F Test Formula

 F-test is a statistical method used in hypothesis testing to determine if the variances of two populations or samples are equal. The data in an F-test follows an F-distribution, and the test compares two variances by calculating the ratio between them using the F statistic. Depending on the problem’s parameters, the F-test can be either one-tailed or two-tailed.

The F value derived from conducting an F-test is essential for performing a one-way ANOVA (Analysis of Variance) test. In this article, we will explore the F-test in detail, including the F statistic, its critical value, the formula, and the steps for conducting an F-test for hypothesis testing.

What is F Test in Statistics?

An F-test is any statistical test that utilizes the F-distribution. The F-value is a specific point on the F-distribution, generated by various statistical tests to determine whether the test results are statistically significant. In statistics, an F-test refers to any test statistic that follows an F-distribution under the null hypothesis. This test is used to compare statistical models based on the available data set. The F-test formula was named by George W. Snedecor in honor of Sir Ronald A. Fisher.

Definition of F Test Formula

The F-test formula is employed to perform a statistical test that allows the tester to determine whether two population sets, which follow a normal distribution, have the same standard deviation.When conducting a hypothesis test, if the F-test results are statistically significant, the null hypothesis can be rejected. If the results are not statistically significant, the null hypothesis cannot be rejected.

F Test Formula

The F-test is employed to assess the equality of variances through hypothesis testing. The F-test formula varies depending on the type of hypothesis test being conducted, as detailed below:

Left-Tailed Test:

• Null Hypothesis (H₀): σ₁₂ =σ₂₂
• ​Alternate Hypothesis (H₁): σ₁₂<σ₂₂ 

Decision Criteria: Reject the null hypothesis if the F statistic is less than the critical F value.

Right-Tailed Test:

• Null Hypothesis (H₀): σ₁₂ =σ₂₂ 
• Alternate Hypothesis (H₁): σ₁₂>σ_{22 ​}

Decision Criteria: Reject the null hypothesis if the F statistic is greater than the critical F value.

Two-Tailed Test:

• Null Hypothesis (H₀): σ₁₂ =σ₂₂
• ​Alternate Hypothesis (H₁): σ₁₂≠σ₂₂ ​

Decision Criteria: Reject the null hypothesis if the F statistic is either greater than or less than the critical F value, depending on the direction of the test.

Formula for F-Test to Compare Two Variances

To compare two variances, you need to calculate the ratio of the two variances as follows:

F Value = Larger Sample Variance/Smaller Sample Variance ​ = σ₁₂/σ₂₂

​Where:

• σ₁₂ is the larger sample variance.
• σ₂₂ is the smaller sample variance.

To compare the variances of two different data sets using the F-test, you need to apply the F-test formula, which is based on the F distribution under the null hypothesis. First, calculate the mean of each set of observations, and then determine their variances using the following formula:

σ² = ∑(x − \(\bar{x}\))²/n−1

Where:

• σ²  is the variance.
• x represents each individual observation.
• \(\bar{x}\) is the mean of the observations.
• n is the number of observations.

How to Use F Test Formula?

While using an F-test with technology, follow these steps:

• Formulate the null hypothesis and the alternative hypothesis.
• Calculate the F-value using the appropriate formula.
• Determine the F-statistic, which is the critical value for this test. This F-statistic is calculated as the ratio of the variance between group means to the variance within groups.
• Based on the F-statistic, decide whether to support or reject the null hypothesis.

F Statistic

The crucial value is compared with the F Test Statistic, also known as the F Statistic, to determine whether or not the null hypothesis should be rejected. The following is the statistic F Test Formula:

For both a right-tailed and a two-tailed  F Test Formula, the numerator will contain the variance with the larger value. As a result, the sample that corresponds to the σ₂₁ will be the first sample. The denominator will come from the second sample and have a lower value variance.

The sample with the smallest variance becomes the numerator (sample 1), while the sample with the highest variance becomes the denominator for a left-tailed test (sample 2).

F Test Critical Value

The critical value for an F-test is determined based on the F-distribution, which depends on two sets of degrees of freedom: one for the numerator and one for the denominator. These degrees of freedom correspond to the variances being compared. The critical value is used to decide whether to reject the null hypothesis in the context of an ANOVA or regression analysis.

1. Identify the Degrees of Freedom:
   • Degrees of Freedom for the Numerator (df1): In ANOVA, this is typically the number of groups minus one (k – 1). In regression, it is the number of predictors.
   • Degrees of Freedom for the Denominator (df2): In ANOVA, this is the total number of observations minus the number of groups (N – k). In regression, it is the total number of observations minus the number of predictors minus one (N – p – 1).
2. Choose a Significance Level ($\alpha$):
   • Common significance levels are 0.05, 0.01, and 0.10. The significance level represents the probability of rejecting the null hypothesis when it is actually true.
3. Use an F-Distribution Table:
   • F-Distribution Table: These tables provide critical values for various combinations of degrees of freedom and significance levels. Look up the critical value that corresponds to your degrees of freedom (df1 and df2) and your chosen significance level.

ANOVA F Test

The F Test is exemplified by the one-way ANOVA. Analysis of variance, or ANOVA, is the term. It is employed to examine the relationship between the group’s observed variability and the group’s mean variability. The ANOVA test is carried out using the F Test Formula statistic. The following is the hypothesis:

• H₀: All groups’ means are equal.
• H₁: No two groups’ means are equal.
• F = explained variance / unexplained variance is the test statistic.
• Decision rule: Reject the null hypothesis if F > F’s critical value.

The degrees of freedom are supplied by df₁=K – 1 and df₁=N – K, where N is the total sample size and K is the number of groups, to determine the critical value of an ANOVA F Test.

F Test vs T-Test

The F-test and t-test are distinct statistical methods used for hypothesis testing, each suited for different types of data distributions and testing scenarios. The table below highlights the key differences between the F-test and the t-test:

Examples on F Test

Example 1: Given the F-statistic obtained by the statistician is 2.38, with degrees of freedom of 8 and 3. Determine whether the null hypothesis can be rejected at a 5% significance level for a one-tailed test, we consult the F-table.

Solution:

Referring to the F-table with degrees of freedom 8 and 3, we find the critical F-value to be 8.845 for a one-tailed test at a 5% significance level.

Since the calculated F-statistic (2.38) is less than the critical F-value (8.845), we do not have sufficient evidence to reject the null hypothesis. Therefore, we fail to reject the null hypothesis, indicating that there is no significant difference between the variances of the two populations.

Example 2: Suppose we have two samples with the following variances:

Sample 1: Variance = 25

Sample 2: Variance = 20

Calculate the F value.

Solution:

To test if the variances are significantly different, we can use the F-test formula:

F=Larger Sample Variance/Smaller Sample Variance

= 25/20​ = 1.25

Example 3: Consider two groups of data with the following variances:

Group 1: Variance = 18

Group 2: Variance = 15

Calculate the F value.

Solution:

Using the F-test formula:

F=Variance of Group 1/Variance of Group 2

=18/15

= 1.2

For the test, a number of assumptions are made. To utilise the test, the students’ population must be roughly normally distributed or have a bell-shaped distribution. Also required are independent events for the samples. Students should also keep in mind the following crucial information:

Example 4: Perform an F-Test on the examples below:

• Sample 1 is 41 and has a variance of 109.63.
• Sample 2 is variance is 65.99 and its sample size is 21.

Solution:

Step 1: First, format the thesis statements as follows:

H₀: There is no variation.

H_a: Variance differences.

Calculate the F-critical value in step two. Here, use the denominator with the lowest variance and the numerator with the highest variance:

F Value= σ₂₁/σ₂₂

F Value= 109.6365.99

F Value= 1.66

Step 3: Calculate the degrees of freedom as follows.

• With sample size 1, the degrees of freedom in the table will be 40 for sample 1 and 20 for sample 2, respectively.
• Choose the alpha level in step four. Since the query did not specify an alpha level, we can utilise the conventional threshold of 0.05. Use 0.025 to divide this in half for the test.

Step 4:

• Using the F-Table, students will determine the important F-Value. Students will take advantage of the 0.025 tables. Critical-F is 2.287 for (40,20) at alpha (0.025).
• Comparing the estimated value to the value from the standard table is step six. Students may reject the null hypothesis if our calculated value is higher than the table value. Here, 1.66 < 2 .287.