The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.
| Monthly consumption (in units) | Number of consumers |
|---|---|
| 65 - 85 | 4 |
| 85 - 105 | 5 |
| 105 - 125 | 13 |
| 125 - 145 | 20 |
| 145 - 165 | 14 |
| 165 - 185 | 8 |
| 185 - 205 | 4 |
Median = 137 units, Mean = 137.06 units, Mode = 135.77 units. All three measures of central tendency are very close to each other, indicating an approximately symmetric distribution.
Explanation
This question from Exercise 13.3 requires students to calculate all three measures of central tendency for grouped data. The median requires finding the median class using cumulative frequency, the mean requires calculating class marks and using the formula, and the mode requires identifying the modal class with the highest frequency. The comparison shows that when the distribution is nearly symmetric, all three measures are approximately equal.
Solution Steps
Step 1: Calculate cumulative frequency to find median class. Total consumers (n) = 68, so n/2 = 34. Cumulative frequencies: 4, 9, 22, 42, 56, 64, 68. Median class is 125-145 (cf = 42 34).
Step 2: Calculate Median using formula: Median = L + [(n/2 - cf)/f] × h = 125 + [(34 - 22)/20] × 20 = 125 + 12 = 137 units
Step 3: Calculate Mean using class marks. Class marks (x): 75, 95, 115, 135, 155, 175, 195. Calculate f × x for each class: 300, 475, 1495, 2700, 2170, 1400, 780. Sum = 9320. Mean = 9320/68 = 137.06 units
Step 4: Calculate Mode. Modal class = 125-145 (highest frequency = 20). Using formula: Mode = L + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h = 125 + [(20-13)/(40-13-14)] × 20 = 125 + (7/13) × 20 = 135.77 units
Step 5: Compare all three measures. Median Mean Mode (137, 137.06, 135.77). The values are very close, indicating the distribution is approximately symmetric.