Analysis of Frequency Distributions

Mean deviation, variance and standard deviation are the measures of dispersion.

The measure of dispersion, which is independent of the units, is called the coefficient of variation (CV).

The coefficient of variation is defined as the standard deviation divided by the mean, and is expressed as a percentage.

Coefficient of variation (CV) = σ/$\stackrel{\_}{\text{x}}$  x 100, where $\stackrel{\_}{\text{x}}$  ≠ 0.

The data in a series with greater CV is said to be more variable.

The data in a series with less CV is said to be more consistent.

Ex: Find the coefficients of variation for the given data.

Name of the Student	Height (in cm)	Weight (in kg)
Ajay	154	48
Kiran	158	49
Vishal	162	50
Mohan	166	51
Shiva	170	52

Mean ($\stackrel{\_}{\text{x}}$) =    $\stackrel{\_}{{\text{x}}_{\text{1}}}\text{= 162 }\stackrel{\_}{{\text{x}}_{\text{2}}}\text{= 50}$ 

Variance (σ²) = σ₁² = 32 σ₂² = 2

CV (heights) = √32/162 x 100 CV (weights) = √2/50 x 100

= 5.66/162 x 100 = 1.41/50 x 100

= 3.5 = 2.82

The CV of the heights is more than the CV of the weights. This implies that the dispersion is more in case of the heights.

Compare two frequency distributions with the same mean:

Let the mean and the standard deviation of the first distribution be $\stackrel{\_}{{\text{x}}_{\text{1}}}$ and σ₁.

Let the mean and the standard deviation of the second distribution be $\stackrel{\_}{{\text{x}}_{\text{2}}}$ and σ₂.

CV (first distribution) = σ₁/$\stackrel{\_}{{\text{x}}_{\text{1}}}$ x 100

CV (second distribution) = σ₂/$\stackrel{\_}{{\text{x}}_{\text{2}}}$ x 100

Now, $\frac{\text{CV (first distribution)}}{\text{CV (second distribution)}}$  =    (σ₁/$\stackrel{\_}{{\text{x}}_{\text{1}}}$ x 100)/( σ₂/$\stackrel{\_}{{\text{x}}_{\text{2}}}$ x 100)   ($$ \because \; $$$\stackrel{\_}{{\text{x}}_{\text{1}}}\text{=}\stackrel{\_}{{\text{x}}_{\text{2}}}$$$ $$) = σ₁/σ₂

The variability of the distributions with the same mean depends on the standard deviation or the variance of the distribution.

The series with the greater standard deviation or variance is more variable. In other words, the data is spread more.

The series with the less standard deviation is more consistent.