In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.
| Number of mangoes | 50 - 52 | 53 - 55 | 56 - 58 | 59 - 61 | 62 - 64 |
|---|---|---|---|---|---|
| Number of boxes | 15 | 110 | 135 | 115 | 25 |
Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?
The mean number of mangoes kept in a packing box is 57.19.
Method Used: I used the Assumed Mean Method (Step-deviation method can also be used).
Reason for choosing this method: Since the class sizes are equal and frequencies are large, the Assumed Mean Method simplifies calculations by reducing the magnitude of numbers we work with. Instead of multiplying large class marks with large frequencies, we work with smaller deviations from the assumed mean.
Calculation Table:
| Class Interval | Number of boxes (fᵢ) | Class Mark (xᵢ) | dᵢ = xᵢ - 57 | fᵢ × dᵢ |
|---|---|---|---|---|
| 50 - 52 | 15 | 51 | -6 | -90 |
| 53 - 55 | 110 | 54 | -3 | -330 |
| 56 - 58 | 135 | 57 | 0 | 0 |
| 59 - 61 | 115 | 60 | 3 | 345 |
| 62 - 64 | 25 | 63 | 6 | 150 |
| Total | Σfᵢ = 400 | Σfᵢdᵢ = 75 |
Step-by-step Calculation:
Step 1: Find class marks (midpoints) for each class interval. xᵢ = (Lower limit + Upper limit)/2
Step 2: Choose assumed mean a = 57 (the class mark of the modal class with highest frequency).
Step 3: Calculate deviations dᵢ = xᵢ - a for each class.
Step 4: Find fᵢ × dᵢ for each class and calculate Σfᵢdᵢ = 75.
Step 5: Apply the formula: Mean (x̄) = a + (Σfᵢdᵢ)/(Σfᵢ) = 57 + (75/400) = 57 + 0.1875 = 57.1875 57.19
Therefore, the mean number of mangoes kept in a packing box is 57.19.
Explanation
This is Exercise 13.1, Question 5 from the NCERT textbook. The question asks students to find the mean of grouped frequency data and identify the method used. The Assumed Mean Method is appropriate here because: (1) the data is grouped with equal class intervals, (2) frequencies are large numbers making direct multiplication cumbersome, and (3) using deviations from an assumed mean simplifies arithmetic. The assumed mean 'a' is chosen as the class mark of the class with highest frequency (modal class) to minimize the magnitude of deviations.
Solution Steps
Step 1: Find class marks (midpoints) for each class: 51, 54, 57, 60, 63
Step 2: Choose assumed mean a = 57 (class mark of modal class)
Step 3: Calculate deviations dᵢ = xᵢ - 57: -6, -3, 0, 3, 6
Step 4: Calculate fᵢ × dᵢ for each class and find sum: -90 - 330 + 0 + 345 + 150 = 75
Step 5: Apply formula Mean = a + (Σfᵢdᵢ)/(Σfᵢ) = 57 + 75/400 = 57.1875 57.19