# TD_FTest Usage Notes | F Test - TD_FTest Usage Notes - Analytics Database

## Database Analytic Functions

Deployment
VantageCloud
VantageCore
Edition
Enterprise
IntelliFlex
VMware
Product
Analytics Database
Release Number
17.20
Published
June 2022
Language
English (United States)
Last Update
2024-04-06
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Product Category
Teradata Vantage™

In statistical analysis, the F-test is a commonly used hypothesis test to determine whether two population variances are equal. It is based on the F distribution, which is a probability distribution that arises in the analysis of variance (ANOVA) models.

The F-test is typically used in situations where we want to compare the variability of two sets of data or test whether a particular variable has a significant effect on the outcome of a study. It works by calculating the ratio of two variances and comparing it to a critical value obtained from the F distribution.

## Assumptions

• Populations from which samples are drawn are normally distributed.
• Populations are independent of each other.
• Data is numeric.

## Test Type

• One-tailed (lower and upper-tailed) or two-tailed (your choice)
• Two-sample
• Unpaired

## Computational Method

The F-test is used to test the Null hypothesis σ2 = in various applications. For example, you might need to test the variability in the measurement of the thickness of a manufactured part in a factory. If the thickness is not equal to a certain thickness ( ) then you can conclude that the manufacturing process is uncontrolled. The types of hypothesis are as follows:

H0: σ2 =

versus

H1: σ2 > (upper-tailed)

or

H1: σ2 < (lower-tailed)

or

H1: σ2 (two-tailed)

Let x1, x2,....xn be a random sample. To test the hypotheses, the test statistic is calculated as:

where

The statistic χ2 follows an F distribution with n-1 degrees of freedom.

For the one-sided upper-tailed test σ2 > the Null hypothesis H0 is rejected if .

For the one-sided lower-tailed test σ2 < , the Null hypothesis H0 is rejected if .

For the two-sided alternative σ2, the Null hypothesis H0 is rejected if

Also, the F-test is used to test if the variances of two populations are equal. The F-test can have the following tests:
• One-tailed test: The test is used to determine if the variance of one population is either greater than (upper-tailed) or less than (lower-tailed) the variance of another population.
• Two-tailed test: The test is used to determine significant differences in variances of the two populations and tests the Null hypothesis (H0) against the alternative hypothesis (H1) to find out if the variances are not equal.
Let x1, x2,....xn1 ~Ɲ (µ1, σ2) and y1, y2,....yn2 ~Ɲ (µ2, σ2) be random samples from two independent populations. The corresponding sample means and variances are as follows:
• Sample Means Formula:
• Sample Variance Formula:
• Sample Variance Formula for and : and

In the following calculation, assume that sample 1 has a larger variance than sample 2. If sample 2 has a larger variance than sample 1, switch the samples and apply the same formula.

H0: =

versus

H1: >

or

<

The test statistic for the one-sided upper tailed test ( > ) is calculated as:

where: n1-1 and n2-1 are degrees of freedom corresponding to sample 1 and sample 2.

The Null hypothesis H0 is rejected if .

The test statistic for the one-sided lower-tailed test ( < ) is calculated as:

The Null hypothesis H0 is rejected if .

For the two-sided hypothesis test:

H0: =

versus

H1:

The Null hypothesis H0 is rejected if:

The two-tailed test is based on the upper tail of the F-distribution.