Critical Values

An Overview of Critical Values:
Definition, Calculations, and Examples

In statistics, the critical value is vital for correctly reflecting a variety of features. In addition to validity and accuracy, the critical value can be useful for disproving hypotheses when they are tested.

Understanding critical value and how to calculate it is vital for evaluating other statistical functions, such as margin of error and significance, whether you're taking a statistics course or just curious about how these concepts operate.

In this post, we'll go over what critical value is, how to calculate it, and an example of calculating t critical value to illustrate the method.

Critical Value - Definition

The critical value in statistics is the measurement statisticians use to quantify the margin of error within a collection of data, and it is represented as:

Critical Value = 1 - (Alpha / 2)


Alpha = 1 - (confidence level / 100).

The critical value can be expressed in two ways: as a Z-score connected to cumulative probability and as a critical t statistic equal to the critical probability. Furthermore, the critical value explains numerous aspects of the margin of error that statisticians may use to assess the quality of the data under consideration.

Assume a statistician is examining population research on the impact of sunshine on mental disorders. There will be a margin of error within a population sample size that specifies the rate at which any differences, such as outliers, will arise within the data set.

Significance of the Critical Value

The critical value is vital in determining validity, accuracy, and the range within which mistakes or inconsistencies within the sample set can occur. This figure is critical in estimating the margin of error. Similarly, the critical value might provide information on the properties of the sample size under consideration.

For example, representing the critical value as a t statistic is vital for precisely assessing small sample sizes or data sets with unknown standard deviations. Expressed as the cumulative probability, or Z-score, the critical value provides for a more precise examination of a larger data set, often with 40 or more samples.

The critical value becomes incredibly significant for examining validity and accuracy, as well as disparities among different population sizes that you research.

How to calculate the Critical Value?

Calculating the critical value of a data set is a simple process. Depending on your sample size, you may also represent the critical value in one of two ways. The following steps will show you how to achieve it:

1. Find Alpha Value

Before computing the critical probability, calculate the alpha value using the formula:

Alpha value = 1 - (the confidence level / 100)

The confidence level shows the likelihood that a statistical parameter is also true for the population being measured. This number is usually expressed as a percentage. A confidence level of 95 percent within a sample group, for example, suggests that the specified criteria have a 95 percent chance of being true for the entire population.

Using a confidence level of 95%, you would complete the following calculation to determine the alpha value:

Alpha = 1 - (95/100) = 1 - (0.95)

Which equals 0.05. The alpha value in this example is 0.05

2. Find the Critical Probability

Calculate the critical probability using the alpha value from the first formula. This is the critical value, which may then be expressed as a t value or a Z-score. You can also use a t value calculator to find t critical value.

Completing the formula to obtain the critical probability using the preceding example's alpha value of 0.05:

1 - (0.05 / 2) = 1 - (0.025) = 0.975 is the critical probability (p*). In this case, the critical probability is 0.975, or 97.5 percent.

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

The critical t statistic is the right formulation for the critical probability when measuring a small sample size. As the t statistic, express the critical probability of 97.5% as follows:

The sample size minus one equals the degree of freedom (df). This implies that dividing the number of samples in your study by one will give you the degree of freedom. So, if your sample size is 25, deduct one from this figure to get the degree of freedom. In this situation, the answer is 24.

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

For population sizes more than 40 samples in a set, the critical value can be expressed as a Z-score. The cumulative probability of the Z-score should be equal to the critical probability. The cumulative probability is the likelihood that a random variable will be less than or equal to a certain value. This probability must be equal to or greater than the critical probability or value.

A z critical value calculator can assist you to calculate z critical value using significance level only.

How many types of critical values are there?

You may use several critical value testing methodologies to assess the statistical significance of a particular population or sample. The statistical significance will inform you whether or not the findings of your tests are valid. The following are the critical value systems that statisticians used when calculating significance:

1. Z Critical Value

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

2. T Critical Value

T critical values are the results of standardized testing. The SATs, for example, is an example of a standardized test that can result in t-scores. In statistics, 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 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 determines 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. It is used to determine the significance of the conducted test. It can be calculating by dividing two mean squares. Mostly, it is used in ANOVA - analysis of variance.

This article is published with permission.

Six Sigma Templates and Calculators

Search active Six Sigma and Lean Manufacturing job openings

Subscribe to get access to entire site

Six Sigma Training Slides

Return to the home page

Recent Articles

  1. Process Capability Indices

    Oct 18, 21 09:32 AM

    Process Capability Overview
    Determing the process capability indices, Pp, Ppk, Cp, Cpk, Cpm

    Read More

  2. Six Sigma Calculator, Statistics Tables, and Six Sigma Templates

    Sep 14, 21 09:19 AM

    MTBF and MTTR Template Picture
    Six Sigma Calculators, Statistics Tables, and Six Sigma Templates to make your job easier as a Six Sigma Project Manager

    Read More

  3. Six Sigma Templates, Statistics Tables, and Six Sigma Calculators

    Aug 16, 21 01:25 PM

    MTBF and MTTR Template Picture
    Six Sigma Templates, Tables, and Calculators. MTBF, MTTR, A3, EOQ, 5S, 5 WHY, DPMO, FMEA, SIPOC, RTY, DMAIC Contract, OEE, Value Stream Map, Pugh Matrix

    Read More

Custom Search

Site Membership

Six Sigma

Templates, Tables & Calculators

Six Sigma Slides


Green Belt Program (1,000+ Slides)

Basic Statistics

Cost of Quality


Process Mapping

Capability Studies



Cause & Effect Matrix


Multivariate Analysis

Central Limit Theorem

Confidence Intervals

Hypothesis Testing

T Tests





Control Plan



Project Pitfalls

Error Proofing

Z Scores


Takt Time

Line Balancing

Yield Metrics

Sampling Methods

Data Classification

Practice Exam

... and more

Need a Gantt Chart?

Click here to get this template