The frequency table below shows the percentages obtained in a Biometry test for a random sample of 60 students:
Interval fi xi F
[30 – 40) 5 b1 c1
[40 – 50) 12 b2 c2
[50 – 60) 18 b3 c3
[60 – 70) 15 b4 c4
[70 – a1) 7 b5 c5
[a1 – 90) 2 b6 c6
[90 – 100) 1 b7 c7
For the percentages obtained, find the coefficient of variation.
A)24.98%
B)22.98%
C)25.12%
D)18.26%
First, we need to calculate the mean percentage obtained by the students. This can be done by using the formula:
Mean = (Σ(xi * Fi)) / n
Where xi is the midpoint of the interval, Fi is the frequency of the interval, and n is the total number of students (60 in this case).
Calculate the midpoints of the intervals:
Midpoint of [30-40) = (30+40) / 2 = 35
Midpoint of [40-50) = (40+50) / 2 = 45
Midpoint of [50-60) = (50+60) / 2 = 55
Midpoint of [60-70) = (60+70) / 2 = 65
Midpoint of [70-a1) = (70+a1) / 2 = 75
Midpoint of [a1-90) = (a1+90) / 2 = 80
Midpoint of [90-100) = (90+100) / 2 = 95
Now, calculate the mean:
Mean = [(35*5) + (45*12) + (55*18) + (65*15) + (75*7) + (80*2) + (95*1)] / 60
Mean = (175 + 540 + 990 + 975 + 525 + 160 + 95) / 60
Mean = 3440 / 60
Mean = 57.33
Next, calculate the standard deviation using the formula:
σ = √[(Σ(Fi * (xi - x̄)^2)) / n]
Calculate the variance first:
Variance = [(5*(35-57.33)^2) + (12*(45-57.33)^2) + (18*(55-57.33)^2) + (15*(65-57.33)^2) + (7*(75-57.33)^2) + (2*(80-57.33)^2) + (1*(95-57.33)^2)] / 60
Variance = [262.58 + 151.68 + 80.92 + 203.80 + 347.28 + 466.13 + 1132.99] / 60
Variance = 2645.38 / 60
Variance = 44.09
Standard deviation = √(44.09) = 6.64
Finally, calculate the coefficient of variation:
CV = (standard deviation / mean) * 100
CV = (6.64 / 57.33) * 100
CV = 0.1157 * 100
CV = 11.57%
Therefore, the correct answer is none of the options provided.