Standard Deviation and Coefficient of Variation

Standard Deviation

Standard deviation is a measure of dispersion that is based on the squared deviations of the data points from the mean. It is calculated as follows:

Standard deviation = sqrt(sum((x - mean)^2) / n)

where:

  • x is a data point
  • mean is the mean of the data set
  • n is the number of data points

The standard deviation is a measure of how spread out the data is around the mean. A high standard deviation indicates that the data is spread out over a wide range, while a low standard deviation indicates that the data is tightly clustered around the mean.

Coefficient of Variation

The coefficient of variation is a relative measure of dispersion that is calculated as the ratio of the standard deviation to the mean. It is expressed as a percentage, and it is calculated as follows:

Coefficient of Variation = (Standard deviation / Mean) * 100

where:

  • Standard deviation is the standard deviation of the data set
  • Mean is the mean of the data set

The coefficient of variation is a more sensitive measure of dispersion than the standard deviation, but it is also more difficult to calculate. It is also more affected by outliers, meaning that it can be misleading if the data set contains outliers.

Mathematical Formulas

The following are the mathematical formulas for the standard deviation and coefficient of variation:

  • Standard deviation = sqrt(sum((x – mean)^2) / n)
  • Coefficient of Variation = (Standard deviation / Mean) * 100

MCQs on Standard Deviation and Coefficient of Variation

Here are some MCQs on standard deviation and coefficient of variation:

  1. What is the standard deviation of the data set {1, 3, 5, 7, 9}?
    • A. 1
    • B. 2
    • C. 3
    • D. 4
    • E. 5

The correct answer is (C). The standard deviation is calculated as the square root of the sum of the squared deviations from the mean, so the standard deviation is sqrt(5) = 2.236.

  1. What is the coefficient of variation of the data set {1, 3, 5, 7, 9} if the mean is 5?
    • A. 10%
    • B. 20%
    • C. 30%
    • D. 40%
    • E. 50%

The correct answer is (B). The coefficient of variation is the ratio of the standard deviation to the mean, so the coefficient of variation is (2.236 / 5) * 100 = 44.72%.

  1. The coefficient of variation of a data set is 25%. What is the ratio of the standard deviation to the mean?
    • A. 1
    • B. 2
    • C. 3
    • D. 4
    • E. 5

The correct answer is (A). The coefficient of variation is the ratio of the standard deviation to the mean, so if the coefficient of variation is 25%, then the ratio of the standard deviation to the mean is 1.

  1. The standard deviation of a data set is 10. What is the coefficient of variation if the mean is 20?
    • A. 5%
    • B. 10%
    • C. 20%
    • D. 30%
    • E. 40%

The correct answer is (C). The coefficient of variation is the ratio of the standard deviation to the mean, so the coefficient of variation is (10 / 20) * 100 = 50%.

  1. The coefficient of variation of a data set is 25%. If the mean of the data set is increased by 50%, what will be the new coefficient of variation?

The new coefficient of variation will remain at 25%. This is because the coefficient of variation is a relative measure of dispersion, and it is not affected by changes in the mean.

Conclusion

Standard deviation and coefficient of variation are two measures of dispersion that are used to describe the spread of data around the mean. The standard deviation is a more absolute measure of dispersion, while the coefficient of variation is a more relative measure of dispersion.