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

Classes Frequency m A = 90 d = X - A |d| f|d| d'= X-A 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 =50 |d|=960 '=24 '^2=354 X =A+ ' =90+24 50 10 =94.8 MD_ x = |d| =960 50 =19.2 = '^2 - ( ' )^2 = 354 50 - (24 50 )^2 10 = 354 50 -576 2500 10 =6.85 10 =26.17 Mean Deviation from mean, MD_ = |D| N =998.4 50 =19.968

Calculate the mean deviation using mean and Standard Deviation fo…

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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

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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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