A critical value in statistics is a point on the sampling distribution of a test statistic that separates the rejection region from the acceptance region. The critical value is a value that is used in hypothesis testing to determine whether the result of a study is statistically significant. Reject the null hypothesis when the test statistic falls within the rejection area. However, the null hypothesis is accepted if the test statistic falls in the acceptance region.

We will discuss the critical values and their common types in this article. We will learn how to determine critical value in different types of tests. We will practice some examples to understand it better. 

Critical Values: Definition 

A critical value refers to a threshold or cutoff point used to make a decision about a hypothesis or determine the significance of a statistical test. It is determined with the help of a specific level of significance and degree of freedom. 

The critical values compare to critical values to decide whether the null hypothesis should be rejected. Reject the null hypothesis when the critical value exceeds (greater than) the test statistic. The null hypothesis is accepted when the test statistic is smaller or equal to the critical value. 

In statistical tests, the critical value used varies according to the test. 

Method to determine the critical value by using a statistical table

Steps are given to determine the critical value in statistical hypothesis testing:

1. Determine the significance level: The significance level, often expressed as alpha (α), represents the probability of rejecting the null hypothesis when it is actually true. It is usually selected in advance and frequently set at 5%. It can be calculated from the confidence level by subtracting the confidence level from 100%.
2. Convert the significance level to a decimal: Divide the significance level (α) by 100 to convert α to decimal form. 
3. Identify one-tailed and two-tailed tests: In hypothesis testing, you need to determine whether you are conducting one-tailed or two-tailed tests. You do not divide the significance level (alpha) by 2 in a one-tailed test. For a two-tailed test, you will divide the significance level by 2 i.e. α /2.  
4. Calculate the degree of freedom: The degree of freedom (df) calculation depends on the sample size and the number of parameters estimated. It is determined as the number of observations minus the number of parameters estimated. For example, if you have a sample size of 8. You are estimating one parameter, and then the degrees of freedom would be 7.
5. Look up the critical value in a statistical table Locate the significance level in the top row and the degrees of freedom in the left column of the tables. The intersection of these values will be critical value. 

Remember that the different statistical tests have different distributions table. Therefore, it is important to observe the appropriate statistical table. 

Types of critical values  

Here are some common types of critical values:

1. T- Critical value
2. Z- Critical value
3. Chi-Square critical value
4. F-critical value

Let’s learn when which types are applicable and how to determine these types of critical values. 

T- Critical value:

This critical value is used in hypothesis tests when the sample size is less than 30 i.e. n < 30, or when the standard deviation is given. It is used in tests that assume a normal distribution but have a smaller sample size. The method to determine T-critical value: 

1. Determine the desired significance level (alpha α). 
2. Determine the degree of freedom, that is equal to the sample size minus one. (df = n – 1)
3. Look up the T-distribution table to determine the t-critical value. Locate the significance level in the top row and the degrees of freedom in the left column of the tables. The intersection of these values will be the t-critical value. 

Z- Critical value:

The z critical value is used when the sample size is greater or equal to thirty i.e. n ≥ 30, or when the standard deviation is known or considered to be known. It is used in tests that assume a normal distribution of population data. Steps to calculate the Z-critical value: 

1. Identify alpha. 
2. For a two-tailed test, minus alpha from 1. For a one-tailed test, minus alpha from 0.5.
3. Look up the z value in the z table that corresponds to the area that is equal to your significance level. 
4. Put a negative sign at the end of the final calculation for the left side tail. 

Chi-square critical value

The chi-square critical value helps determine whether the observed date significantly deviates from the expected distribution. Steps to calculate the chi-squared critical value: 

1. Determine the desired significance level (alpha α). 
2. Determine the degree of freedom, which is equal to the sample size minus 1. (df = n – 1)
3. Look up the chi-square distribution table to find the chi-square value. Locate the significance level in the top row and the degrees of freedom in the left column of the tables. The intersection of these values will be the chi-square critical value. 

F-critical value: 

The F-critical value is used to compare two or more variances. Steps to calculate the F-critical value:

1. Determine the desired significance level (alpha), degree of freedom for the numerator (df₁ = n₁ – 1), and degree of freedom for the denominator (df₂ = n₂ – 1).
2. Look up the f-distribution table to find the f-critical value, the intersection of df₁ (Column) and df₂ (Row) will give the f-critical value.

The t, z, f, and chi-square critical values can be evaluated with the help of a critical value calculator (https://www.criticalvaluecalculator.com/)

Example of finding critical value

Find a one-tailed t-critical value with a significance level of 0.05 and a sample size of 6. 

Solution: 

Here, sample size = n = 6

Significance level α = 0.05

Degree of freedom = df = n – 1 = 6 – 1 = 5

Look up the T-distribution table for a one-tailed test. 

The intersection of the alpha (0.05) and degree of freedom (5) is 2.0150. Hence, 2.0150 is our t-critical value.

Conclusion 

In this article, we have explored the definition of critical values. We learned how to find the critical values for different statistical tests using statistical tables. We covered different types of critical values along with methods of determining these types. We did solve the example of finding the critical values. After reading this article, you can calculate a critical value for any statistical test.