8.5 Chi Squared Distribution

Gordon E. Sarty

8. Confidence Intervals

8.5 Chi Squared Distribution

The $\chi^{2}$ (chi squared) distribution is a consequence of a random process based on the normal distribution. It is derived from the normal distribution as the result of the following stochastic process :

Suppose you have a population that has variance $\sigma^{2}$ and is normally distributed.
Take a sample of size $n$ from the population and compute $x_{1} =\frac{(n-1)s_{1}^{2}}{\sigma^{2}}$ using the sample standard deviation $s_{1}$ from that sample.
Put the sample back into the population.
Take another sample of size $n$ from the population and compute $x_{2} = \frac{(n-1)s_{2}^{2}}{\sigma}$ using the sample standard deviation $s_{2}$ from that sample.
etc.
The distribution of the values of $x_{i} = \frac{(n-1)s_{i}^{2}}{\sigma^{2}}$ values will be a $\chi^{2}$ distribution with $\nu = n-1$ degrees of freedom.

Like the $t$ -distributions, the $\chi^2$ distributions are a family, see Figure 8.10.

Figure 8.10 : The $\chi^{2}$ distributions are enumerated by degrees of freedom.

The $\chi^2$ distribution underlies why $s$ is the best estimate for $\sigma$ . It mean, or expected value is $\nu = n-1$ so the expected value of $s$ is $\sigma$ . The expected value of $\sum (x - \bar{x})/n$ in a random sample of size $n$ is not $\sigma$ .

Confidence Intervals on $\sigma$ and $\sigma^{2}$

The $\chi^{2}$ distribution is already normalized in its definition through including $s$ in its definition. Therefore no $z$ -transforms are needed and we can work directly with a table that gives right tail areas under the $\chi^{2}$ distribution. That table is the Chi-squared Distribution Table, in the Appendix, and it gives values of $\chi^2$ for given values of area to the right of $\chi^2$ , see Figure 8.11.

Figure 8.11 : The Chi-squared Distribution Table gives $\chi^2$ associated with given right tail areas.

We’ll need $\chi^2_{\rm left}$ and $\chi^2_{\rm right}$ such that the tail areas are equal and such that the area between them is $\cal{C}$ , see Figure 8.12.

Figure 8.12 : The values $\chi^2_{\rm left}$ and $\chi^2_{\rm right}$ define the confidence region $\cal{C}$ .

Notation : Let’s call the $\alpha$ in the Chi-squared Distribution Table $\alpha_{T}$ and let $\chi^2(\alpha_{T})$ be the table value that corresponds to $\alpha_T$ . In other words $\chi^2(\alpha_{T})$ is the $\chi^{2}$ value that corresponds to a right tail area of $\alpha_{T}$ .

So given $\cal{C}$ , the appropriate $\chi^{2}_{\rm left}$ and $\chi^{2}_{\rm right}$ are the following values from the Chi-squared Distribution Table:

$\chi^{2}_{\rm right} = \chi^2 \left( \frac{1 - {\cal{C}}}{2} \right)$

$\chi^{2}_{\rm left} = \chi^2 \left( 1- \left[ \frac{1 - {\cal{C}}}{2} \right] \right).$

Note the symmetry of the Chi-squared Distribution Table. If $\chi^{2}_{\rm right}$ comes from the column 3 columns from the right edge of the table then $\chi^{2}_{\rm left}$ comes from a column 3 columns from the left edge of the table. Only small and large areas appear in the table, there are no intermediate values.

Finally, the confidence interval for $\sigma^2$ is given by

$\frac{(n-1)s^{2}}{\chi^{2}_{\rm right}} < \sigma^2 < \frac{(n-1)s^2}{\chi^{-2}_{\rm left}}$

and for $\sigma$ by:

$\sqrt{\frac{(n-1)s^2}{\chi^2_{\rm right}}} < \sigma < \sqrt{\frac{(n-1)s^2}{\chi^2_{\rm left}}}$

Where the $\chi^2$ distribution with $\nu = n-1$ degrees of freedom (giving the line to use in the Chi-squared Distribution Table) is used.

Example 8.5 : Find the 90 $\%$ confidence interval on $\sigma$ and $\sigma^2$ for the following data

$59, 54, 53, 52, 51, 39, 49, 46, 49, 48$

Solution : Compute, using your calculator :

$s^2 = 28.2$

$\nu =n-1 = 9.$

From the Chi-squared Distribution Table, in the $\nu = 9$ line, find :

$\chi^2_{\rm right} = \chi^2 \left( \frac{1-0.90}{2}\right) = \chi^2(0.05) = 16.919$

and

$\chi^2_{\rm left} = \chi^2 (1-0.05) = \chi^2(0.95) = 3.325$

So

$\begin{align*} \frac{(n-1)s^2}{\chi^2_{\rm right}} &< \sigma^{2} < \frac{(n-1)s^2}{\chi^2_{\rm left}}\\ \frac{9 \cdot 28.2}{16.919} &< \sigma^{2} < \frac{9 \cdot 28.2}{3.325}\\ 15.0 &< \sigma^2 < 76.3 \hspace{1in} \mbox{with 90\% confidence.} \end{align*}$

Taking square roots:

$3.87 < \sigma < 8.73 \hspace{1in} \mbox{with 90\% confidence.}$

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Introduction to Applied Statistics for Psychology Students Copyright © 2022 by Gordon E. Sarty is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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