Question

The following is the frequency distribution of heights of 200 male adults in a factory:

Height (in cm.)	No. of male adults
145 – 150	4
150 – 155	6
155 – 160	25
160 – 165	57
165 – 170	64
170  – 175	30
175 – 180	8
180 – 185	6

Find the central height.

Answer 1

To find the central height, we have to find Q₂.

We construct the less than cumulative frequency table as given below:

Height (in cm.)	No. of male adults (f)	Less than Cumulative frequency (c.f.)
145 – 150	4	4
150 – 155	6	10
155 – 160	25	35
160 – 165	57	92
165 – 170	64	156 ← Q2
170  – 175	30	186
175 – 180	8	194
180 – 185	6	200
Total	200

Here, N = 200

Q₂ class = class containing `((2"N")/4)^"th"` observation

∴ `(2"N")/4=(2xx200)/4` = 100

Cumulative frequency which is just greater than (or equal to) 100 is 156.

∴ Q₂ lies in the class 165 – 170

∴ L = 165, f = 64, c.f. = 92; h = 5

Q₂ = `"L"+"h"/"f"((2"N")/4 - "c.f.")`

= `165 + (5)/(64) (100 - 92)`

=  `165 + (5)/(64) xx 8`

= `165+5/8`

= 165 + 0.625

= 165.625

∴ The central height is 165.625 cm.