Answer :
The Standard Deviation is approximately 8.59.
To calculate the Standard Deviation, we need to follow a series of steps. First, find the midpoint for each class interval. The midpoint is the average of the lower and upper class limits. Then, calculate the deviation of each midpoint from the mean of all midpoints. Next, square each deviation, multiply it by its corresponding frequency, and sum these values. Divide the sum by the total frequency to get the variance. Finally, take the square root of the variance to obtain the standard deviation.
Midpoints:
[tex]M_1 & = \frac{20 + 25}{2} = 22.5 \\M_2 & = \frac{25 + 30}{2} = 27.5 \\M_3 & = \frac{30 + 35}{2} = 32.5 \\M_4 & = \frac{35 + 40}{2} = 37.5 \\M_5 & = \frac{40 + 45}{2} = 42.5 \\M_6 & = \frac{45 + 50}{2} = 47.5 \\[/tex]
Mean of Midpoints:
[tex]\bar{X} = \frac{8 \cdot 22.5 + 12 \cdot 27.5 + 6 \cdot 32.5 + 9 \cdot 37.5 + 15 \cdot 42.5 + 10 \cdot 47.5}{60} = 35[/tex]
Deviations from the Mean:
[tex]D_1 & = 22.5 - 35 = -12.5 \\D_2 & = 27.5 - 35 = -7.5 \\D_3 & = 32.5 - 35 = -2.5 \\D_4 & = 37.5 - 35 = 2.5 \\D_5 & = 42.5 - 35 = 7.5 \\D_6 & = 47.5 - 35 = 12.5 \\[/tex]
Squared Deviations multiplied by Frequency:
[tex]D_1^2 \cdot f_1 & = 156.25 \cdot 8 = 1250 \\D_2^2 \cdot f_2 & = 56.25 \cdot 12 = 675 \\D_3^2 \cdot f_3 & = 6.25 \cdot 6 = 37.5 \\D_4^2 \cdot f_4 & = 6.25 \cdot 9 = 56.25 \\D_5^2 \cdot f_5 & = 56.25 \cdot 15 = 843.75 \\D_6^2 \cdot f_6 & = 156.25 \cdot 10 = 1562.5 \\[/tex]
Sum of Squared Deviations multiplied by Frequency:
[tex]1250 + 675 + 37.5 + 56.25 + 843.75 + 1562.5 = 4425[/tex]
Variance:
[tex]\text{Variance} = \frac{4425}{60} = 73.75[/tex]
Standard Deviation:
[tex]\text{Standard Deviation} \approx \sqrt{73.75} \approx 8.59[/tex]
