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 ₹) | 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 |
(Mean monthly expenditure) ₹ 2662.5
(Modal monthly expenditure) ₹ 1847.83 (approximately)
Explanation
This question requires finding both mode and mean from grouped frequency distribution. The mode is found using the formula for grouped data, identifying the modal class first. The mean is calculated using the step-deviation method which simplifies calculations when class sizes are equal.
Solution Steps
Step 1: Identify the modal class. The maximum frequency is 40, which lies in the class 1500 - 2000. So, modal class = 1500 - 2000.
Step 2: Apply the mode formula: Mode = l + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h, where l = 1500, f₁ = 40, f₀ = 24, f₂ = 33, h = 500.
Step 3: Calculate Mode = 1500 + [(40-24)/(-24-33)] × 500 = 1500 + [16/23] × 500 = 1500 + 347.83 = ₹ 1847.83
Step 4: For mean, use step-deviation method. Take assumed mean a = 2750 (mid-point of class 2500-3000), class size h = 500.
Step 5: Calculate ui = (xi - a)/h for each class. Find fi × ui for each class: -72, -80, -33, 0, 30, 44, 48, 28.
Step 6: Sum of fi × ui = -35. Apply formula: Mean = a + (Σfiui/Σf) × h = 2750 + (-35/200) × 500 = 2750 - 87.5 = ₹ 2662.5