Find the mean deviation about the mean for the data.
$4, 7, 8, 9, 10, 12, 13, 17$

1. Dispersion:

Dispersion means scattered ness around the central value.

2. Measures of dispersion:

(i) Range:

Range is the difference between the greatest and the least values of the variable.

Range$=X_{\max }-X_{\min }$

(ii) Quartile deviation:

Quartile deviation is the product of half of the difference between the third and first quartiles.

Quartile deviation$=\frac{Q_3-Q_1}{2}$

(iii) Mean deviation:

Mean deviation is the arithmetic mean of the absolute values of deviations about some point (mean or median or mode).

(a) Mean deviation for ungrouped data, we have

$\mathrm{M}.\mathrm{D}.\left(a\right)=\frac{1}{n}\sum _{i=1}^n\left|x_i-a\right|$, where $a=$mean or median or mode

(b) For a discrete frequency distribution, we have

$\mathrm{M}.\mathrm{D}.\left(a\right)=\frac{1}{N}\mathrm{ }\sum _{i=1}^n{f}_i\left|x_i-a\right|$, where $a=$mean or median or mode

(iv) Standard deviation$\left(\mathit{\sigma }\right)$:

Standard deviation is the positive square root of variance.

3. Variance$\left({\mathit{\sigma }}^{\mathbf{2}}\right)$:

Variance is the arithmetic mean of the squares of deviations about mean.

(i) Variance for ungrouped data, we have

(a) ${\sigma }^2=\frac{1}{n}\mathrm{ }\sum _{\mathrm{i}=1}^n{\left(x_i-\overline{x}\right)}^2$

(b) ${\sigma }^2=\left(\frac{1}{n}\mathrm{ }\stackrel{n}{\sum _{i=1}}\mathrm{ }{x}_i^2\right)-{\left(\frac{1}{n}\mathrm{ }\stackrel{n}{\sum _{i=1}}\mathrm{ }{x}_i\right)}^2$

(c) ${\sigma }^2=h^2\left\{\left(\frac{1}{n}\mathrm{ }\stackrel{n}{\sum _{i=1}}\mathrm{ }{u}_i^2\right)-{\left(\frac{1}{n}\mathrm{ }\stackrel{n}{\sum _{i=1}}\mathrm{ }{u}_i\right)}^2\right\},$ where $u_i=\frac{x-a}{h}$

(ii) For a discrete frequency distribution, we have

(a) ${\sigma }^2=\frac{1}{N}\mathrm{ }\sum _{i=1}^n\mathrm{ }{f}_i{\left(x_i-\overline{x}\right)}^2\mathrm{ }$

(b) ${\sigma }^2=\left(\frac{1}{N}\mathrm{ }\stackrel{n}{\sum _{i=1}}\mathrm{ }{f}_i{x}_i^2\right)-{\left(\frac{1}{N}\mathrm{ }\stackrel{n}{\sum _{i=1}}\mathrm{ }{f}_i{x}_i\right)}^2$

4. In order to compare two or more frequency distributions we compare their coefficients of variations. The coefficient of variation represents the ratio of standard deviation to mean. It is defined as

$C.V.=\frac{\sigma }{\overline{x}}\times 100$

5. The distribution having a greater coefficient of variation has more variability around the central value than the distribution having a smaller value of the coefficient of variation.