Answer :
To find the Standard Deviation and its coefficient of variation using the Step Deviation Method, follow these steps:
Step 1: Create a Frequency Table
For the given data of ages and number of students:
- Age 20-25: Frequency (f) = 170
- Age 25-30: Frequency (f) = 110
- Age 30-35: Frequency (f) = 80
- Age 35-40: Frequency (f) = 45
- Age 40-45: Frequency (f) = 40
- Age 45-50: Frequency (f) = 30
- Age 50-55: Frequency (f) = 25
Step 2: Determine the Midpoint (x) for Each Class Interval
- Midpoint for 20-25: [tex]\frac{20 + 25}{2} = 22.5[/tex]
- Midpoint for 25-30: [tex]\frac{25 + 30}{2} = 27.5[/tex]
- Midpoint for 30-35: [tex]\frac{30 + 35}{2} = 32.5[/tex]
- Midpoint for 35-40: [tex]\frac{35 + 40}{2} = 37.5[/tex]
- Midpoint for 40-45: [tex]\frac{40 + 45}{2} = 42.5[/tex]
- Midpoint for 45-50: [tex]\frac{45 + 50}{2} = 47.5[/tex]
- Midpoint for 50-55: [tex]\frac{50 + 55}{2} = 52.5[/tex]
Step 3: Choose an Assumed Mean (A)
Commonly, the midpoint of the class interval with the highest frequency is chosen as the assumed mean. Here, we choose 22.5.
Step 4: Calculate "d", the deviation from the assumed mean
- [tex]d = \frac{x - A}{c}[/tex], where [tex]c[/tex] is the class interval (5 in this case).
Calculations for d:
- d for 22.5: [tex]d = \frac{22.5 - 22.5}{5} = 0[/tex]
- d for 27.5: [tex]d = \frac{27.5 - 22.5}{5} = 1[/tex]
- d for 32.5: [tex]d = \frac{32.5 - 22.5}{5} = 2[/tex]
- d for 37.5: [tex]d = \frac{37.5 - 22.5}{5} = 3[/tex]
- d for 42.5: [tex]d = \frac{42.5 - 22.5}{5} = 4[/tex]
- d for 47.5: [tex]d = \frac{47.5 - 22.5}{5} = 5[/tex]
- d for 52.5: [tex]d = \frac{52.5 - 22.5}{5} = 6[/tex]
Step 5: Find [tex]fd[/tex] and [tex]f(d^2)[/tex]
- Calculate [tex]fd[/tex] and [tex]f(d^2)[/tex] for each class.
Age Interval
d
Frequency (f)
fd
f(d^2)
20-25
0
170
0
0
25-30
1
110
110
110
30-35
2
80
160
320
35-40
3
45
135
405
40-45
4
40
160
640
45-50
5
30
150
750
50-55
6
25
150
900
Sum of frequencies ([tex]N[/tex]) = 500
[tex]\sum fd = 865[/tex]
[tex]\sum f(d^2) = 3125[/tex]
Step 6: Calculate Mean and Standard Deviation
Mean, [tex]\bar{x} = A + (\frac{\sum fd}{N})\times c[/tex]
[tex]\bar{x} = 22.5 + (\frac{865}{500})\times 5 = 22.5 + 8.65 = 31.15[/tex]
Standard Deviation ([tex]\sigma[/tex]):
[tex]\sigma = c \times \sqrt{\frac{\sum f(d^2) - (\frac{(\sum fd)^2}{N})}{N}}[/tex]
[tex]\sigma = 5 \times \sqrt{\frac{3125 - (\frac{865^2}{500})}{500}}[/tex]
[tex]\sigma = 5 \times \sqrt{\frac{3125 - 1497.62}{500}}[/tex]
[tex]\sigma = 5 \times \sqrt{3.2556} = 5 \times 1.80434 = 9.02[/tex]
Step 7: Coefficient of Variation (CV)
[tex]CV = \frac{\sigma}{\bar{x}} \times 100\%[/tex]
[tex]CV = \frac{9.02}{31.15} \times 100\% = 28.96\%[/tex]
Thus, the Standard Deviation is 9.02 and the Coefficient of Variation is 28.96%.
