The weights, x kg, of 120 students in a sports college are recorded. The results are summarised in the following table.
| Weight (x kg) | x ≤ 40 | x ≤ 60 | x ≤ 65 | x ≤ 70 | x ≤ 85 | x ≤ 100 |
|---|
| Cumulative frequency | 0 | 14 | 38 | 60 | 106 | 120 |
Calculate estimates for the mean and standard deviation of the weights of the 120 students.
Solution
First, calculate the frequencies for each class interval:
- 40 < x ≤ 60: 14
- 60 < x ≤ 65: 24
- 65 < x ≤ 70: 22
- 70 < x ≤ 85: 46
- 85 < x ≤ 100: 14
Next, determine the midpoints for each class interval:
- 40 < x ≤ 60: 50
- 60 < x ≤ 65: 62.5
- 65 < x ≤ 70: 67.5
- 70 < x ≤ 85: 77.5
- 85 < x ≤ 100: 92.5
Calculate the mean:
\(\text{Mean} = \frac{0 \times 20 + 14 \times 50 + 24 \times 62.5 + 22 \times 67.5 + 46 \times 77.5 + 14 \times 92.5}{120} = \frac{8545}{120} = 71.2 \text{ kg}\)
\(Calculate the variance:\)
\(\text{Variance} = \frac{0 \times 20^2 + 14 \times 50^2 + 24 \times 62.5^2 + 22 \times 67.5^2 + 46 \times 77.5^2 + 14 \times 92.5^2}{120} - 71.2^2\)
\(= \frac{625062.5}{120} - 71.2^2 = 138.23\)
Calculate the standard deviation:
\(\text{Standard deviation} = \sqrt{138.23} = 11.8 \text{ kg}\)
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