The daily expenditure of 100 families are given below. Calculate f1 and f2 if the mean daily expenditure is Rs.188.
Expenditure (in Rs.) | 140 - 160 | 160 - 180 | 180 - 200 | 200 - 220 | 220 - 240 |
Number of families | 5 | 25 | f1 | f2 | 5 |
For equal class intervals, we will solve by finding mid points of these classes using direct method.
EXPENDITURE (Rs.) | MID - POINT(xi) | NUMBER OF FAMILIES(fi) | fixi |
140 - 160 | 150 | 5 | 750 |
160 - 180 | 170 | 25 | 4250 |
180 - 200 | 190 | f1 | 190f1 |
200 - 220 | 210 | f2 | 210f2 |
220 - 240 | 230 | 5 | 1150 |
TOTAL | 35 + f1 + f2 | 6150 + 190f1 + 210f2 |
We have got
Σfi = 35 + f1 + f2 and Σfixi = 6150 + 190f1 + 210f2
∵ mean is given by
⇒ (∵ given: mean of pocket allowance is 188)
⇒ 6580 + 188f1 + 188f2 = 6150 + 190f1 + 210f2
⇒ 190f1 – 188f1 + 210f2 – 188f2 = 6580 – 6150
⇒ 2f1 + 22f2 = 430 …(i)
As given in the question, frequency(Σfi) = 100
And as calculated by us, frequency (Σfi) = 35 + f1 + f2
Comparing them, we get
35 + f1 + f2 = 100
⇒ f1 + f2 = 65 …(ii)
We will now solve equations (i) and (ii), multiply eq.(ii) by 2 and then subtracting it from eq.(i), we get
(2f1 + 22f2) – [2(f1 + f2)] = 430 – 130
⇒ 2f1 + 22f2 – 2f1 – 2f2 = 300
⇒ 20 f2 = 300
⇒ f2 = 15
Substitute f2 = 15 in equation (ii),
f1 + 15 = 65
⇒ f1 = 65 – 15
⇒ f1 = 50
Thus, f1 = 50 and f2 = 15.