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 onetailed and twotailed hypothesis test. The onetailed test gives one critical value while a twotailed 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.
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 (onetailed or twotailed). 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)
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 twotailed test then it ensures that 2.5% of the data lies in each tail. It concludes that the critical value shows a 97.5thpercentile in one tail curve and a 2.5thpercentile 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.
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 (ttest, ztest, Ftest, and chisquare test) discussed below:
There are many methods to calculate the critical value for their respective test such as: Using Statistical Tables for each test, Using Statistical Software (Rlanguage, 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, & Rcritical 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.
For this value perform the Ttest (one sample, two samples) according to the given condition to evaluate the final critical results. The Ttest Formula for onesample and twosample tests is stated as:
Formula for one sample test 
Formula for two sample test 
The formula of a onesample ttest is stated as: T= (X̄µ) / (σ/√ n) Where:

The formula of a twosample ttest is written as: T= (X̄1− X̄2)/[(S12/n1) +(S22/n2)] where:

Steps to calculate the Tcritical value from TTable
The Tcritical value can be calculated by following the belowdescribed steps:
In this technique, the tscore is compared with the critical value obtained from the Ttable. If the tscore is smaller that shows the group is similar. On the other hand, if the tscore is larger that shows the group is different.
Note: The Ttest is only applied in that situation if the sample size is less than 30 and the standard deviation is not known.
Example: Suppose a onetailed Ttest is applied to a sample whose size is 9 and α = 0.025, then find the Tcritical 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 Ttable for one sample test.
T (8, 0.025) = 2.306
Tcritical value = 2.306
This value is found by the ztest and uses the normal distribution table. The ztest 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 = ( X̄µ) / (σ/√ n)
Where:
Steps to calculate the Zcritical value from ZTable
The Zcritical value can be calculated by following the below steps with the help of the table.
Note: For the lefttailed test negative sign multiply by the critical value at the end of the calculation.
Example: Find Zcritical Value, if the righttailed Ztest is applied to the sample whose αlevel is 0. 0067.
Solution:
Step 1: Determined the αlevel.
α level = 0. 0067
Step 2: For a onetailed test subtract the αlevel from 0.5.
Region indication value = 0.5 0.0067 = 0. 4933
Step 3: Find the Zinterval by Ztable.
We note from the table the value lies between 2.5 and 0.00.
Step 4: Adding the interval value.
Zcritical value = 2.5 + 0.00 = 2.5
Zcritical value = 2.5
This value is evaluated with the help of the Ftest or ANOVAtest. The Ftest 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
Steps to calculate the Fcritical value from FTable
To calculate the Fcritical value with the ftable follow the bellows steps:
Note: If the calculated Fstatistic exceeds the Ftable 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 Fcritical 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 Ftest.
Solution:
Step 1: Note the αlevel 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 1stsample and 2ndsample.
P = 12 – 1 = 11
Q = 15 – 1 = 14
Step 3: Locate the Fcritical value from the table by the intersection of the row and the column under given αlevel.
F Critical value = 2.81
The Chisquare critical value is used in hypothesis testing, particularly in tests of independence and goodnessoffit. For this, performs the Chisquare test that compares the observed frequencies in a categorical dataset for expected frequencies.
The formula to calculate the Chisquare statistic is stated as:
χ2 = ∑(Oi−Ei)2/Ei
Where:
Steps to Calculate the ChiSquare Critical Value from ChiSquare Table
The steps for Chisquare critical value are discussed as below:
The critical value helps determine whether there is a significant difference between the observed and expected frequencies.
Note: If the calculated Chisquare statistic exceeds the critical value then reject the null hypothesis that indicates a significant difference.
Example: Calculate the Chisquare critical value for a 2x3 contingency table and the significance level (α) is 0.05.
Solution:
Step 1: Determine the αlevel 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 Chisquare critical value for the respective significance level in the table.
Chisquare critical value = 5.991
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.