Testing variance or standard deviation

In this tutorial we will discuss a method for testing a claim made about the population variance $\sigma^2$ or population standard deviation $\sigma$. To test the claim about the population variance or population standard deviation we use chi-square test.

In this tutorial we will explain the six steps approach used in hypothesis testing to test hypothesis about the population variance or population standard deviation.

Chi-square Test for Variance

Let $X_1, X_2, \cdots, X_n$ be a random sample from a normal population with mean $\mu$ and variance $\sigma^2$.

Let $\overline{x}=\frac{1}{n} \sum x_i$ be the sample mean and $s^2=\frac{1}{n-1} \sum (x_i-\overline{x})^2$ be the sample variance.


a. The sample must be randomly selected from the population.

b. The population must be normally distribution for the variable under study.

c. The observations must be independent.

Step by Step Procedure

We wish to test the null hypothesis $H_0 : \sigma^2 = \sigma^2_0$, where $\sigma^2_0$ is the specified value of the population variance.

Step 1 State the hypothesis testing problem

The hypothesis testing problem can be structured in any one of the three situations as follows:

Situation Hypothesis Testing Problem
Situation A : $H_0: \sigma^2=\sigma^2_0$ against $H_a : \sigma^2 < \sigma^2_0$ (Left-tailed)
Situation B : $H_0: \sigma^2=\sigma^2_0$ against $H_a : \sigma^2 > \sigma^2_0$ (Right-tailed)
Situation C : $H_0: \sigma^2=\sigma^2_0$ against $H_a : \sigma^2 \neq \sigma^2_0$ (Two-tailed)

Step 2 Define the test statistic

The test statistic for testing above hypothesis is

$$ \chi^2 =\frac{(n-1)s^2}{\sigma^2} $$

The test statistic $\chi^2$ follows $\chi^2$ distribution with $n-1$ degrees of freedom.

Step 3 Specify the level of significance $\alpha$

Step 4 Determine the critical values

For the specified value of $\alpha$ determine the critical region depending upon the alternative hypothesis.

  • For left-tailed alternative hypothesis: Find the $\chi^2$-critical value using

$$ P(\chi^2\leq \chi^2 _{1-\alpha,n-1}) = \alpha. $$

  • For right-tailed alternative hypothesis: $\chi^2_\alpha$.

$$ P(\chi^2\geq\chi^2_{\alpha, n-1}) = \alpha. $$

  • For two-tailed alternative hypothesis: $\chi^2_{\alpha/2}$.

$$ P(\chi^2\leq \chi^2 _{1-\alpha/2,n-1} \text{ or } \chi^2\geq \chi^2_{\alpha/2,n-1}) = \alpha. $$

Step 5 Computation

Compute the test statistic under the null hypothesis $H_0$ using $$ \chi^2_{obs} = \frac{(n-1)s^2}{\sigma^2_0} $$

Step 6 Decision (Traditional Approach)

Based on the critical values.

  • For left-tailed alternative hypothesis: Reject $H_0$ if $\chi^2_{obs}\leq \chi^2_{1-\alpha,n-1}$.
  • For right-tailed alternative hypothesis: Reject $H_0$ if $\chi^2_{obs}\geq \chi^2_{\alpha,n-1}$.
  • For two-tailed alternative hypothesis: Reject $H_0$ if $\chi^2_{obs}\leq \chi^2_{1-\alpha/2, n-1}$ or $\chi^2_{obs}\geq \chi^2_{\alpha/2, n-1}$.


Step 6 Decision ($p$-value Approach)

It is based on the $p$-value.

Alternative Hypothesis Type of Hypothesis $p$-value
$H_a: \sigma^2<\sigma^2_0$ Left-tailed $p$-value $= P(\chi^2\leq \chi^2_{obs})$
$H_a: \sigma^2>\sigma^2_0$ Right-tailed $p$-value $= P(\chi^2\geq \chi^2_{obs})$
$H_a: \sigma^2\neq \sigma^2_0$ Two-tailed $p$-value $= 2P(\chi^2\geq \chi^2_{obs})$

If $p$-value is less than $\alpha$, then reject the null hypothesis $H_0$ at $\alpha$ level of significance, otherwise fail to reject $H_0$ at $\alpha$ level of significance.

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