Answer :
To find the standard deviation of the given frequency distribution, follow these steps:
Identify the Class Midpoints: The midpoint of each class interval is calculated by taking the average of the lower and upper limits of each class.
- For 20-25: Midpoint is [tex]\frac{20 + 25}{2} = 22.5[/tex]
- For 25-30: Midpoint is [tex]\frac{25 + 30}{2} = 27.5[/tex]
- For 30-35: Midpoint is [tex]\frac{30 + 35}{2} = 32.5[/tex]
- For 35-40: Midpoint is [tex]\frac{35 + 40}{2} = 37.5[/tex]
- For 40-45: Midpoint is [tex]\frac{40 + 45}{2} = 42.5[/tex]
- For 45-50: Midpoint is [tex]\frac{45 + 50}{2} = 47.5[/tex]
Calculate Deviations: Subtract the assumed average (32.5) from each class midpoint.
Deviation from Assumed Average:
- For 20-25: [tex]22.5 - 32.5 = -10[/tex]
- For 25-30: [tex]27.5 - 32.5 = -5[/tex]
- For 30-35: [tex]32.5 - 32.5 = 0[/tex]
- For 35-40: [tex]37.5 - 32.5 = 5[/tex]
- For 40-45: [tex]42.5 - 32.5 = 10[/tex]
- For 45-50: [tex]47.5 - 32.5 = 15[/tex]
Calculate Squared Deviations and multiply by the frequency of each class:
- For 20-25: [tex](-10)^2 \times 170 = 17000[/tex]
- For 25-30: [tex](-5)^2 \times 110 = 2750[/tex]
- For 30-35: [tex](0)^2 \times 80 = 0[/tex]
- For 35-40: [tex]5^2 \times 45 = 1125[/tex]
- For 40-45: [tex]10^2 \times 40 = 4000[/tex]
- For 45-50: [tex]15^2 \times 35 = 7875[/tex]
Sum Up the Squared Deviations:
- Total = [tex]17000 + 2750 + 0 + 1125 + 4000 + 7875 = 32750[/tex]
Find the Total Number of Persons (N):
- [tex]170 + 110 + 80 + 45 + 40 + 35 = 480[/tex]
Calculate Variance:
[tex]\text{Variance} = \frac{32750}{480} \approx 68.23[/tex]Calculate Standard Deviation:
[tex]\text{Standard Deviation} = \sqrt{68.23} \approx 8.26[/tex]
So, the standard deviation of the distribution is approximately 8.26 years.
