To calculate the standard deviation and coefficient of variation, we need to follow these steps:
- Step 1: Calculate the mean (average) of the data set.
- Step 2: Calculate the squared difference between each data point and the mean.
- Step 3: Calculate the variance by taking the average of the squared differences.
- Step 4: Calculate the standard deviation as the square root of the variance.
- Step 5: Calculate the coefficient of variation by dividing the standard deviation by the mean and multiplying by 100.
Given data:
- | Mark secured |No. of stds|
- | 0-20 |2 |
- | 20-40 |8 |
- | 40-60 |16 |
- | 60-80 |10 |
- | 80-100 |4 |
Step 1: Calculate the mean:
Mean = (0 + 20 + 40 + 60 + 80) / 5 = 40
Step 2: Calculate the squared difference between each data point and the mean:
- (0 - 40)^2 = 1600
- (20 - 40)^2 = 400
- (40 - 40)^2 = 0
- (60 - 40)^2 = 400
- (80 - 40)^2 = 1600
Step 3: Calculate the variance by taking the average of the squared differences:
Variance = (1600 + 400 + 0 + 400 + 1600) / 5 = 800
Step 4: Calculate the standard deviation:
Standard Deviation = √Variance = √800 ≈ 28.28
Step 5: Calculate the coefficient of variation:
Coefficient of Variation = (Standard Deviation / Mean) * 100 = (28.28 / 40) * 100 ≈ 70.70%
So, the standard deviation is approximately 28.28 and the coefficient of variation is approximately 70.70%.