Critical values are essential in statistical hypothesis testing and help us to decide whether a null hypothesis is rejected or not. It cuts off the points that divide the area under the probability distribution curve into regions, where the null hypothesis is either rejected or not rejected. 

 

The critical value of any hypothesis is evaluated by the significance level and the distribution used in test statistics. It relied on a one-tailed and two-tailed hypothesis test. The one-tailed test gives one critical value while a two-tailed test gives two critical values.

 

In this comprehensive guide, we will explain what critical values are, how they are used, and the steps to find them for various distributions. For better understanding perform the detailed steps example.

 

What is Critical value?

 

A critical value is a point in the test statistic distribution that is used to compare the calculated test statistic value with the tabular value to select or reject the null hypothesis in the hypothesis test. Simply it can be defined as:

 

“It is a range of acceptance regions for any hypothesis.”

 

If the test statistic value exceeds the critical value, the null hypothesis is rejected. On the other hand, if the test statistics value is greater than the critical value then the null hypothesis is rejected and the other hypothesis is accepted.  

 

Critical values depend on the significance level (alpha) and the type of test being performed (one-tailed or two-tailed). The formula of the critical is changed according to the selection of the test. While generally can be stated as:

 

Critical value = 1 – (α/2)

 

Where:

 

α = Significance level = 1 – (confidence level / 100)

 

Understanding of the Significance Level (α)

 

The significance level gives the probability out of “100” to determine how much data lies in the tail or selected region. If α = 0.05 for a two-tailed test then it ensures that 2.5% of the data lies in each tail. It concludes that the critical value shows a 97.5th-percentile in one tail curve and a 2.5th-percentile in the other.

 

The significance level is mathematically denoted by the Greek letter alpha (α) which represents the probability of rejecting the null hypothesis when it is true. Common alpha levels that are used to determine the critical values are 0.05, 0.01, and 0.10. 

 

Types of critical value

 

There are different types of critical values due to sample size, using different distributions, or using a specific type of hypothesis test to determine the critical value. 

 

The different types of critical values that are determined by using different distribution tests such as are (t-test, z-test, F-test, and chi-square test) discussed below:

 

1. T-critical value
2. Z-critical value
3. F-critical value 
4. Chi-square critical value
5. r-critical value

 

How to find the Critical values?

 

There are many methods to calculate the critical value for their respective test such as: Using Statistical Tables for each test, Using Statistical Software (R-language, SPSS, SAS, or Python), Using Microsoft Excel, and using many online tools. 

 

Moreover, the manual calculation is also performed by the statistician and mathematician to determine its value. However, these methods are lengthy, ambiguous, and difficult to use. The performing of separate methods for each test in critical value calculation increases the chances of error.

 

To overcome this difficulty use Criticalvaluecalculator.com, which provides the online Critical value calculator that helps to find all types of critical values (T, Z, F, & R-critical values) with a single click. 

 

 

 

 

This site provides many other statistical tools for students, researchers, and statisticians to make statistical analysis easy. These tools not only provide the final values but also provide the complete steps that help to understand the methods more clearly.

 

Here we perform some manual calculations for different critical values (like T & Z critical value) by using its tables in practical examples.

 

1. T-critical value

 

For this value perform the T-test (one sample, two samples) according to the given condition to evaluate the final critical results. The T-test Formula for one-sample and two-sample tests is stated as:

 

Formula for one sample test	Formula for two sample test
The formula of a one-sample t-test is stated as: T= (X̄-µ) / (σ/√ n) Where: X̄ is the sample mean  µ is the population mean σ is the standard deviation n is the sample size	The formula of a two-sample t-test is written as: T= (X̄1− X̄2)/[(S12/n1) +(S22/n2)] where: X̄1​ and X̄2 ​ are the sample means, S1 and S2 are the sample variances, n1 and n2 are the sample sizes of the given groups.

 

Steps to calculate the T-critical value from T-Table

 

The T-critical value can be calculated by following the below-described steps:

 

• Firstly, determine the Alpha level.
• Find the df (degree of freedom) by subtracting 1 from the sample size.
• To find the t-value use the one-tailed T-table for the one-tailed hypothesis while for the two-tailed hypothesis use the two-tailed T-table.
• Finally, match the value of df (left side column) and the α-value (top row) of the table, thus intersection of both sections is required T-critical value.
•  

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. On the other hand, if the t-score is larger that shows the group is different.

 

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

 

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

 

Solution:

 

Step 1: Note the data from the above statement.

 

Sample size = n = 9 

 

df = Degree of freedom = 9 – 1 = 8

 

Step 2: Find the value with the help of the T-table for one sample test.

 

            T (8, 0.025) = 2.306 

 

T-critical value = 2.306

 

1. Z-Critical Value:

2.  

This value is found by the z-test and uses the normal distribution table. 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. Use the below formula to determine its value.

 

Z = ( X̄-µ) / (σ/√ n)

 

Where:

 

• X̄ is the sample mean 
• µ is the population mean
• “σ” is the standard deviation 
• “n” is the sample size.

 

Steps to calculate the Z-critical value from Z-Table

 

The Z-critical value can be calculated by following the below steps with the help of the table.

 

• First, determine the α-level.
• For the one-tailed test (right or left-tailed) subtract the α-level from 0.5 and for the two-tailed α-level subtract from 1.
• Finally, note the value from the Z-table to evaluate the Z-critical value.

 

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

 

Example: Find Z-critical Value, if the right-tailed Z-test is applied to the sample whose α-level is 0. 0067. 

 

Solution:

 

Step 1: Determined the α-level.

 

   α -level = 0. 0067 

 

Step 2: For a one-tailed test subtract the α-level from 0.5.

 

Region indication value = 0.5 -0.0067 = 0. 4933

 

Step 3: Find the Z-interval by Z-table.

 

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

 

Step 4: Adding the interval value.

 

Z-critical value = 2.5 + 0.00 = 2.5 

 

Z-critical value = 2.5

 

1. F-Critical value

2.  

This value is evaluated with the help of the F-test or ANOVA-test. 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 mathematical formula used for this critical value is stated as:

 

F = (P1)2 / (P2)2 

 

Where 

 

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

 

Steps to calculate the F-critical value from F-Table

 

To calculate the F-critical value with the f-table follow the bellows steps:

 

• First, determine 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 the F-value. (Intersection of P row and Q column)

 

Note: If the calculated F-statistic exceeds the F-table value then the null hypothesis is rejected and selects the other hypothesis. It shows a significant difference between the variances of the two groups.

 

Example: Find the F-critical value if the first group sample size of 12 and the second group sample size of 15 with the significance level (α) of 0.05. Apply the ANOVA or F-test. 

 

Solution:

 

Step 1: Note the α-level and sample sizes of the groups from the above statement. 

 

α = 0.05, N1 = 12, N2 = 15

 

Step 2: Find the degrees of freedom for the 1st-sample and 2nd-sample.

 

P = 12 – 1 = 11

 

Q = 15 – 1 = 14

 

Step 3: Locate the F-critical value from the table by the intersection of the row and the column under given α-level.

 

F- Critical value = 2.81 

 

1. Chi-square critical value

 

The Chi-square critical value is used in hypothesis testing, particularly in tests of independence and goodness-of-fit. For this, performs the Chi-square test that compares the observed frequencies in a categorical dataset for expected frequencies. 

 

The formula to calculate the Chi-square statistic is stated as:

 

χ2 = ∑(Oi−Ei)2/Ei

 

Where:

 

• Oi ​= Observed frequency
• Ei = Expected frequency

 

Steps to Calculate the Chi-Square Critical Value from Chi-Square Table

 

The steps for Chi-square critical value are discussed as below:

 

• First, determine the α-level.
• For df, take the product by subtracting “1” from the rows “r” and columns “c”. i.e., df= (r−1)×(c−1)
• With the help of the Chi-square table find the chi-critical value. (Intersection of df (row) and α-value (column))

 

The critical value helps determine whether there is a significant difference between the observed and expected frequencies.

 

Note: If the calculated Chi-square statistic exceeds the critical value then reject the null hypothesis that indicates a significant difference.

 

Example: Calculate the Chi-square critical value for a 2x3 contingency table and the significance level (α) is 0.05. 

 

Solution:

 

Step 1: Determine the α-level and table size:

 

α = 0.05, r = 2, c = 3

 

Step 2: Calculate the degrees of freedom:

 

df = (2−1) × (3−1)

 

df = 1 × 2 =2

 

Step 3: Now, locate the Chi-square critical value for the respective significance level in the table.

 

 Chi-square critical value = 5.991

 

Wrap up

 

Critical value is essential for statistical hypothesis testing that helps us to decide whether the null hypothesis is rejected or not. In this guide discussed the comprehensive explanation of critical values, their role, and how to find them for various distributions.

 

By understanding critical values and how to find them, anyone can effectively conduct hypothesis testing and make conclusions for their data analysis. I hope this comprehensive guide will help to find the critical values in any statistical analysis and make remarkable results.