What is Critical Value? Explained with its types and formulas

Critical value played a crucial role in the testing of the statistical hypothesis which tells us about the region in the sampling distribution of test statistics. In hypothesis testing, the critical value is compared with the statistical value and tells us null hypothesis is rejected or not.       

The critical value of any hypothesis is deduced from the significance level and the distribution of test statistics. it relied on two tests one-tailed hypothesis test and a two-tailed hypothesis test. It’s clear from the name one-tailed test has one critical value and on the other hand, a two-tailed test has two critical values.                                   

What is Critical Value? Explained with its types and formulas

Definition of Critical value:

Critical value is a range of acceptance regions of the hypothesis.”

The critical value is used to predict whether the hypothesis is significant or not. Critical value confidence interval helps us to determine the lower and upper limits of the distribution. Furthermore, with the help of critical value in graphical analysis split the graph into two regions (acceptance & rejection).

Note: If the value of test statistics is less extreme than the critical value, then the null hypothesis cannot be rejected. On the other hand, if the test statistics value is more than the critical value then the null hypothesis is rejected and the other hypothesis is accepted.  


Critical value = 1 –(α/2)

Where α = 1 – (confidence level / 100)

Types of critical value:

There are different testing techniques to find the critical value of any population or sample. Most used techniques to find different types of critical value by many statisticians are discussed below:

  • T-critical value
  • Z-critical value
  • F-critical value
  • Chi-square critical value

1. T-critical value:

For this value we used T-test, and the value indicates by T-test Formula. In this technique, the t-score is compared with the critical value obtained from the T-table. If the t-score is smaller that shows the group is similar and on the other hand if the t-score is larger that shows the group is different.

T-value can be calculated by this step described below:

  • Firstly, determine the Alpha level.
  • Find the degree of freedom (df) by subtracting 1 from the sample size.
  • Find the value by the T-table (for the one-tailed hypothesis use a one-tailed T-table and similarly, for the two-tailed hypothesis used a two-tailed T-table)
  • Finally, match the value of df (left side column) and the α-value (top row) of the table. The intersection of both values is the T-critical value.

The formula for one sample test:

T= (Y-µ) / (σ/√ n)

Where y is the sample mean

  • µ is the population mean
  • σ is the standard deviation
  • n is the sample size

Note: T-test is used only in that situation if the sample size is less than 30 and the standard deviation is not known.

2. Z-Critical Value:

This value finds out by the z-test that lies on the normal distribution and the z-test applies only if the value of the sample size is more than or equal to 30 and the standard deviation is known.

Z-value can be calculated as follows:

  • Firstly, find the α-level.
  • For the one-tailed test subtract the α-level from 0.5 and for the two-tailed α-level subtract from 1.
  • Finally, from the Z-table find the Z-critical value.


Z=(Y-µ) / (σ/√ n)

Where “σ” is the standard deviation and “n” is the sample size.

Note: For the left-tailed test negative sign multiply by the critical value at the end of the calculation.

3. F-Critical value:

This value finds out with the help of the F-test and the F-test is used for the comparison of the variances of two samples. In this sample, Test statistics are obtained using regression analysis.

The steps for F-value are described as follows:

  • Firstly, determined the α-level.
  • Secondly, subtract the 1 from the size of the 1st sample which gives the df of the 1st sample. Assign the name p.
  • In the same process, subtract 1 from the size of the 2nd sample which gives the df of the 2nd sample. Assign the name Q.
  • With the help of the F-table, we find F-value. (Intersection of P row and Q column)


F = (P1)2/(P2)2

Where P1 and P2 are the standard deviations 1st and 2nd samples respectively.

4. Chi-square critical value:

This value finds out by the chi-square test, in this test we compare the sample data with the population data. This test is used to determine the comparison of two variables and how they are related to each other.

The steps for Chi-value are described as follows:

  • Firstly, determined the α-level.
  • To determine the value of df subtract 1 from the sample size.
  • With the help of the Chi-square, the table finds the chi-critical value.(Intersection of df (row) and α-value (column))

A critical value calculator is an online resource to evaluate the above types of critical values with table.


In this section, we learn how to find the critical value with the help of an example.

Example 1:

Suppose a one-tailed T-test apply to a sample whose size is 9 and α=0.025, then find the T-critical value.           



Let, sample size = n = 9

Degree of freedom (df) = 9 – 1 = 8


Find the value with the help of the T-table.

T (8, 0.025) = 2.306

T-critical value = 2.306

Example 2:

Suppose a right-tailed Z-test is applied to the sample whose α-level is 0.0067. Then find Z-critical Value.


Step1:Determine the α-level.

α -level = 0. 0067


For a one-tailed test subtract the α-level from 0.5.

Region indication value = 0.5 -0.0067 = 0.4933

Step3:Find the Z-interval by Z-table.

We note from the table the value lies between 2.5 and 0.00.

Step4:Adding the interval value.

Z-critical value = 2.5 + 0.00 = 2.5

Z-value = 2.5


In this article, we discussed the critical value, its types, and formulas of different techniques i.e. (T-test formula, Z-test formula, etc.). Furthermore, discussed how to find the critical value with the help of examples.

The critical value is a very interesting topic, now you can easily solve the related problems by reading this article.

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