An Introduction to Critical Value and Its Types

The critical value is the statistical term that plays a very important role in the field of statistics. It tells us where the threshold value of the test statistic falls in a certain range of the given significant value.

Critical values can also be used in confidence interval estimation to determine the range of values the true population parameter is likely to fall. Taking into account the sample size and confidence level, the critical value is calculated. 

In hypothesis testing, the critical value is used to reject or accept the region where the values of given data lie under the given significance level. The degrees of freedom are related to the test statistic and the critical value can be discovered on the significance level.

In this article, we will discuss the basic definition of critical value, discuss its different types, and for a better understanding of the concept of critical value solve different examples.

What is the critical value?

The critical value is used to reject or accept the statistical hypothesis that is determined by degrees of freedom connected to the test statistic as well as its level of significance. If the calculated test statistic value remains beyond the critical value then the null hypothesis is rejected and moves toward the alternative hypothesis which represents there is a particular difference between the observed data and the expected values under the null hypothesis.

 Critical values are used in various statistical tests such as t-tests, F-tests, and chi-square tests which play an important role in concluding the valid results from statistical analysis.

Types of Critical Value:

There are four types of critical value depending on the statistical test which is applied to the statistical data. Types of critical value are given below.

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

T- Critical value:

T-critical value is finding out using the t-test on the defined statistical data. T-value helps us to separate the accepted or rejected region for the null or alternative hypothesis. T- Value is finding out using the value of the level of significance and degree of freedom related to the T-distribution. 

Now furthermore, the t-test is further characterized into two types.

1. 1-Sample test
2. 2-Sample test

1-Sample test

If the mean of a single sample equals a particular amount then a 1-sample t-test is applied. The test is conducted by comparing the sample mean to the hypothesized population mean and determining whether the difference between the two samples is statistically significant or not.

t- Critical value = (x̄ – μ)/(s / √n)

x̄ = sample mean μ = Hypothesized population mean 

s = sample standard deviation n = sample size

2-Sample test

The 2-sample t-test is used if the means of two independent samples are equal. The test is conducted by comparing the means of the two samples and determining whether the difference between them is statistically significant or not.

t = (x̄1 – x̄2)/(s * √(1/n₁ + 1/n₂))

• x̄₁ = mean of the 1^st sample
• x̄₂ = mean of the 2^nd sample 
• s = standard deviation of the two samples 
• n₁ = sample size for the 1^st sample
• n₂ = sample size for the 2^nd sample

F-critical value:

The f-critical value is finding out using the f-test on the defined statistical data. F-value helps us to separate the accepted or rejected region for the null or alternative hypothesis. 

F- Value is finding out using the value of the level of significance and degree of freedom of the numerator and denominator related to the F-distribution. F-tests are used to compare the variances of two or more populations. Its formula is stated as.

F-critical value = (α, df₁, df₂)

• df₁ = degrees of freedom of the numerator
• df₂ = degrees of freedom of the denominator
• α = level of significance

Examples:

In this section, we will discuss some examples of critical values using the above types and their formulas.

Example 1:

Consider a 1-sample T-test applies on a sample whose size is 6 and α=0.05, then evaluate the T-value.

Solution:

Step 1:

Write the given data from a sample.

Sample size = n = 6, α=0.05

Then, df = Degree of freedom = 6 – 1 = 5

Step 2:

Write the t-value formula.

T-value = T (df, α)

α = level of significance, df = degree of freedom

Step 3:

Put the values in the above formula and evaluate the T-value using the T-table. 

df = 5, α = 0.05 

T-value = T (5, 0.05) = 2.015

T-value = 2.015

T value of the critical value can also be calculated with the help of a t critical value calculator to avoid calculations.

Example 2:

Consider a 1-sample T-test applies to a sample whose size is 7 and α=0.05. While the sample and population mean are 25 and 45 respectively. The standard deviation of the sample is 15 then evaluate the T-value.

Solution:

Step 1:

Write the given data from a sample.

Sample mean = 25, population mean = 45 

Sample standard deviation = 15, sample size = 7

T-value =?

Step 2:

Write the t-value formula for the 1-sample test.

t- Critical value = (x̄ – μ)/(s / √n)

x̄ = sample mean μ = Hypothesized population mean 

s = sample standard deviation n = sample size

Step 3:

Put the values in the above formula and evaluate the T-value.

x̄ = 25, μ =45, s = 15, n = 7

t- Critical value = (x̄ – μ)/(s / √n)

t- Critical value = [(25 – 45)]/[(15/ √7)]

t- Critical value = [(-20)] / [(15/ 2.65)]

t- Critical value = [-20] / [5.66]

t- Critical value = – 3.53

Summary of the Critical Value

In this article, we discussed the definition of critical value and its different types. Moreover, the detailed discussion of all types and their formulas. For a better understanding of critical value solved different examples with detailed discussion.