Critical Value: Definition, Relations, Uses, and Calculations

In mathematics, critical values play a crucial role in various fields, including calculus, optimization, and statistical analysis. A critical value is a point in the domain of a function where there is a significant change or transition in its behavior. It is often associated with important features such as extrema, points of inflection, or solutions to equations.

In the context of calculus, critical values are typically found by determining where the derivative of a function is equal to zero or undefined. These points are called critical points and can indicate the presence of a local minimum, maximum, or inflection point.

They are vital in analyzing the behavior of functions and finding optimal solutions to optimization problems. In this article, basic definition and comparison of critical values with confidence interval and its significance level will discussed.

Critical Value


A critical value is a point in the domain of a function where its behavior undergoes a significant change or transition. It is often associated with extrema, inflection points, or decision boundaries in statistical analysis. In hypothesis, testing, critical values are used to define rejection regions for a given significance level.

By comparing test statistics to critical values, researchers can make decisions about accepting or rejecting the null hypothesis.

Relation of Critical Values With Confidence Intervals

Confidence intervals and critical values are closely related concepts in statistical analysis. A confidence interval is an estimate of an unknown population parameter, such as a mean or proportion, along with an associated level of confidence. Critical values, on the other hand, are specific values that define the boundaries or endpoints of a confidence interval.

The critical values are determined based on the desired confidence level and the underlying distribution of the data. For example, in the case of a normal distribution, critical values are derived from the standard normal distribution table or calculated using statistical software.

To construct a confidence interval, the critical values are used in conjunction with the sample data and the estimated standard error of the parameter being estimated. The confidence interval is then computed as a range of values within which the true population parameter is expected to lie with the specified level of confidence.

The critical values help determine the width of the confidence interval, which reflects the precision of the estimate. Wider intervals are produced by higher confidence levels (such as 95% or 99%), which offer more certainty but at the expense of some precision.

The Relationship between Critical Values and Significance Level

The critical value designates the point beyond which the null hypothesis is rejected, and the significance level (alpha) establishes the likelihood that the null hypothesis will be rejected even if it turns out to be true.

Uses of critical value

Critical values have several important uses in mathematics and statistics. Here are some common applications:

  • Hypothesis Testing:

Critical values are extensively used in hypothesis testing to make decisions about accepting or rejecting a null hypothesis. By comparing the test statistic (such as a z-score or t-score) to the critical value associated with a specified significance level, researchers can determine whether the observed data provide sufficient evidence to support or refute the null hypothesis.

  • Confidence Intervals:

 Critical values play a crucial role in constructing confidence intervals. These intervals provide an estimate of the range within which an unknown population parameter is likely to fall, given a certain level of confidence. Critical values help determine the width of the confidence interval, ensuring that the desired level of confidence is achieved.

  • Optimization:

 In optimization problems, critical values are used to identify the optimal solutions. By analyzing the critical points (where the derivative is zero or undefined) of a function, one can determine whether these points correspond to local minima, maxima, or points of inflection. Critical values help in optimizing functions in various fields, such as economics, engineering, and operations research.

  • Statistical Distributions:

Critical values are also essential in determining critical regions or rejection regions in statistical distributions. These regions are used to define the boundaries beyond which a null hypothesis is rejected. Specific distributions relates to the critical values such as the standard normal distribution or t-distribution, are used to establish these regions for hypothesis testing or confidence intervals.

  • Significance of Statistical:

Critical values help in determining statistical significance. By comparing the test statistic to the critical value, researchers can assess whether the observed data are statistically significant or simply due to random chance. This is crucial in interpreting the results of experiments or studies and drawing meaningful conclusions.

Examples of critical value

Here are a few examples of critical values

Example 1:

Find z critical value by right-tailed when α = 0.652.


Extract the given data

 Alpha (α) = 0.652

Step 1:

Taking an idea from cumulative probability in the right tail of the standard distribution table for α = 0.652.

Observing from the standard normal distribution table the z-score of a cumulative probability of 0.652 for the right tail for 0.652 is almost -0.3907.

Step 2:

The critical value for the right-tailed z-test is the negative z-score given.

Hence you can say for the right-tailed z-test critical value α = 0.652 is -0.3907.

Example 2:

Evaluate the critical value when the sample size is 5 and the alpha is 0.05.


Now check the significance level to determine the critical value for a one-tailed t-test with alpha is 0.05 and a sample size is 5

Step 1:

Initially calculate the degree of freedom for the t-distribution for this sample size is 5 so the degree of freedom will be df = 5-1 = 4

Step 2:

Now looking at the t-distribution table the critical value for a one-tailed t-test with alpha is 0.05 and it relates to the degrees of freedom (df = 4)

If taking a look at the result when alpha is 0.05 and the degree of freedom is 4

For 5 samples and for a one-tailed t-test having alpha 0.05 So the “critical value is 2.132”


In this article, a basic definition of critical value and its comparison with confidence interval and its relation with significance level is discussed. Moreover, for easy understanding with the help of examples critical value is explained. Hopefully, you will easily defend this topic after a complete understanding of this article.