To analyze the monthly electricity consumption data, we will calculate the median, mean, and mode, and then compare these measures of central tendency.
Data Overview
The frequency distribution of monthly consumption is as follows:
- 65-85: 4 consumers
- 85-105: 5 consumers
- 105-125: 13 consumers
- 125-145: 20 consumers
- 145-165: 14 consumers
- 165-185: 8 consumers
- 185-205: 4 consumers
Calculating the Median
The median is the middle value when the data is ordered. First, we find the cumulative frequency:
- 65-85: 4
- 85-105: 9 (4+5)
- 105-125: 22 (9+13)
- 125-145: 42 (22+20)
- 145-165: 56 (42+14)
- 165-185: 64 (56+8)
- 185-205: 68 (64+4)
Since there are 68 consumers, the median position is at (68/2) = 34. The median falls in the 125-145 range, where the cumulative frequency reaches 42. We can use the formula:
Median = L + [(N/2 - CF) / f] * c
Where:
- L = lower boundary of the median class (125)
- N = total number of consumers (68)
- CF = cumulative frequency of the class before the median class (22)
- f = frequency of the median class (20)
- c = class width (20)
Substituting the values:
Median = 125 + [(34 - 22) / 20] * 20 = 125 + 12 = 137
Calculating the Mean
The mean is calculated using the formula:
Mean = Σ(f * x) / N
Where:
- f = frequency
- x = midpoint of each class
Calculating midpoints:
- 65-85: 75
- 85-105: 95
- 105-125: 115
- 125-145: 135
- 145-165: 155
- 165-185: 175
- 185-205: 195
Now, calculate Σ(f * x):
- 4 * 75 = 300
- 5 * 95 = 475
- 13 * 115 = 1495
- 20 * 135 = 2700
- 14 * 155 = 2170
- 8 * 175 = 1400
- 4 * 195 = 780
Summing these values gives:
Σ(f * x) = 300 + 475 + 1495 + 2700 + 2170 + 1400 + 780 = 10920
Now, calculate the mean:
Mean = 10920 / 68 ≈ 160
Finding the Mode
The mode is the class with the highest frequency, which is 125-145 with 20 consumers. Thus, the mode is:
Mode = 125-145
Comparison of Measures
In summary:
- Median: 137
- Mean: 160
- Mode: 125-145
The mean is higher than both the median and mode, indicating a right-skewed distribution. The median is a better measure of central tendency in this case, as it is less affected by extreme values.
