Critical Values: An Introduction with Its Definition, Methods, Types, and Examples

The critical value plays an important role in correctly reflecting a category of features. Further to stability and accuracy, the critical value plays an important role in disproving hypotheses when hypotheses are tested.

A detailed understanding of the critical value and how to evaluate it is important for finding other similar functions, such as margin of error and significance, it does not matter whether you’re working in a statistics course or just interested in how this work.

A critical value tells us about regions in the sampling distribution of a test statistic. These values play a vital role in both test hypotheses and confidence intervals. In hypothesis tests, critical values tell us whether the results are statistically significant. On the other hand, confidence intervals help us to calculate the upper and lower limits.

In both cases, hypothesis tests and confidence intervals critical values use for uncertainty in the sample data which is used to make inferences about a population.

Here we will discuss what critical value is, methods to calculate it, and a detailed explanation of this topic with an example of calculating t critical value to understand the method.

Definition Of Critical Values

महत्वपूर्ण बिन्दू

“In statistical hypothesis testing, the critical values of a statistical test are the boundaries of the acceptance region of the test”

It is shown as:

Critical Value = 1 – (a / 2)


= 1 – (confidence level / 100)

There are two ways to express the critical value a Z-score connected to cumulative probability and a critical t-statistic equal to the critical probability. Moreover, the critical value discusses the different scenarios of the margin of error that statisticians may apply to estimate the quality of the data under consideration.

Let’s suppose a statistician is obtaining population research on the effect of sunlight on mental disorders. Now there will be a margin of error within a population no. of the sample that specifies the rate at which any change, such as outliner, will be expressed within the data set.

Significance of the Critical Value

The critical value is useful in checking validity, accuracy, and the specific place within which mistakes or inconveniences within the sample set can occur. This value is critical in finding the margin of error. Similarly, the critical value gives information on the bases of the no. of samples under discussion.

For Instance, showing the critical value as a t statistic is useful for precisely approaching small sample sizes or data sets with unknown standard deviations which is the square root of Variance. 

The critical value becomes very significant for checking validity and accuracy, as well as a treatment among different population sizes that are under discussion.

Methods to find the Critical Value

Evaluating the critical value of a data set is a simple process. Depending on your no. of samples, you may also express the critical value in one of two methods. A few steps are shown below on how to deal with it:

1. Calculate Alpha Value

Initially find the alpha value by using the formula given below before evaluating the critical probability

Alpha value = confidence level / 100

The confidence level expresses the likelihood that a statistical parameter is also true for the population being measured. This number is commonly shown as a percentage. A confidence level of 99 percent within a sample group, for instance, for specific criteria you have a 99 percent chance of being true for the whole population.

Having a confidence level of 99%, you would complete the following evaluation to find the alpha value:

Alpha = 1 – (99/100) = 1 – (0.99)

This is equal to 0.01. The alpha value will be 0.01 in your relevant example 

2. Find the Critical Probability

Estimate the critical probability by using the alpha value 0.01 above found from the first formula. This critical value may then be shown as a Z-score or t-value. You can also use a t-value table to check the t critical value.

To find the critical probability using the preceding example value where we find = 0.01 to complete the formula:

1 – (0.01 / 2) = 1 – (0.005) = 0.995 

Here the critical probability is expressed as (p*) and it is equal to 0.995.

3. For fewer sample sets, use the t critical value

The critical t statistic is the perfect formulation for the critical probability while measuring a sample size with less size. The t statistic uses to show the critical probability of 99.5% as follows:

The sample size -1 equals the degree of freedom. It means that dividing the number of samples in your study by 1 provides a degree of freedom. For example, if you want a degree of freedom from sample size just minus one from sample size like sample size is 13 then the degree of freedom would be 12.

4. For greater data sets, use the z critical value

Z-score test is basically used for more than 40 samples in a set so the critical would be expressed in a z-score. one more thing critical probability should be equal to the cumulative probability of the Z-score. Basically, the cumulative probability is the likelihood that a random variable will be < or = to a certain value. This probability must be equal to or greater than the critical probability.

How many types of critical values are there?

There are several critical value testing techniques to check the statistical significance of a specific population or sample. The statistical significance will inform you, where test finding is necessary or not in the given situation. There are a few critical value systems discussed below that statisticians used for calculating significance:

1. Z Critical Value

Z critical values are the standard scores that may be evaluated from data collection. The Z-score identifies how far a specific data point deviates from the sample mean. This sort of critical value will tell you how many standard deviations your population means are above or below the raw score.

2. T Critical Value

T critical values are the results of standardized testing. The SAT, for instance, is an example of a standardized test that can result in t-scores. Its role in the statistic is the t-score allows you to turn an individual test score into a standardized form that you can subsequently compare to other test results.

T critical values can also be calculated using a table. If you are not comfortable using tables, you can use the t table calculator to find the critical value of t.

3. Chi-square Value

Chi-squares are derived from two types of chi-square tests: goodness of fit chi-square tests and independence chi-square tests. The goodness of fit chi-square test evaluates whether a small collection of sample data is representative of the entire population. In the independence chi-square test, you will compare two variables to discover their connection.

4. F Critical Value

F critical value is a value on f distribution. The basic purpose of its use is to determine the significance of the test conducted. It can be evaluated by dividing two mean squares. Mostly, it is used in ANOVA – analysis of variance.

Try a critical value calculator ( for evaluating the problems of z critical value, t critical value, chi square critical value, and f critical value.

Example Section

In this section with the help of examples, a brief discussion of critical value is discussed.

Example 1
Find the critical value for a left-tailed z-test where α = 0.012.


Step 1

First, subtract α from 0.5. 

Thus, 0.5 – 0.012 = 0.488

Step 2

Now using the z distribution table, z = 2.26

However, as this is a left-tailed z test thus, z = -2.26

Critical value = -2.26

Example 2: 

Calculate the critical value for a two-tailed f test under observation on the following samples at an α = 0.025

Variance = 110, Sample size = 41

Variance = 70, Sample size = 21


Step 1:

n1 = 41, 

n2 = 21,

n1 – 1= 40, 

n2 – 1 = 20,

For Sample 1 

Step 2: 

df = 40, 

For Sample 2

 df = 20

Now using the F distribution table for α = 0.025, the value at the intersection of the 40th column and 20th row is F(40, 20) = 2.287

Critical Value = 2.287

Example 3: 

Let a one-tailed t-test is under observation on data with a sample size of 7 at 

α = 0.05. 

Then calculate the critical value.


Step 1:

Given data 

n = 7

Step 2: 

To calculate the degree of freedom

df = 6 – 1 = 7

Using the one-tailed t distribution table t (6, 0.05) = 1.943.

Critical Value = 1.943


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It is mostly used in Hypothesis testing; the critical value is a famous method. Data with its uncertainty and inconsistency stops here and is completely measured and applied to evaluation using its famous technique. 

The critical value is the mostly used method to check risk in finance and gives the data with accuracy for more predictions. Lastly, it is a fine thing to decide before the test how short a p-value is required to reject the test.

The critical value is basically a number based on the type of test (one-tailed vs. two-tailed), degrees of freedom, and alpha level. The critical value gives information regarding how probable a result would be, given that the null hypothesis was in fact true.

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