Concept of Critical Value with the help of Different Techniques to find it

A critical value is the test statistic’s value that establishes a confidence interval’s upper and lower boundaries. It explains how far away from the distribution’s mean you must be to account for a specific percentage of the overall variation in the data. For instance, 90%, 95%, and 99%.

Your critical value will be the same in both scenarios if you are creating a 90% confidence interval and utilizing a p = 0.1 as a level of significance. In this article, the basic definition of critical value with formula and also further techniques to find it will be discussed with the help of examples.

What is the Critical Value?

The critical values of a statistical test in statistical hypothesis testing are the limits of the test’s acceptance region. The set of test statistic values for which the null hypothesis is not rejected is known as the acceptance zone. There may be one or many critical values, depending on the geometry of the acceptance region.

In mathematical form it is written as:

Critical value = 1 – (α/2)


 α = 1 – (confidence level / 100)

In order to test the statistical hypothesis, which provides information about the region in the sampling distribution of test statistics, the critical value was an important factor. In hypothesis testing, the critical value and statistical value are compared to determine whether or not the null hypothesis is accepted.

The significance level and test statistic distribution are used to determine any hypothesis’ crucial value. It utilized two tests. Two-tailed and one-tailed hypothesis tests are also used. One-tailed tests have one critical value, as implied by their name, while two-tailed tests have two critical values.

Tests to Determine Critical Values

There are a few important techniques to evaluate the critical values of any specific sample or population. Commonly used techniques to determine the critical values by statisticians are discussed below:

For T-critical Value:

We utilized the T-test for this value, and the T-test Formula shows the value. This method compares the t-score to the crucial value derived from the T-table. If the t-score is smaller, it indicates that the group is similar, and if it is greater, it indicates that the group is different.

T-value can be evaluated by this step shown below:

  • First of all, the alpha level will be calculated.
  • Subtract 1 from the sample size to get the degree of freedom.
  • With the help of a one-tailed T-table, you can evaluate a one-tailed hypothesis and similarly for a two-tailed hypothesis, a two-tailed T-Table will be used.
  • Now to find the T-Critical value check the box of df (Left side column) and α-value (top row) of the table. Then the selected box will be its critical value.

The formula for a sample test

T= (Y-µ) / (σ/√ n)

Where y shows the sample mean

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

For Z-critical Value:

It calculates the z-test that lies on the normal distribution if the sample size is more than or equal to 30 and the standard deviation is known then z-test will apply.

Z-value can be evaluated by following the steps below:

  • First of all, the alpha level will be calculated.
  • Subtract the α-level from 0.5 for the one-tailed test and 1 from the α-level for two-tailed tests.
  • Now using the Z table z critical value will be calculated.


Z=(Y-µ) / (σ/√ n)

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

For F-critical Value:

It can be calculated with F-test and it is used for the comparison of two samples and test statistics are obtained using regression analysis.

The steps for F-value are mentioned below:

  • First of all, the alpha level will be calculated.
  • To get the degree of freedom subtract 1 from sample size 1st.
  • Similarly, subtract 1 from 2nd sample size to get the degree of freedom.
  • Using the F-table you can easily determine F-value.


F = (K1)2 / (K2)2

Where K1 and K2 are the standard deviations 1st and 2nd samples respectively.

Chi-square Critical Value:

The chi-square test, which compares sample data with population data, is used to determine this number. This test is used to compare two factors and establish their relationships with one another.

The steps for Chi-value are mentioned below:

  • First of all, the alpha level will be calculated.
  • Subtract 1 from the sample size to get the degree of freedom.
  • With the help of a table, the chi-square value can be calculated.


In this section, with the help of examples, the topic will be explained.

Example 1:

Let the one-tailed T-test apply to a sample whose size is 7 and α=0. 5, then calculate the T-critical value.  



Extract the given data

Sample size = n = 7

Subtract 1 from the sample size to get a degree of freedom (df) = 7 – 1 = 6

Step 2:

To Calculate the value with the help of the T-table.

            T (6, 0.5) = ± 0.7176

So, T-critical value = ± 0.7176

Use online tools to cross-check the result of calculated problems. Let us check the result of the above problem.

Example source:

Example 2:

Let a right-tailed Z-test is applied to the sample whose α-level is 0. 0052. Then find Z-critical Value.


Step 1:

Extract the given data

   α -level = 0. 0052

Step 2:

Subtract the α-level from 0.5 for a one-tailed test.

Region indication value = 0.5 -0.0052 = 0. 4948

Step 3:

Calculate the Z-interval with the help of the Z-table.

This value lies between an interval of 2.5 and 0.00

Step 4:

Now Add the intervals to get the Z-value

Z-critical value = 2.5 + 0.00 = 2.5

Z-value = 2.5


In this article, the basic definition with its formula and a few important techniques are discussed with their formulas and methods to apply them. Furthermore, with the help of examples topic is explained hope you can easily calculate critical values after a complete understanding of this article.

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