How to Calculate Critical Value in Chi-Square?

The chi-square critical value is a threshold against which the chi-square test statistic is compared to determine statistical significance. Here's how to calculate and find it:

Steps to Calculate the Chi-Square Critical Value

1. Define the Significance Level (Alpha):

   • The significance level, denoted as α (alpha), represents the probability of rejecting the null hypothesis when it is actually true (Type I error).
   • Commonly used values for alpha are 0.05 (5%) and 0.01 (1%).
   • Example: If you set α = 0.05, it means you're willing to accept a 5% chance of incorrectly rejecting the null hypothesis.

2. Determine the Degrees of Freedom (df):

   • The degrees of freedom depend on the specific chi-square test being performed.
   • Chi-Square Goodness-of-Fit Test: df = (Number of categories) - 1
   • Chi-Square Test of Independence: df = (Number of rows - 1) * (Number of columns - 1)
   • Example: In a goodness-of-fit test with 4 categories, df = 4 - 1 = 3. In a test of independence with a 3x2 contingency table, df = (3-1)*(2-1) = 2.

3. Find the Critical Value Using a Chi-Square Distribution Table or Calculator:

   • Chi-Square Distribution Table: Chi-square tables provide critical values for different degrees of freedom and alpha levels. Look up the intersection of your calculated degrees of freedom and chosen alpha level.
   • Statistical Software/Calculator: Programs like R, Python (with SciPy), Excel, or online calculators can directly compute the critical value given the alpha level and degrees of freedom.

Example

Let's say you have a chi-square test with α = 0.05 and df = 2.

1. Alpha (α): 0.05
2. Degrees of Freedom (df): 2

Using a chi-square distribution table, you would look for the value corresponding to α = 0.05 and df = 2. The critical value would be approximately 5.991.

This means that if your calculated chi-square test statistic is greater than 5.991, you would reject the null hypothesis at the 0.05 significance level.

Chi-Square Test Statistic Calculation

While finding the critical value is crucial, remember that you also need to calculate the chi-square test statistic itself. The formula for this is:

χ² = ∑ [(O_i – E_i)² / E_i]

Where:

• χ² is the chi-square test statistic
• O_i is the observed frequency for category i
• E_i is the expected frequency for category i
• ∑ denotes the sum across all categories

The calculated chi-square test statistic is then compared to the critical value obtained as described above.