One of the primary ways that you will find yourself interacting with the chi-square distribution, primarily later in Stat 415, is by needing to know either a chi-square value or a chi-square probability in order to complete a statistical analysis. For that reason, we'll now explore how to use a typical chi-square table to look up chi-square values and/or chi-square probabilities. Let's start with two definitions.

**Definition.** Let \(\alpha\) be some probability between 0 and 1 (most often, a small probability less than 0.10). The upper \(100\alpha^{th}\)** percentile** of a chi-square distribution with \(r\) degrees of freedom is the value \(\chi^2_\alpha (r)\) such that the area under the curve and to the right of \(\chi^2_\alpha (r)\) is \(\alpha\):

The above definition is used, as is the one that follows, in Table IV, the chi-square distribution table in the back of your textbook.

**Definition.** Let \(\alpha\) be some probability between 0 and 1 (most often, a small probability less than 0.10). The \(100\alpha^{th}\)** percentile** of a chi-square distribution with \(r\) degrees of freedom is the value \(\chi^2_{1-\alpha} (r)\) such that the area under the curve and to the right of \(\chi^2_{1-\alpha} (r)\) is \(1-\alpha\):

With these definitions behind us, let's now take a look at the chi-square table in the back of your textbook.

#### Solution

In summary, here are the steps you should use in using the chi-square table to find a chi-square value:

- Find the row that corresponds to the relevant degrees of freedom, \(r\) .
- Find the column headed by the probability of interest... whether it's
**0.01**,**0.025**,**0.05**,**0.10**,**0.90**,**0.95**,**0.975**, or**0.99**. - Determine the chi-square value where the \(r\) row and the probability column intersect.

Now, at least theoretically, you could also use the chi-square table to find the probability associated with a particular chi-square value. But, as you can see, the table is pretty limited in that direction. For example, if you have a chi-square random variable with 5 degrees of freedom, you could only find the probabilities associated with the chi-square values of 0.554, 0.831, 1.145, 1.610, 9.236, 11.07, 12.83, and 15.09:

What would you do if you wanted to find the probability that a chi-square random variable with 5 degrees of freedom was less than 6.2, say? Well, the answer is, of course... statistical software, such as SAS or Minitab! For what we'll be doing in Stat 414 and 415, the chi-square table will (mostly) serve our purpose. Let's get a bit more practice now using the chi-square table.

##
Example 15-5
Section* *

Let \(X\) be a chi-square random variable with 10 degrees of freedom. What is the upper fifth percentile?

#### Solution

The upper fifth percentile is the chi-square value *x* such that the probability to the right of \(x\) is 0.05, and therefore the probability to the left of \(x\) is 0.95. To find *x* using the chi-square table, we:

- Find \(r=10\) in the first column on the left.
- Find the column headed by \(P(X\le x)=0.95\).

Now, all we need to do is read the chi-square value where the \(r=10\) row and the \(P(X\le x)=0.95\) column intersect. What do you get?

The table tells us that the upper fifth percentile of a chi-square random variable with 10 degrees of freedom is 18.31.

What is the tenth percentile?

#### Solution

The tenth percentile is the chi-square value \(x\) such that the probability to the left of \(x\) is 0.10. To find *x* using the chi-square table, we:

- Find \(r=10\) in the first column on the left.
- Find the column headed by \(P(X\le x)=0.10\).

Now, all we need to do is read the chi-square value where the \(r=10\) row* * and the \(P(X\le x)=0.10\) column intersect. What do you get?

The table tells us that the tenth percentile of a chi-square random variable with 10 degrees of freedom is 4.865.

What is the probability that a chi-square random variable with 10 degrees of freedom is greater than 15.99?

#### Solution

There I go... just a minute ago, I said that the chi-square table isn't very helpful in finding probabilities, then I turn around and ask you to use the table to find a probability! Doing it at least once helps us make sure that we fully understand the table. In this case, we are going to need to read the table "backwards." To find the probability, we:

- Find \(r=10\) in the first column on the left.
- Find the value
**15.99**in the \(r=10\) row. - Read the probability headed by the column in which the 15.99 falls.

What do you get?

The table tells us that the probability that a chi-square random variable with 10 degrees of freedom is *less* than 15.99 is 0.90. Therefore, the probability that a chi-square random variable with 10 degrees of freedom is *greater* than 15.99 is 1−0.90, or 0.10.