The use of statistical tables is a common topic in many statistics courses. Although software does calculations, the skill of reading tables is still an important one to have. We will see how to use a table of values for a chi-square distribution to determine a critical value. The table that we will use is located here, however other chi-square tables are laid out in ways that are very similar to this one.

## Critical Value

The use of a chi-square table that we will examine is to determine a critical value. Critical values are important in both hypothesis tests and confidence intervals. For hypothesis tests, a critical value tells us the boundary of how extreme a test statistic we need to reject the null hypothesis. For confidence intervals, a critical value is one of the ingredients that goes into the calculation of a margin of error.

To determine a critical value, we need to know three things:

- The number of degrees of freedom
- The number and type of tails
- The level of significance.

## Degrees of Freedom

The first item of importance is the number of degrees of freedom. This number tells us which of the countably infinitely many chi-square distributions we are to use in our problem. The way that we determine this number depends upon the precise problem that we are using our chi-square distribution with. Three common examples follow.

- If we are doing a goodness of fit test, then the number of degrees of freedom is one less than the number of outcomes for our model.
- If we are constructing a confidence interval for a population variance, then the number of degrees of freedom is one less than the number of values in our sample.
- For a chi-square test of the independence of two categorical variables, we have a two-way contingency table with
*r*rows and*c*columns. The number of degrees of freedom is (*r*- 1)(*c*- 1).

In this table, the number of degrees of freedom corresponds to the row that we will use.

If the table that we are working with does not display the exact number of degrees of freedom our problem calls for, then there is a rule of thumb that we use. We round the number of degrees of freedom down to the highest tabled value. For example, suppose that we have 59 degrees of freedom. If our table only has lines for 50 and 60 degrees of freedom, then we use the line with 50 degrees of freedom.

## Tails

The next thing that we need to consider is the number and type of tails being used. A chi-square distribution is skewed to the right, and so one-sided tests involving the right tail are commonly used. However, if we are calculating a two-sided confidence interval, then we would need to consider a two-tailed test with both a right and left tail in our chi-square distribution.

## Level of Confidence

The final piece of information that we need to know is the level of confidence or significance. This is a probability that is typically denoted by alpha. We then must translate this probability (along with the information regarding our tails) into the correct column to use with our table. Many times this step depends upon how our table is constructed.

## Example

For example, we will consider a goodness of fit test for a twelve-sided die. Our null hypothesis is that all sides are equally likely to be rolled, and so each side has a probability of 1/12 of being rolled. Since there are 12 outcomes, there are 12 -1 = 11 degrees of freedom. This means that we will use the row marked 11 for our calculations.

A goodness of fit test is a one-tailed test. The tail that we use for this is the right tail. Suppose that the level of significance is 0.05 = 5%. This is the probability in the right tail of the distribution. Our table is set up for probability in the left tail. So the left of our critical value should be 1 – 0.05 = 0.95. This means that we use the column corresponding to 0.95 and row 11 to give a critical value of 19.675.

If the chi-square statistic that we calculate from our data is greater than or equal to19.675, then we reject the null hypothesis at 5% significance. If our chi-square statistic is less than 19.675, then we fail to reject the null hypothesis.