The arithmetic mean of the following frequency distribution is 53. Find the value of x.

Class 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100

Frequency 12 15 32 x 13

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Solution

Class Mid-Values(x_i) Frequency (f_i) f_i x_i
0−20 10 12 120
20−40 30 15 450
40−60 50 32 1600
60−80 70 x 70x
80−100 90 13 1170
$\sum f_{i}$ =72+x $\sum f_{i} x_{i}$ =3340+70x

Given,
Mean of the given frequency distribution = 53
Now, $M e a n = \frac{\sum f_{i} x_{i}}{\sum f_{i}} \Rightarrow 53 = \frac{3340 + 70 x}{72 + x} \Rightarrow 53 (72 + x) = 3340 + 70 x \Rightarrow 53 x - 70 x = 3340 - 3816 \Rightarrow 476 = 17 x \Rightarrow x = 28$
Thus, the value of x is 28.

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120

Class	0-20	20-40	40-60	60-80	80-100	100-120	120-140
Frequency	12	18	15	25	26	15	9

Class	0-20	20-40	40-60	60-80	80-100	100-120
Frequency	7	f₁	12	f₂	8	5

Class	0 – 20	20 – 40	40 – 60	60 – 80	80 – 100
Frequency	12	15	32	x	13

Classes	0−20	20−40	40−60	60−80	80−100
Frequency	17	28	32	p	19

Class :	0−20	20−40	40−60	60−80	80−100
Frequency :	7	p	10	9	13