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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.
Solution
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.
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