Following information pertains to the daily income of 150 families. Calculate the arithmetic mean. Income (in Rs) Number of families More than 75 150 ,, 85 140 ,, 95 115 ,, 105 95 ,, 115 70 ,, 125 60 ,, 135 40 ,, 145 25
To calculate the arithmetic mean, the given 'more than' cumulative frequency distribution must first be converted into a simple frequency distribution. The class interval is determined by the difference between the limits (e.g., 85 - 75 = 10).
Table: Calculation of Arithmetic Mean
| Class Interval (Income) | Frequency (f) | Mid-point (X) | f × X |
|---|---|---|---|
| 75 – 85 | 10 | 80 | 800 |
| 85 – 95 | 25 | 90 | 2,250 |
| 95 – 105 | 20 | 100 | 2,000 |
| 105 – 115 | 25 | 110 | 2,750 |
| 115 – 125 | 10 | 120 | 1,200 |
| 125 – 135 | 20 | 130 | 2,600 |
| 135 – 145 | 15 | 140 | 2,100 |
| 145 – 155 | 25 | 150 | 3,750 |
| Total | N = 150 | ∑fX = 17,450 |
Calculation: Using the formula for Arithmetic Mean:
Substituting the values:
Arithmetic Mean = Rs 116.3
Explanation
The question provides a cumulative frequency series ('more than' type). The first step is to convert this into a normal frequency distribution by subtracting successive cumulative frequencies (e.g., frequency for 75-85 is 150 - 140 = 10). The class size is 10. The arithmetic mean is calculated using the direct method where the mean is the sum of the product of frequencies and mid-points divided by the total number of observations. The calculated mean of Rs 116.3 matches the answer key provided in the context.
Solution Steps
Step 1: Convert cumulative 'more than' frequencies into simple frequencies. For the first class (75-85), frequency = 150 - 140 = 10; for the second (85-95), frequency = 140 - 115 = 25, and so on.
Step 2: Determine class mid-points (X) for each interval (e.g., for 75-85, X = 80).
Step 3: Calculate the product of frequency and mid-point (f × X) for each class and sum them to get ∑fX = 17,450.
Step 4: Apply the arithmetic mean formula: .