\displaystyle \textbf{1. Dispersion}
\displaystyle \text{Dispersion means scatteredness around the central value.}
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\displaystyle \textbf{2. Measures of Dispersion}
\displaystyle \text{Following are the measures of dispersion:}
\displaystyle \text{(i) Range \qquad (ii) Quartile deviation \qquad (iii) Mean deviation \qquad (iv) Standard deviation}
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\displaystyle \textbf{3. Range}
\displaystyle \text{Range is the difference between the greatest and the least values of the variable.}
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\displaystyle \textbf{4. Mean Deviation}
\displaystyle \text{Mean deviation is the arithmetic mean of the absolute values of deviations about some}
\displaystyle \text{point (mean or median or mode).}
\displaystyle \text{(i) For individual observations, we have}
\displaystyle \text{M.D.}=\frac{1}{n}\sum_{i=1}^{n}|x_i-a|,\;\text{where }a=\text{mean, median, mode}
\displaystyle \text{Also, M.D.}=a+h\left\{\frac{1}{N}\sum_{i=1}^{n}|u_i|\right\},\;\text{where }u_i=\frac{x_i-a}{h}
\displaystyle \text{(ii) For a discrete frequency distribution, we have}
\displaystyle \text{M.D.}=\frac{1}{N}\sum_{i=1}^{n}f_i|x_i-a|,\;a=\text{mean, median, mode}
\displaystyle \text{M.D.}=a+h\left\{\frac{1}{N}\sum_{i=1}^{n}f_i|u_i|\right\},\;\text{where }u_i=\frac{x_i-a}{h}
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\displaystyle \textbf{5. Standard Deviation}
\displaystyle \text{Standard deviation is the positive square root of variance.}
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\displaystyle \textbf{6. Variance}
\displaystyle \text{Variance is the arithmetic mean of the squares of deviations about mean }\overline{X}.
\displaystyle \text{(i) For individual observations, we have}
\displaystyle \text{Variance }(X)=\frac{1}{n}\sum_{i=1}^{n}(x_i-\overline{X})^2
\displaystyle \text{Also, }\mathrm{Var}(X)=\left(\frac{1}{n}\sum_{i=1}^{n}x_i^2\right)-\left(\frac{1}{n}\sum_{i=1}^{n}x_i\right)^2
\displaystyle \text{and, }\mathrm{Var}(X)=h^2\left\{\left(\frac{1}{n}\sum_{i=1}^{n}u_i^2\right)-\left(\frac{1}{n}\sum_{i=1}^{n}u_i\right)^2\right\},
\displaystyle \text{where }u_i=\frac{x_i-a}{h}
\displaystyle \text{(ii) For a discrete frequency distribution, we have}
\displaystyle \mathrm{Var}(X)=\frac{1}{N}\sum_{i=1}^{n}f_i(x_i-\overline{X})^2
\displaystyle \text{Also, }\mathrm{Var}(X)=\left(\frac{1}{N}\sum_{i=1}^{n}f_i x_i^2\right)-\left(\frac{1}{N}\sum_{i=1}^{n}f_i x_i\right)^2
\displaystyle \text{and, }\mathrm{Var}(X)=h^2\left\{\left(\frac{1}{N}\sum_{i=1}^{n}f_i u_i^2\right)-\left(\frac{1}{N}\sum_{i=1}^{n}f_i u_i\right)^2\right\}
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\displaystyle \textbf{7. Coefficient of Variation}
\displaystyle \text{In order to compare two or more frequency distributions we compare their coefficients of}
\displaystyle \text{variation. The coefficient of variation is defined as}
\displaystyle \mathrm{C.V.}=\frac{\sigma}{\overline{X}}\times100
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\displaystyle \textbf{8. Comparison Using Coefficient of Variation}
\displaystyle \text{The distribution having greater coefficient of variation has more variability around the}
\displaystyle \text{central value than the distribution having smaller value of the coefficient of variation.}
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