*39*

The Chi-square distribution table** **is a table that shows the critical values of the Chi-square distribution. To use the Chi-square distribution table, you only need two values:

- A significance level (common choices are 0.01, 0.05, and 0.10)
- Degrees of freedom

The Chi-square distribution table is commonly used in the following statistical tests:

When you conduct each of these tests, you’ll end up with a test statistic *X ^{2}*. To find out if this test statistic is statistically significant at some alpha level, you have two options:

- Compare the test statistic
*X*to a critical value from the Chi-square distribution table.^{2 } - Compare the p-value of the test statistic
*X*^{2}

Let’s walk through an example of how to use each of these approaches.

**Examples**

Suppose we conduct some type of Chi-Square test and end up with a test statistic *X ^{2 }*of

**27.42**and our degrees of freedom is

**14**. We would like to know if these results are statistically significant.

**Compare the test statistic ***X*^{2}* *to a critical value from the Chi-square distribution table

*X*

^{2}*to a critical value from the Chi-square distribution table*

The first approach we can use to determine if our results are statistically significant is to compare the test statistic *X ^{2 }*

*of*

**27.42**to the critical value in the Chi-square distribution table. The critical value is the value in the table that aligns with a significance value of

**0.05**and a degrees of freedom of

**14**. This number turns out to be

**23.685**:

Since out test statistic *X ^{2}*

*(*

**27.42**) is larger than the critical value (

**23.685**), we reject the null hypothesis of our test. We have sufficient evidence to say that our results are statistically significant at alpha level 0.05.

**Compare the p-value of the test statistic ***X*^{2}* *to a chosen alpha level

*X*

^{2}*to a chosen alpha level*

The second approach we can use to determine if our results are statistically significant is to find the p-value for the test statistic *X ^{2}*

*of*

**27.42**. In order to find this p-value,

**we can’t use the Chi-square distribution table because it only provides us with critical values, not p-values**.

So, in order to find this p-value we need to use a Chi-Square Distribution Calculator with the following inputs:

** Note**:

*Fill in the values for “Degrees of Freedom” and “Chi-square critical value”, but leave “cumulative probability” blank and click the “Calculate P-value” button.*

The calculator returns the cumulative probability, so to find the p-value we can simply use 1 – 0.98303 = **0.01697**.

Since the p-value **(0.01697) **is less than our alpha level of **0.05**, we reject the null hypothesis of our test. We have sufficient evidence to say that our results are statistically significant at alpha level 0.05.

**When to Use the Chi-square Distribution Table**

If you are interested in finding the Chi-square critical value for a given significance level and degrees of freedom, then you should use the Chi-square Distribution Table.

Instead, if you have a given test statistic *X ^{2}* and you simply want to know the p-value of that test statistic, then you would need to use a Chi-Square Distribution Calculator to do so.