Calculate mean deviation about mean for the following data:

Age in years	$1-5$	$6-10$	$11-15$	$16-20$	$21-25$	$26-30$	$31-35$	$36-40$	$41-45$
Number of persons	$7$	$10$	$16$	$32$	$24$	$18$	$10$	$5$	$1$

1. Absolute measure of dispersion:

(i) Range:

The range is the difference between the largest and smallest observations.

(ii) Variance:

${\sigma }^2=\frac{\sum _{}^{}{\left(X-\mu \right)}^2}{N}$

(iii) Standard Deviation:

$\mathrm{S}.\mathrm{D}=\sqrt{\mathrm{\sigma }}$

(iv) Quartiles and Quartile Deviation:

The quartile deviation is half of the distance between the third and the first quartile.

(v) Mean:

$\mu =\frac{\text{Sum of all observations}}{\text{Number of observations}}$

(vi) Mean deviation:

The arithmetic mean of the absolute deviations of the observations from a measure of central tendency is known as the mean deviation.

2. Relative measure of dispersion:

(i) Coefficient of range:

Coefficient of range $=\frac{X_{\max }-X_{\mathit{min}}}{X_{\max }+X_{min}}$

(ii) Coefficient of variation:

Coefficient of variation $=\frac{\sigma }{\mu }$

(iii) Coefficient of standard deviation:

Coefficient of standard deviation $=\frac{\sqrt{\sigma }}{\mu }$

(iv) Coefficient of quartile deviation:

Coefficient of quartile deviation $=\frac{Q_3-Q_1}{Q_3+Q_1}$

(v) Coefficient of mean deviation:

Coefficient of mean deviation $=\frac{\text{Mean deviation}}{\text{Average}}$

3. Moments:

(i) Raw Moments:

(ii) Central Moments:

4. Measures of skewness:

(i) Karl-Pearson coefficient of skewness:

Karl- Pearson coefficient of skewness $=\frac{\text{ Mean }-\text{ Mode }}{\text{ S.D }}$

(ii) Bowley's coefficient of skewness:

Bowley's coefficient of skewness $=\frac{Q_3+Q_1-2Q_2}{Q_3-Q_1}$

(iii) Coefficient of skewness based on moments:

${\beta }_1=\frac{{\mu }_3^2}{{\mu }_2^3}$

5. Measures of Kurtosis:

${\beta }_2=\frac{{\mu }_4}{{\mu }_2^2}$