Question

The monthly expenditure on fruits in 200 families of a Housing Society is given below. Find the value of $x$ and also find the mode and mean expenditure on fruits.

Hint:

Using the step-deviation method for mean calculations with large class intervals saves you from dealing with large products ($f_i \cdot y_i$), keeping calculations rapid and error-free.

Answer 1

Step 1: Find $x$ from the total number of families, same as before.
The frequencies must add up to 200:
\[ 24 + 40 + 33 + 28 + x + 22 + 16 + 7 = 200 \] \[ 170 + x = 200 \implies x = 30 \]
Step 2: Find the mean using the Direct Method instead of the step-deviation method.
The Direct Method uses each class mark (mid-value) $x_i$ multiplied directly by its frequency $f_i$, without shifting to an assumed mean:
\[ \bar{X} = \frac{\sum f_i x_i}{\sum f_i} \] The class marks for the eight intervals of width 500 are $1250, 1750, 2250, 2750, 3250, 3750, 4250, 4750$, with frequencies $24, 40, 33, 28, 30, 22, 16, 7$.

Step 3: Multiply each class mark by its frequency.
\[ 24(1250) = 30000, \quad 40(1750) = 70000, \quad 33(2250) = 74250, \quad 28(2750) = 77000 \] \[ 30(3250) = 97500, \quad 22(3750) = 82500, \quad 16(4250) = 68000, \quad 7(4750) = 33250 \]
Step 4: Add these products and divide by the total frequency.
\[ \sum f_i x_i = 30000+70000+74250+77000+97500+82500+68000+33250 = 532500 \] \[ \bar{X} = \frac{532500}{200} = 2662.5 \] This is the same mean value as the step-deviation method would give, obtained here through direct multiplication instead of shifting to an assumed mean.

Step 5: Find the mode using the standard grouped-data mode formula.
The highest frequency, 40, falls in the class $1500\text{-}2000$, making this the modal class. Here $L = 1500$, $f_1 = 40$, $f_0 = 24$, $f_2 = 33$, $h = 500$:
\[ \text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h = 1500 + \left(\frac{40-24}{80-24-33}\right) \times 500 \] \[ = 1500 + \frac{16}{23} \times 500 = 1500 + \frac{8000}{23} \approx 1847.83 \]
Final Answer:
The value of $x$ is 30, the mean expenditure is Rs. 2662.50, and the mode is about Rs. 1847.83. \[ \boxed{x = 30,\ \text{Mean} = 2662.5,\ \text{Mode} \approx 1847.83} \]