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 approximately equal, indicating a symmetric distribution.
Explanation
This question from Exercise 13.3 requires calculating all three measures of central tendency using standard formulas for grouped data. The median requires finding the median class using n/2, the mean uses assumed mean method, and the mode uses the modal class with highest frequency.
Solution Steps
Step 1: Finding Median
n = 68, n/2 = 34
Cumulative frequencies: 4, 9, 22, 42, 56, 64, 68
Median class is 125-145 (cf = 42 > 34)
Median = L + [(n/2 - cf)/f] × h
Median = 125 + [(34 - 22)/20] × 20 = 125 + 12 = 137 units
Step 2: Finding Mean (Assumed Mean Method)
Let Assumed Mean A = 135, h = 20
Class f x u=(x-A)/h fu 65-85 4 75 -3 -12 85-105 5 95 -2 -10 105-125 13 115 -1 -13 125-145 20 135 0 0 145-165 14 155 1 14 165-185 8 175 2 16 185-205 4 195 3 12 Σf = 68, Σfu = 7
Mean = A + (Σfu × h)/Σf = 135 + ()/68 = 135 + 2.06 = 137.06 units
Step 3: Finding Mode
Modal class = 125-145 (highest frequency = 20)
Mode = L + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h
Mode = 125 + [(20 - 13)/(40 - 13 - 14)] × 20
Mode = 125 + [7/13] × 20 = 125 + 10.77 = 135.77 units
Step 4: Comparison
Median = 137, Mean = 137.06, Mode = 135.77
All three measures are approximately equal (around 136-137 units). This indicates the distribution is nearly symmetric.