### 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

### Formulas for critical values

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.

### Example section:

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