## What is critical value in statistics?

In statistics, the critical value is a numerical value that is used to calculate the level of significance, the degree of freedom, and the distribution of the test statistic. The critical value is denoted by `‘c’`

.

## Types of critical value:

Generally, there are five types of the term "**Critical **`value`

"

- T critical value
- Z critical value
- Chi-square value
- F value
- R-value

## Critical Value Formula

Formulas vary depending on the type of critical value being calculated.

**Formula for T-Value**

**T = (α/2, df)**

Df = n – 1 (n is the sample size)

where:

- Alpha
**(α)** is the significance level - df represents the degree of freedom

**Formula for Z-Value**

**Z = (α / 2)**

where:

- α / 2 gives us the Z-Value known as the normal standard distribution

**Formula for Chi-Square value**

**X**^{2} = ∑ (O – E)^{2} / E

Where:

**χ**^{2 }is the chi-square test statistic**∑** is the sum of values**O** is the observed frequency for each category in the contingency table**E** is the expected frequency for each category in the contingency table, calculated as (row total * column total) / grand total

**Formula for F Value**

**F-Value = (σ**_{1})^{2} / (σ_{2})^{2}

Where:

**(σ**_{1})^{2} is the larger sample variance**(σ**_{2})^{2 }is the smaller sample variance

## Applications of Critical Value:

The critical value helps us in various fields, some real-life applications of this term are as follows:

**Quality control:**

In manufacturing and production, critical values are used to determine if a product meets certain quality standards.

**Medical testing:**

Critical values are used in medical testing to determine if a result is abnormal or requires immediate attention.

**Finance and investments:**

Critical values are used in finance and investments to determine if a particular investment or financial decision is sound.

**Environmental monitoring:**

Critical values are used in environmental monitoring to determine if certain levels of pollutants or other substances are hazardous to human health or the environment.

**Sports:**

In sports, critical values can be used to determine if a particular athlete's performance meets a certain standard or record.

**Calibration of instruments:**

In scientific research, critical values are used to calibrate instruments and equipment.

## How to find critical value? Example:

The calculation of critical value specifically depends on the specific statistical test being performed. In this section, we’ll cover the method how to find critical value.

**Example 1:**

Find the critical value of a sample size of 5 with a level of significance of 0.1.

**Solution:**

**Step 1:**

α = 0.1

n = 5

df = n – 1 = 4

**Step 2:**

Calculate the t-value of the left critical probability using the t-critical value calculator or t-table:

t-value = 1.5332 (right-tailed probability)

t-value = -1.5332 (left-tailed probability)

t-value = ±2.1318 (two-tailed probability)

## Frequently Asked Questions About Critical Value Calculator

**What are the types of critical value?**

There are five types depending on the** **data size and use of distribution to determine the critical value.

**T-Critical Value**: It finds for small samples and unknown population standard deviation values or uses the T-distribution.**Z-Critical Value**: It determines if a sample is large and known population standard deviation values or using z-distribution (normal distribution).**Chi-Square Critical Value**: In this value using the chi-square distribution for tests of independence or goodness-of-fit of the model.**F-Critical Value**: For this value use the F-distribution to compare variances or means of three or more groups of data in ANOVA analysis.**R-Critical Value**: For this value use the Pearson correlation coefficient to find the significance of the correlation between two variables.

**How to find Z-Critical value?**

To find the Z-critical value with a given confidence level “α” follow the below steps.

- Select the test that you performed either a one- or two-tailed test.

**One-tailed test**: In a one-tailed test, the critical region can be divided into two parts: left tail or right tail distribution.**Left-Tailed**: The critical value is negative or based on the **α-th** part of the normal distribution and determining its value using z-table.**Right-Tailed**: The z-critical value is positive or based on the (**1- α)-th **part of the confidence level.

**Two-tailed test**: The critical value is based on the **± (1-** α/2**)-**th part of the confidence level and applied on both sides of the distribution.

To verify the result using our above critical value calculator.

**What is the difference between T-critical value and Z-critical value?**

Both are the statistical measures used in hypothesis testing for the acceptance or rejection of the null hypothesis.

**T-critical value: **determines if the sample size is less than 30 & the standard deviation of data is unknown for the t-distribution in the t-test. On the other hand, the**Z-critical value:** is associated with z-distribution (normal distribution) and is determined when the sample size is larger than 30 or the value of the standard deviation of data is known.

**What is two-tailed critical value?**

In statistics, the two-tailed critical value helps us to split the region into two parts that determine whether the sample value is greater or less than the range values of the data.

**Is a T-critical value same as a Z-critical value?**

**No,** both are not the same in practice but the process of t-distribution from a table is approximately similar to the standard normal distribution from the table. Moreover, if the sample size is greater than 30 then there are chance the numerical t-critical value is the same as the Z-critical value.

**What is the z-critical value for the “95%” confidence level?**

The Z-critical value for a 95% confidence level such as:

- For right-tailed test = -1.64
- For left-tailed test = 1.64
- For two-tailed test = 1.96

To verify the values use our critical value calculator.