Statistics: Mean, Median, and Mode for Grouped Data
Understanding Central Tendency Measures for Frequency Distributions
Problem Statement
The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality:
| 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 |
What is Given?
We are given a grouped frequency distribution with:
- 7 class intervals
- Frequencies for each class
- Total number of consumers (N) = 68
What We Need to Find
We need to calculate three measures of central tendency:
- Mean (Average) monthly consumption
- Median monthly consumption
- Mode (most frequent) monthly consumption
Concepts and Formulas
Mean for Grouped Data
The mean for grouped data is calculated using:
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]
Where:
- \( f_i \) = frequency of the i-th class
- \( x_i \) = midpoint of the i-th class
- \( \sum f_i \) = total frequency (N)
Median for Grouped Data
The median for grouped data is calculated using:
\[ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h \]
Where:
- L = lower boundary of the median class
- N = total frequency
- F = cumulative frequency before the median class
- f = frequency of the median class
- h = class width
Mode for Grouped Data
The mode for grouped data is calculated using:
\[ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \]
Where:
- L = lower boundary of the modal class
- f₁ = frequency of the modal class
- f₀ = frequency of the class before the modal class
- f₂ = frequency of the class after the modal class
- h = class width
Step-by-Step Calculations
Step 1: Prepare the Data Table
We need to calculate class midpoints, cumulative frequencies, and other necessary values:
| Class Interval | Frequency (f) | Midpoint (x) | f × x | Cumulative Frequency |
|---|---|---|---|---|
| 65 - 85 | 4 | 75 | 300 | 4 |
| 85 - 105 | 5 | 95 | 475 | 9 |
| 105 - 125 | 13 | 115 | 1495 | 22 |
| 125 - 145 | 20 | 135 | 2700 | 42 |
| 145 - 165 | 14 | 155 | 2170 | 56 |
| 165 - 185 | 8 | 175 | 1400 | 64 |
| 185 - 205 | 4 | 195 | 780 | 68 |
| Total | ∑f = 68 | ∑fx = 9320 |
Step 2: Calculate the Mean
Using the formula: \[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]
\[ \bar{x} = \frac{9320}{68} = 137.06 \text{ units} \]
Step 3: Calculate the Median
First, find the median class where cumulative frequency ≥ N/2 = 34
The median class is 125 - 145 (cumulative frequency = 42, which is ≥ 34)
L = 125, N = 68, F = 22, f = 20, h = 20
\[ \text{Median} = 125 + \left( \frac{34 - 22}{20} \right) \times 20 \]
\[ \text{Median} = 125 + \left( \frac{12}{20} \right) \times 20 = 125 + 12 = 137 \text{ units} \]
Step 4: Calculate the Mode
The modal class is the one with the highest frequency, which is 125 - 145 (f = 20)
L = 125, f₁ = 20, f₀ = 13, f₂ = 14, h = 20
\[ \text{Mode} = 125 + \left( \frac{20 - 13}{2 \times 20 - 13 - 14} \right) \times 20 \]
\[ \text{Mode} = 125 + \left( \frac{7}{40 - 27} \right) \times 20 = 125 + \left( \frac{7}{13} \right) \times 20 \]
\[ \text{Mode} = 125 + \frac{140}{13} = 125 + 10.77 = 135.77 \text{ units} \]
Results
Mean: 137.06 units
Median: 137 units
Mode: 135.77 units
Step 5: Comparison and Interpretation
All three measures of central tendency are relatively close to each other:
- The mean (137.06) represents the mathematical average
- The median (137) represents the middle value
- The mode (135.77) represents the most frequent consumption range
The similarity of these values suggests that the data is fairly symmetric with no significant outliers.