Calculate the mean deviation using mean and Standard Deviation fo… - Vidyadip

Question

Calculate the mean deviation using mean and Standard Deviation for the following distribution.

Classes	Frequencies
20-40	3
40-80	6
80-100	20
100-120	12
120-140	9
	50

✓

Answer

Classes	Frequency	m	A = 90 d = X - A	$\|d\|$	$f\|d\|$	$\text{d}'=\frac{\text{X}-\text{A}}{\text{i}}$	$fd'$	$d'^2$	$fd'^2$
20–40	3	30	-60	60	180	-6	-18	36	108
40–80	6	60	-30	30	180	-3	-18	9	54
80–100	20	90	0	0	0	0	0	0	0
100–120	12	110	20	20	240	2	24	4	48
120–140	9	130	40	40	360	4	36	16	144
	$\sum\text{f}=50$				$\sum\text{f}\|\text{d}\|=960$		$\sum\text{fd}'=24$		$\sum\text{fd}'^2=354$

$\overline{\text{X}}=\text{A}+\frac{\sum\text{fd}'}{\sum\text{f}}\times\text{i}$
$=90+\frac{24}{50}\times10$
$=94.8$
$\text{MD}_\bar{\text{x}}=\frac{\sum\text{f}|\text{d}|}{\sum\text{f}}$
$=\frac{960}{50}=19.2$
$\sigma=\sqrt{\frac{\sum\text{fd}'^2}{\sum\text{f}}-\bigg(\frac{\sum\text{fd}'}{\sum\text{f}}\bigg)^2}\times\text{i}$
$=\sqrt{\frac{354}{50}-\Big(\frac{24}{50}\Big)^2}\times10$
$=\sqrt{\frac{354}{50}-\frac{576}{2500}}\times10$
$=\sqrt{6.85}\times10$
$=26.17$
Mean Deviation from mean,
$\text{MD}_\sigma=\frac{\sum\text{f}|\text{D}|}{\text{N}}$
$=\frac{998.4}{50}=19.968$

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