Notes On Analysis of Frequency Distributions - CBSE Class 11 Maths
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 ($\stackrel{_}{\text{x}}$) =      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 $\stackrel{_}{{\text{x}}_{\text{1}}}\text{}$ and σ1. Let the mean and the standard deviation of the second distribution be $\stackrel{_}{{\text{x}}_{\text{2}}}\text{}$ and σ2. CV (first distribution) = σ1/$\stackrel{_}{{\text{x}}_{\text{1}}}\text{}$ x 100 CV (second distribution) = σ2/$\stackrel{_}{{\text{x}}_{\text{2}}}\text{}$ x 100 Now, $\frac{\text{CV (first distribution)}}{\text{CV (second distribution)}}$  =    (σ1/$\stackrel{_}{{\text{x}}_{\text{1}}}\text{}$ x 100)/( σ2/$\stackrel{_}{{\text{x}}_{\text{2}}}\text{}$ x 100)   ($\stackrel{_}{{\text{x}}_{\text{1}}}\text{=}\stackrel{_}{{\text{x}}_{\text{2}}}$) = 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.

#### Summary

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 ($\stackrel{_}{\text{x}}$) =      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 $\stackrel{_}{{\text{x}}_{\text{1}}}\text{}$ and σ1. Let the mean and the standard deviation of the second distribution be $\stackrel{_}{{\text{x}}_{\text{2}}}\text{}$ and σ2. CV (first distribution) = σ1/$\stackrel{_}{{\text{x}}_{\text{1}}}\text{}$ x 100 CV (second distribution) = σ2/$\stackrel{_}{{\text{x}}_{\text{2}}}\text{}$ x 100 Now, $\frac{\text{CV (first distribution)}}{\text{CV (second distribution)}}$  =    (σ1/$\stackrel{_}{{\text{x}}_{\text{1}}}\text{}$ x 100)/( σ2/$\stackrel{_}{{\text{x}}_{\text{2}}}\text{}$ x 100)   ($\stackrel{_}{{\text{x}}_{\text{1}}}\text{=}\stackrel{_}{{\text{x}}_{\text{2}}}$) = 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.

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