To calculate the variance of the scores, we'll need to follow these steps:
1. Determine the mid-point of each interval:
The given score intervals are [tex]\(16-20\)[/tex], [tex]\(21-25\)[/tex], and [tex]\(26-30\)[/tex]. We find their mid-points as follows:
- Mid-point of [tex]\(16-20\)[/tex] is [tex]\(\frac{16+20}{2} = 18\)[/tex],
- Mid-point of [tex]\(21-25\)[/tex] is [tex]\(\frac{21+25}{2} = 23\)[/tex],
- Mid-point of [tex]\(26-30\)[/tex] is [tex]\(\frac{26+30}{2} = 28\)[/tex].
2. Calculate the mean score:
The number of students in each interval is given as 6, 3, and 1 respectively. Thus, the total number of students is 10.
The mean score is calculated by the formula:
[tex]\[
\text{Mean} = \frac{\sum (\text{mid-point} \times \text{number of students})}{\text{total number of students}}
\][/tex]
[tex]\[
= \frac{(18 \times 6) + (23 \times 3) + (28 \times 1)}{10} = \frac{108 + 69 + 28}{10} = \frac{205}{10} = 20.5
\][/tex]
3. Calculate the variance:
Variance is calculated by the formula:
[tex]\[
\text{Variance} = \frac{\sum (\text{number of students} \times (\text{mid-point} - \text{mean})^2)}{\text{total number of students}}
\][/tex]
Substituting the values, we get:
[tex]\[
\text{Variance} = \frac{6 \times (18 - 20.5)^2 + 3 \times (23 - 20.5)^2 + 1 \times (28 - 20.5)^2}{10}
\][/tex]
[tex]\[
= \frac{6 \times (-2.5)^2 + 3 \times 2.5^2 + 1 \times 7.5^2}{10}
\][/tex]
[tex]\[
= \frac{6 \times 6.25 + 3 \times 6.25 + 1 \times 56.25}{10}
\][/tex]
[tex]\[
= \frac{37.5 + 18.75 + 56.25}{10} = \frac{112.5}{10} = 11.25
\][/tex]
Thus, the variance of the scores is 11.25. Therefore, the correct answer is not given directly in the multiple-choice options. Adding the results:
A. 1125 is not correct.
B. 7.5 is not correct.
C. 3.35 is not correct.
D. 273 is not correct.
So, there must have been an error in the choices provided in context with the actual variance calculation result 11.25.
