Critical values in statistical analysis are important for drawing meaningful conclusions from data. It helps researchers, whether to accept or reject a hypothesis by conducting a t-test, a z-test, or a chi-square test. 

Critical value acts as the cutoff point in hypothesis testing and indicates the boundary where the results are considered statistically significant. Having knowledge about how to find critical values is important for students, researchers, and professionals who use statistics in their work. 

In this blog post, we’ll explain how to find critical values and provide a step-by-step process for using a critical value table. Let’s start by discussing the definition of critical values. 

What Are Critical Values?

Critical values are specific points on a statistical distribution that help you decide whether to reject or fail to reject the null hypothesis.  You can think of them as cutoff points that separate the region where your results are considered significant from the region where they are not. 

For example, 

In a t-test, if your test statistic exceeds the critical value, you reject the null hypothesis.

Critical values are used in various statistical tests, including t-tests, z-tests, F-tests, and chi-square tests. They are determined based on the following concepts that are necessary for the calculation of critical values: 

Steps to Find Critical Values

Now, let’s walk through the process of finding critical values in different types of statistical tests.

Step 1: The first step is to identify the type of statistical test you’re conducting. The test you choose will determine which distribution (e.g., t-distribution, z-distribution) and the table you’ll use to find the critical values.

Step 2: After selecting the test, choose your significance Level (α). This will help you locate the correct column in the statistical table or find the critical values.

Step 3: Next, identify degrees of freedom (if applicable). Degrees of freedom help you find the correct row in the statistical table.  The way to find this depends on the test types:

• In a t-test: df = n – 1 (where n is the sample size).  
• In a chi-square test: df = (rows – 1) * (columns – 1).  

Step 4:  Determine whether your hypothesis is one-tailed or two-tailed. This decision will affect the critical value:  

• For a one-tailed test, use the α value directly.  
• For a two-tailed test, divide α by 2 before looking up the critical value.  

Step 5: Now it’s time to find the critical values. The method to find critical values depends on the statistical test used. 

Step 6: Once you have found the critical value, compare it with your test statistic:

• If the test statistic is greater than (or less than) the critical value, reject the null hypothesis.
• If the test statistic falls within the critical value range then do not reject the null hypothesis.

How to Find Critical Values?

Here we explain how to find critical values for different distribution tests by using their respective tables such as t-table, z-table, and f-table. 

• Z-Test (Standard Normal Distribution)

It is a statistical test that is used to determine whether two population means are different when variances are known and the sample size is large. A z-test is a hypothesis test for data that follows a normal distribution. 

How to find z critical value? 

This statistical test is used when sample sizes are large (n ≥ 30) and the population standard deviation is known. The z critical value can be calculated as follows: 

• Calculate the alpha level 
• For the one-tailed test, subtract the alpha level from 0.5, and for the two-tailed from 1.
• Locate the area from the z distribution table to get the z critical value. 

Example:

If the Z-test result is 2.10 and the critical value for two-tailed by using the z-table at α = 0.05 is ±1.96, then we reject the null hypothesis because 2.10 is beyond the critical value.

• T-Test (Student’s t-Distribution)

This type of statistical test is used when sample sizes are small (n < 30) and the population standard deviation is unknown. It is conducted when our data follows a student’s t distribution.

Steps to find the t critical value

Follow the below steps to find the t critical value:

• Locate the alpha level.
• To calculate the degrees of freedom, subtract 1 from the sample size. 
• You used the one-tailed t-distribution table test when the hypothesis test is one-tailed. Otherwise, use the two-tailed distribution table. 
• Now, match the corresponding df value and alpha value of the table. The intersection of this row and column gives the t critical value. 

Example:

If the t-test result is 2.8, and the critical value for two-tailed is ±2.1009 by t-table at α = 0.05, df = 18, then we reject the null hypothesis because 2.8 is beyond the critical value.

• F Critical value

The f test is commonly used for comparing variances between two samples. It can also be used for regression analysis. F-critical value can be calculated as: 

• Find significance level alpha. 
• Get your 1st df by subtracting 1 from the sample size and say it x, and do this for 2nd df and call it y.
• Use the f distribution table and get the f critical value with the intersection of the row and column.

Example:

If the F-test result is 3.5, and the critical values from the f-table are approximately 0.1580 & 3.2891 at α = 0.05, df₁ = 5, df₂ = 20. Then we reject the null hypothesis because 3.5 is beyond the upper critical value.

• Chi square Critical Value

The chi square test is used to check if the sample data matches the population data. This can also be used to compare 2 variables to see if they are related. In this chi-square critical value can be found by below steps:

• Notify alpha level
• Calculate df from the sample size. 
• The intersection of the df’s row and the column of the alpha in the chi-square distribution table yields the chi-square critical value. 

Pro Tips:

Sometimes, finding critical values from a table can be time-consuming and difficult for beginners who have no strong understanding of this topic. But with the use of a critical value calculator, you can get quick and accurate critical values of any type such as t, z, r, f, and chi-square.

Conclusion

Finding critical values is a fundamental skill for any academic research. It enables you to make informed decisions for any supposed hypotheses. Then it is necessary to find the accurate value of the critical value for their respective test. By following our above-detailed guide you can determine the critical value of any test. Also, you can confidently make an informed decision to do perfect research.