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 |
Modal monthly expenditure = ₹ 1847.83 Mean monthly expenditure = ₹ 2662.50
Explanation
This question requires finding both mode and mean from grouped frequency data. The mode is found using the formula Mode = l + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h, where the modal class is identified by the highest frequency. The mean is calculated using the step-deviation method (assumed mean method) which simplifies calculations when class sizes are uniform.
Solution Steps
Step 1: Identify the modal class. The maximum frequency is 40, corresponding to class 1500-2000. So, modal class = 1500-2000.
Step 2: Apply the mode formula. l = 1500, f₁ = 40, f₀ = 24, f₂ = 33, h = 500. Mode = 1500 + [(40-24)/(-24-33)] × 500 = 1500 + [16/23] × 500 = 1500 + 347.83 = ₹ 1847.83
Step 3: For mean, use step-deviation method. Take assumed mean A = 3250 (class mark of 3000-3500), class size h = 500.
Step 4: Calculate class marks (x): 1250, 1750, 2250, 2750, 3250, 3750, 4250, 4750. Calculate u = (x - A)/h: -4, -3, -2, -1, 0, 1, 2, 3.
Step 5: Calculate f × u for each class: -96, -120, -66, -28, 0, 22, 32, 21. Sum (Σfu) = -235.
Step 6: Apply mean formula: Mean = A + (Σfu/N) × h = 3250 + (-235/200) × 500 = 3250 - 587.50 = ₹ 2662.50