To find the coefficient of variation from the given data, we can follow these steps:
1. Calculate the midpoints for each class interval:
- For the interval [tex]\(5-15\)[/tex], the midpoint is [tex]\((5 + 15) / 2 = 10\)[/tex].
- For the interval [tex]\(15-25\)[/tex], the midpoint is [tex]\((15 + 25) / 2 = 20\)[/tex].
- For the interval [tex]\(25-35\)[/tex], the midpoint is [tex]\((25 + 35) / 2 = 30\)[/tex].
- For the interval [tex]\(35-45\)[/tex], the midpoint is [tex]\((35 + 45) / 2 = 40\)[/tex].
- For the interval [tex]\(45-55\)[/tex], the midpoint is [tex]\((45 + 55) / 2 = 50\)[/tex].
So, the midpoints are [tex]\(10.0, 20.0, 30.0, 40.0, 50.0\)[/tex].
2. Calculate the mean:
- We need to multiply each midpoint by the number of students in the corresponding interval and then divide by the total number of students.
- The total number of students is [tex]\(8 + 11 + 6 + 5 + 10 = 40\)[/tex].
[tex]\[
\text{Mean} = \frac{(10 \times 8) + (20 \times 11) + (30 \times 6) + (40 \times 5) + (50 \times 10)}{40}
= \frac{80 + 220 + 180 + 200 + 500}{40}
= \frac{1180}{40}
= 29.5
\][/tex]
3. Calculate the variance:
- We need to compute the sum of the squared differences between each midpoint and the mean, weighted by the number of students in each interval, and then divide by the total number of students.
[tex]\[
\text{Variance} = \frac{8(10 - 29.5)^2 + 11(20 - 29.5)^2 + 6(30 - 29.5)^2 + 5(40 - 29.5)^2 + 10(50 - 29.5)^2}{40}
= \frac{8 \times 380.25 + 11 \times 90.25 + 6 \times 0.25 + 5 \times 110.25 + 10 \times 420.25}{40}
= \frac{3042 + 992.75 + 1.5 + 551.25 + 4202.5}{40}
= \frac{7790}{40}
= 194.75
\][/tex]
4. Calculate the standard deviation:
- The standard deviation is the square root of the variance.
[tex]\[
\text{Standard Deviation} = \sqrt{Variance} = \sqrt{219.75} \approx 14.823967080373594
\][/tex]
5. Calculate the coefficient of variation (CV):
- The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage.
[tex]\[
\text{Coefficient of Variation} = \left(\frac{\text{Standard Deviation}}{\text{Mean}}\right) \times 100
= \left(\frac{14.823967080373594}{29.5}\right) \times 100
\approx 50.2507358656732\%
\][/tex]
So, the coefficient of variation is approximately [tex]\(50.25\%\)[/tex].
