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 :

1. Suppose you have a population that has variance \(\sigma^{2}\) and is normally distributed.
2. 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.
3. Put the sample back into the population.
4. 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.
5. etc.
6. 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 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:

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

▢