Chapter 2, Problem 174UP

To determine

Explain the reason behind preferring standard deviation over range to measure variation in a data set.

Explanation of Solution

Variability:

Measure of variability infers the amount of dispersion in the dataset, it defines the distance between the data points from the center. That is, it defines the spread of the data set. Variability describes the variation or diversity among the data points in the data set.

Range:

Range of a data set is the difference between maximum and minimum values of the data set.

The general formula to obtain range is given below:

$\text{Range }=\text{maximum}-\text{minimum}$

Some of the flaws for using the range to compare the variability of data sets:

• Even though, both the data sets have same range, the data will quickly vary among themselves, and the spread will be different.
• The extreme measures will affect the range very much.

Standard deviation:

Standard deviation is based on how much each observation deviates from a central point of the data set. It is known that, the greater the distances between the individual observations and the central point, the greater the variability in the data set.

Consider $x_1,x_2,\mathrm{...},x_n$ be the values of the data set.

The general formula to obtain standard deviation is given below:

$\text{Standard deviation }=\sqrt{\frac{{\left(x_i-\overline{x}\right)}^2}{n-1}}$

The value of standard deviation is based on all the values of the data set and the range is based on only the maximum and minimum values of the data set.

Since, standard deviation is based on each value in the data set, it is preferable to use standard deviation over range to measure variability in the data set.