Chapter 14.3, Problem 26E

To determine

To show: Whether the variance can be less than the standard deviation for a set of data. Also, when would the variance be equal to the standard deviation for a set of data.

Explanation of Solution

Given information:

Variance and Standard deviation.

Formula used:

A measure of variability associated with the arithmetic mean is the standard deviation.

  $\sigma =\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2}}$

The variance is the mean of the squares of the deviations from $\overline{X}$ .

The standard deviation is the positive square root of the variance.

Proof:

Consider the definition of the variance and standard deviation ,

Since the variation is a squared quantity , it cannot be compared with standard deviation.

If the data values are all similar , then the standard deviation will be low and if the data values are highly variable then the standard variation is high.

Also, both the terms could be equal when the data represented as $x$ is equal to $x^2$

The possibilities are :-

1. when x is not negative.

2. when they are zero; means no variation.

3. When they are 1: for example normal distribution.

Hence Proved.