Question 3
The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure:
| Expenditure (in Rs.) | Number of families |
|---|---|
| 1000-1500 | 24 |
| 1500-2000 | 40 |
| 2000- 2500 | 33 |
| 2500-3000 | 28 |
| 3000-3500 | 30 |
| 3500- 4000 | 22 |
| 4000-4500 | 16 |
| 4500-5000 | 7 |
Solution:
Given data:
Modal class = 1500-2000,
l = 1500,
Frequencies:
$f_m$ = 40 $f_{1}$ = 24, $f_{2}$ = 33 and
h = 500
Mode formula:
| Mode = $l + [\dfrac{(f_m – f_{1})}{(2f_m – f_{1} – f_{2})}] × h$ |
Substitute the values in the formula, we get
Mode = 1500 + $[\dfrac{(40 – 24)}{(80 – 24 – 33)}]$ × 500
Mode = 1500 + $[\dfrac{(16 × 500)}{23}]$
Mode = 1500 + $\dfrac{8000}{23}$ = 1500 + 347.83
Therefore, modal monthly expenditure of the families = Rupees 1847.83
Calculation for mean:
First find the midpoint using the formula, $x_{i} =\dfrac{(upper \: limit + lower \: limit)}{2}$
Let us assume a mean, (a) be 2750.
| Class Interval | $f_i$ | $x_i$ | $d_{i} = x_{i} – a$ | $u_{i} =\dfrac{ d_{i}}{h}$ | $f_{i}x_{i}$ |
|---|---|---|---|---|---|
| 1000-1500 | 24 | 1250 | -1500 | -3 | -72 |
| 1500-2000 | 40 | 1750 | -1000 | -2 | -80 |
| 2000-2500 | 33 | 2250 | -500 | -1 | -33 |
| 2500-3000 | 28 | 2750 = a | 0 | 0 | 0 |
| 3000-3500 | 30 | 3250 | 500 | 1 | 30 |
| 3500-4000 | 22 | 3750 | 1000 | 2 | 44 |
| 4000-4500 | 16 | 4250 | 1500 | 3 | 48 |
| 4500-5000 | 7 | 4750 | 2000 | 4 | 28 |
| $f_i$=200 | $f_iu_i$ = -35 |
The formula to calculate the mean,
| Mean = $\overline{x} = a +(\dfrac{∑f_iu_i}{∑f_i}) × h$ |
Substitute the values in the given formula
= 2750 + $(\dfrac{-35}{200})$ × 500
= 2750 – 87.50
= 2662.50
So, the mean monthly expenditure of the families = Rs. 2662.50