In Statistics, the critical value played a crucial role to find the many features of population and sample data. In the testing of the hypothesis, it can help to determine whether a hypothesis is rejected or not. In the graphical analysis, critical value helped us to split the graph into two regions, which are stated as accepted and rejected regions.

The critical value of any test can be interpreted from the significance level and the allocation of the test statistics. The critical value was based on two tests stated as a one-tailed and a two-tailed hypothesis test. The name of the test makes it obvious that a one-tailed test has one critical value, while a two-tailed test has two critical values.

In this article, we will discuss the definition, different techniques to find the critical value, and for a better understanding of the idea of critical value solving different examples.

## Definition of the critical value:

“Critical value is used to calculate the error range for a set of data.” It is represented as

**Critical value = 1 – (alpha/2)**

Where, **alpha = 1 – (confidence level / 100)**

The critical value used to compare the test statistics in the hypothesis testing. It tells us whether the null hypothesis is rejected or not. If the critical value is greater than the value of test statistics then the null hypothesis cannot reject. While the critical value is less than the test statistics value then the null hypothesis is rejected and accepted the other hypothesis.

## Calculate the Critical value by the confidence interval:

The critical value of both tailed tests (one & two-tailed tests) is determined with the help of confidence interval. Let the confidence level of 90 % be specified for the hypothesis test. The critical can be determined by the following steps.

- Firstly, subtract the value of the confidence level from 100 %. That is 100 – 90 = 10.
- For the value of “
**alpha**” convert it into a decimal. i.e., alpha= 10 %. - If the one-tailed hypothesis test is performed then the value of “
**alpha**” same as that defined in step 2. - If the two-tailed hypothesis test is performed then the value of “
**alpha**” gets divided by 2. - The critical value finds by the table (i.e., T-table & Z-table) depends on the type of test conducted (i.e., t-test, z-test).

A t-critical value calculator can be a handy tool to solve the critical values according to one-tailed and two-tailed t-distribution tables with the help of significance level and degree of freedom.

## Calculate the critical value by T-test:

This test was used to compare the average value of two data sets and determined whether the values belong to the same or different populations. The t-test applies by comparing the values of mean, variance, and standard deviation of the given data set. It is applied only at that time if the sample size is less than 30 and the standard deviation of a population is not known.

It is further characterized into three types on the “n” sample size. These types are described as follows.

- 1-sample T-test.
- Independent Sample T-test.
- Paired Sample T-test.

1-Sample T-test:

It is applied by comparing the theoretical mean “μ” with the mean of a population “y” sample size “n”. Its formula is represented below.

T = (y – μ) / (σ / √n)

Where “σ” is the standard deviation of the sample.

Independent Sample T-test:

This test is used to compare the means of the two group samples. It is used to evaluate the means of the two data sets which are statistically significant and different from each other. Its formula is given as.

T = (y_{1} – y_{2}) / √ (σ_{1}^{2}/n_{1}) + (σ_{2}^{2}/n_{2})

Where y_{1} & y_{2} are the means of the 1^{st} & 2^{nd} groups respectively.

And σ_{1 }& σ_{2} are the standard deviation of the 1^{st} & 2^{nd} groups respectively.

The n_{1 }& n_{2 }are the sample size taken under observation 1^{st} & the 2^{nd} groups respectively.

Paired Sample T-test:

This test applies only if the two distributions are correlated to each other, these are the pre and post-test results from the observer. Its formula is given as.

T = ∑ (y_{1} – y_{2}) / (σ/ √n)

Where y_{1} – y_{2} = difference of the means, “σ” is the standard deviation, and “n” is the sample size.

## How to find the Critical Value:

In this section, the critical value finds with the help of examples that are described below.

**Example 1:**

**A company wants to exceed its product sales. The previous sales data showed the average sale of 20 salesmen was 30$ per transaction. After training the salesman, the average sale of 50$ per transaction. If the standard deviation is 10$, find the t-score (T-critical value).**

**Solution:**

Step 1: Write the hypothesis under observation.

H_{0}: population means and the statistic means are equal, μ = μ_{0}

H_{1}: population means and the statistic means are not equal, μ ≠ μ_{0}

Step 2: write the given data.

Population mean = y = 50, Sample mean = μ = 30,

Standard deviation = σ = 10, Sample size = n = 20

Step 3: write the formula of the 1-sample T-test.

T = (y – μ) / (σ / √n)

Step 4: Put the value carefully in the formula.

T = (50 – 30) / (10 / √20)

T = 20 / 2.24

T = 8.92

We note from the T-table “8.92 > 1.711” at alpha = 0.05. The statistical value is greater than the T-table value so rejected hypothesis H_{0}.

**Example 2:**

Apply the T-test on a sample whose size is 7 and α = 0.05, then find the critical value.

**Solution:**

Step 1: write the data carefully from the given question.

Sample size = n = 7

Degree of freedom (df) = 7 – 1 = 6

Step 2: Find the value with the help of the T-table.

T (6, 0.05) = 1. 94

Critical value = 1. 94

## Summary:

In this article, we discussed the definition of critical value and techniques to find a critical value. To understand how to find the critical value we performed different examples. Moreover, with the help of examples you can understand easily this topic and be able to solve related problems.