Critical value is a cut-off value that is used to mark the start of a region where the test statistic, obtained in hypothesis testing, is unlikely to fall in. In hypothesis testing, the critical value is compared with the obtained test statistic to determine whether the null hypothesis has to be rejected or not.

Graphically, the critical value splits the graph into the acceptance region and the rejection region for hypothesis testing. It helps to check the statistical significance of a test statistic. In this article, we will learn more about the critical value, its formula, types, and how to calculate its value.

1. | What is Critical Value? |

2. | Critical Value Formula |

3. | T Critical Value |

4. | Z Critical Value |

5. | F Critical Value |

6. | Chi-Square Critical Value |

7. | Critical Value Calculation |

8. | FAQs on Critical Value |

## What is Critical Value?

A critical value can be calculated for different types of hypothesis tests. The critical value of a particular test can be interpreted from the distribution of the test statistic and the significance level. A one-tailed hypothesis test will have one critical value while a two-tailed test will have two critical values.

### Critical Value Definition

Critical value can be defined as a value that is compared to a test statistic in hypothesis testing to determine whether the null hypothesis is to be rejected or not. If the value of the test statistic is less extreme than the critical value, then the null hypothesis cannot be rejected. However, if the test statistic is more extreme than the critical value, the null hypothesis is rejected and the alternative hypothesis is accepted. In other words, the critical value divides the distribution graph into the acceptance and the rejection region. If the value of the test statistic falls in the rejection region, then the null hypothesis is rejected otherwise it cannot be rejected.

## Critical Value Formula

Depending upon the type of distribution the test statistic belongs to, there are different formulas to compute the critical value. The confidence interval or the significance level can be used to determine a critical value. Given below are the different critical value formulas.

### Critical Value Confidence Interval

The critical value for a one-tailed or two-tailed test can be computed using the confidence interval. Suppose a confidence interval of 95% has been specified for conducting a hypothesis test. The critical value can be determined as follows:

**Step 1:**Subtract the confidence level from 100%. 100% - 95% = 5%.**Step 2:**Convert this value to decimals to get \(\alpha\). Thus, \(\alpha\) = 5%.**Step 3:**If it is a one-tailed test then the alpha level will be the same value in step 2. However, if it is a two-tailed test, the alpha level will be divided by 2.**Step 4:**Depending on the type of test conducted the critical value can be looked up from the corresponding distribution table using the alpha value.

The process used in step 4 will be elaborated in the upcoming sections.

## T Critical Value

A t-test is used when the population standard deviation is not known and the sample size is lesser than 30. A t-test is conducted when the population data follows a Student t distribution. The t critical value can be calculated as follows:

- Determine the alpha level.
- Subtract 1 from the sample size. This gives the degrees of freedom (df).
- If the hypothesis test is one-tailed then use the one-tailed t distribution table. Otherwise, use the two-tailed t distribution table for a two-tailed test.
- Match the corresponding df value (left side) and the alpha value (top row) of the table. Find the intersection of this row and column to give the t critical value.

**Test Statistic for one sample t test:** t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, s is the sample standard deviation and n is the size of the sample.

**Test Statistic for two samples t test:** \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).

**Decision Criteria:**

- Reject the null hypothesis if test statistic > t critical value (right-tailed hypothesis test).
- Reject the null hypothesis if test statistic < t critical value (left-tailed hypothesis test).
- Reject the null hypothesis if the test statistic does not lie in the acceptance region (two-tailed hypothesis test).

This decision criterion is used for all tests. Only the test statistic and critical value change.

## Z Critical Value

A z test is conducted on a normal distribution when the population standard deviation is known and the sample size is greater than or equal to 30. The z critical value can be calculated as follows:

- Find the alpha level.
- Subtract the alpha level from 1 for a two-tailed test. For a one-tailed test subtract the alpha level from 0.5.
- Look up the area from the z distribution table to obtain the z critical value. For a left-tailed test, a negative sign needs to be added to the critical value at the end of the calculation.

**Test statistic for one sample z test:** z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\sigma\) is the population standard deviation.

**Test statistic for two samples z test:** z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

## F Critical Value

The F test is largely used to compare the variances of two samples. The test statistic so obtained is also used for regression analysis. The f critical value is given as follows:

- Find the alpha level.
- Subtract 1 from the size of the first sample. This gives the first degree of freedom. Say, x
- Similarly, subtract 1 from the second sample size to get the second df. Say, y.
- Using the f distribution table, the intersection of the x column and y row will give the f critical value.

**Test Statistic for large samples:** f = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\). \(\sigma_{1}^{2}\) variance of the first sample and \(\sigma_{2}^{2}\) variance of the second sample.

**Test Statistic for small samples:** f = \(\frac{s_{1}^{2}}{s_{2}^{2}}\). \(s_{1}^{1}\) variance of the first sample and \(s_{2}^{2}\) variance of the second sample.

## Chi-Square Critical Value

The chi-square test is used to check if the sample data matches the population data. It can also be used to compare two variables to see if they are related. The chi-square critical value is given as follows:

- Identify the alpha level.
- Subtract 1 from the sample size to determine the degrees of freedom (df).
- Using the chi-square distribution table, the intersection of the row of the df and the column of the alpha value yields the chi-square critical value.

**Test statistic for chi-squared test statistic: **\(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\).

## Critical Value Calculation

Suppose a right-tailed z test is being conducted. The critical value needs to be calculated for a 0.0079 alpha level. Then the steps are as follows:

- Subtract the alpha level from 0.5. Thus, 0.5 - 0.0079 = 0.4921
- Using the z distribution table find the area closest to 0.4921. The closest area is 0.4922. As this value is at the intersection of 2.4 and 0.02 thus, the z critical value = 2.42.

**Related Articles:**

- Probability and Statistics
- Data Handling
- Data

**Important Notes on Critical Value**

- Critical value can be defined as a value that is useful in checking whether the null hypothesis can be rejected or not by comparing it with the test statistic.
- It is the point that divides the distribution graph into the acceptance and the rejection region.
- There are 4 types of critical values - z, f, chi-square, and t.

## FAQs on Critical Value

### What is the Critical Value in Statistics?

Critical value in statistics is a cut-off value that is compared with a test statistic in hypothesis testing to check whether the null hypothesis should be rejected or not.

### What are the Different Types of Critical Value?

There are 4 types of critical values depending upon the type of distributions they are obtained from. These distributions are given as follows:

- Normal distribution (z critical value).
- Student t distribution (t).
- Chi-squared distribution (chi-squared).
- F distribution (f).

### What is the Critical Value Formula for an F test?

To find the critical value for an f test the steps are as follows:

- Find the alpha level.
- Determine the degrees of freedom for both samples by subtracting 1 from each sample size.
- Find the corresponding value from a one-tailed or two-tailed f distribution at the given alpha level.
- This will give the critical value.

### What is the T Critical Value?

The t critical value is obtained when the population follows a t distribution. The steps to find the t critical value are as follows:

- Determine the alpha level.
- Subtract the sample size number by 1 to get the df.
- Use the t distribution table for the alpha value to get the required critical value.

### How to Find the Critical Value Using a Confidence Interval for a Two-Tailed Z Test?

The steps to find the critical value using a confidence interval are as follows:

- Subtract the confident interval from 100% and convert the resultant into a decimal value to get the alpha level.
- Subtract this value from 1.
- Find the z value for the corresponding area using the normal distribution table to get the critical value.

### Can a Critical Value be Negative?

If a left-tailed test is being conducted then the critical value will be negative. This is because the critical value will be to the left of the mean thus, making it negative.

### How to Reject Null Hypothesis Based on Critical Value?

The rejection criteria for the null hypothesis is given as follows:

- Right-tailed test: Test statistic > critical value.
- Left-tailed test: Test statistic < critical value.
- Two-tailed test: Reject if the test statistic does not lie in the acceptance region.