Question

In the frequency distribution of families given below, the number of families corresponding to expenditure group 2000 - 4000 is missing from the table. However value of 25^th percentile is 2880. Find the missing frequency.

Weekly Expenditure (₹1000)	0 – 2	2 – 4	4 – 6	6 – 8	8 – 10
No. of families	14	?	39	7	15

Answer 1

Let x be the missing frequency of expenditure group 2000 – 4000.
We construct the less than cumulative frequency table as given below:

Weekly Expenditure	No. of families (f)	Less than cumulative frequency (c.f.)
0 – 2000	14	14
2000 – 4000	x	14 + x ← P25
4000 – 6000	39	53 + x
6000 – 8000	7	60 + x
8000 – 10000	15	75 + x
Total	75 + x

Here, N = 75 + x
Given, P₂₅ = 2880
∴ P₂₅ lies in the class 2000 – 4000.
∴ L = 2000, h = 2000, f = x, c.f. = 14

∴ P₂₅ = `"L"+"h"/"f"((25"N")/100-"c.f.")`

∴ 2880 = `2000+2000/"x"((75+"x")/4-14)`

∴ 2880 – 2000 =`2000/"x"((75+"x"-56)/4)`

∴ 880x = 500(x + 19)
∴ 880x = 500x + 9500
∴ 880x – 500x = 9500
∴ 380x = 9500

∴ x = `9500/380` = 25
∴ 25 is the missing frequency of the expenditure group 2000 – 4000.