Find the arithmetic mean of each of the following frequency distr… - Vidyadip

Question

Find the arithmetic mean of each of the following frequency distributions using step-deviation method:

Age (in years)	$18-24$	$24-30$	$30-36$	$36-42$	$42-48$	$48-54$
Number of workers	$6$	$8$	$12$	$8$	$4$	$2$

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Answer

Class interval	Frequency $f_i$	Mid-value $x_i$	$\text{u}_\text{i}=\frac{\text{x}_\text{i}-\text{A}}{\text{h}}=\frac{\text{x}_\text{i}-203.5}{1}$	$f_i× u_i$
$18-24$	$6$	$21$	$-3$	$-18$
$24-30$	$8$	$27$	$-2$	$-16$
$30-36$	$12$	$33$	$-1$	$-12$
$36-42$	$8$	$39 =A$	$0$	$0$
$42-48$	$4$	$45$	$1$	$4$
$48-54$	$2$	$51$	$2$	$4$
	$\sum\text{f}_\text{i}=40$			$\sum\text{f}_\text{i}\text{u}_\text{i}=-38$

Thus, $\text{A}=39,\ \text{h}=6,\ \sum\text{f}_\text{i}=40$ and $\sum\text{f}_\text{i}\text{u}_\text{i}=-38$
Mean $=\text{A}+\Big\{\text{h}\times\frac{\sum\text{f}_\text{i}\text{u}_\text{i}}{\sum\text{f}_\text{i}}\Big\}$
$=39+\Big\{16\times\frac{-38}{40}\Big\}$
$=39-5.7$
$=33.3$

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