent of change of origin.

Property II:

The variance is not independent of the change of scale.

Property III:

Prove that the standard deviation is the least possible root mean square deviation.

Property IV:

The standard deviations of two sets containing , and numbers are and respectively being measured from their respective means and . If the two sets are grouped together as one of numbers, then the standard deviation of this set, measured from its mean m is given by

## Analysis of Frequency Distribution with Equal Mean:

The variability of two series with same mean can be compared when the measures of variation are absolute and are free of units. For this, coefficient of variation (C.V.) is obtained which is defined as

C.V.

Where and are standard deviation and mean of the data. The coefficients of variation are compared to compare the variability of two series. The series with greater C.V. is said to be more variable than the other. The series having less C.V. is said to be more consistent than the other.

For series with same means, we can have

C.V. (1st distribution)

C.V. (2nd distribution)

Where, are standard deviation of the 1st and 2nd distribution respectively, is the equal mean of the distributions.

From (1) and (2), we can conclude that two C.V.’s can be compared on the basis of the values and only.

Example:

Which of the following series or is more consistent?

Solution:

From the given data we have following table

C.V.

C.V.

Clearly C.V C.V. Series is more consistent.