Question

If the mode of the following distribution is 55, then find the value of x. Hence, find the mean.

Class interval	0 – 15	15 – 30	30 – 45	45 – 60	60 – 75	75 – 90
Frequency	10	7	x	15	10	12

Answer 1

Class interval	Frequency
0 – 15	10
15 – 30	7
30 – 45	x → f0
45 – 60	15 → f1
60 – 75	10 → f2
75 – 90	12

Mode of the frequency distribution = 55

∴ Modal class is 45 – 60.

Mode = `l + ((f_1 - f_0)/(2f_1 - f_0 - f_2)) xx h`

Where l = Lower limit of modal class = 45

h = Class interval = 15 – 0 = 15

f₁ = Frequency of the modal class = 15

f₀ = Frequency of the class before modal class = x

f₂ = Frequency of the class after modal class = 10

Now, Mode = `l + ((f_1 - f_0)/(2f_1 - f_0 - f_2)) xx h`

`55 = 45 + (15 - x)/(2(15) - x - 10) xx 15`

`55 − 45 = (15 - x)/(30 - x - 10) xx 15`

`10 = (15 - x)/(20 - x) xx 15`

10(20 – x) = (15 – x) × 15

200 – 10x = 225 – 15x

15x – 10x = 225 – 200

5x = 25

x = `25/5`

x = 5

Class interval	Frequency (fi)	Class mark (xi)	fixi
0 – 15	10	7.5	75
15 – 30	7	22.5	157.5
30 – 45	5	37.5	187.5
45 – 60	15	52.5	787.5
60 – 75	10	67.5	675
75 – 90	12	82.5	990
	∑fi = 59		∑fixi = 2872.5

Mean `(barx)` = `(sumf_ix_i)/(sumf_i)`

= `2872.5/59`

= 48.68