# F Test Formula

## F Test Formula

All tests that make use of the F-distribution are collectively referred to as “F Tests.” The F Test Formula to Compare Two Variances is often what is meant when the term “F-Test” is used. However, a number oF Tests, including regression analysis, the Chow test, and the Scheffe test, use the f-statistic. If students are conducting an F Test Formula, they might employ a variety of technological tools since doing an F Test Formula manually while accounting for variations is a difficult and time-consuming operation. Students will learn about the F Test Formula in this article through examples. Here students will learn about it. The Extramarks website provides study materials for students interested in learning more about this topic.

An F Test Formula is a statistical test statistic that has an F-distribution under the null hypothesis. It is used to compare statistical models according to the provided data set. The F Test Formula was given by George W. Snedecor as a tribute to Sir Ronald A. Fisher.

The steps below are present if students use technology to do an F Test:

• Describe the alternative hypothesis and the null hypothesis.
• Apply the formula to determine the F-value.
• Discover the F Statistic, which is the test’s critical value. The ratio of the variance of the group means divided by the mean of the within-group variances is the F-statistic formula.
• Decide whether to accept or reject the null hypothesis.

## What is F Test in Statistics?

A statistical F Test divides two variances, 1 and 2, and compares them using the F statistic. Due to the fact that variances are always positive, the outcome will always be a positive number. The formula for using the F-test to compare two variances is as follows:

F−value=variance1/variance2

i.e. F−value=σ21/σ22

The F Test Formula is used to compare the variances of two separate sets of values. Under the null hypothesis, students must first compute the mean of the two given observations before computing their variance. This is applied to the F distribution.

The F Test Formula is used to compare the variances of two separate sets of values. Under the null hypothesis, students must first compute the mean of the two given observations before computing their variance. This is applied to the F distribution.

σ2=∑(x–x¯¯¯)2n−1

σ2 Variance

X Values given in a set of data

X¯ Mean of the data

n The total number of values.

When doing an F-Test, students always verify that the population variances are equal. To put it another way, students always consider the variances to be equal to 1. Therefore, the equality of the variances will always be our null hypothesis.

It is common knowledge that the gathering, classification, and display of data fall under the umbrella of Statistics, a subfield of Mathematics. The F Test is the single word used in Statistics to describe tests that make use of the F-distribution. A generic statement for comparing two variances is the F Test. Other tests including Regression Analysis, the Chow Test, and the Scheffe Test use the statistics F Test Formula. The F Test Formula can be performed with the help of many technical tools. However, manual computation requires some time and is somewhat difficult. The F Test Formula and its use are thoroughly explained in this article.

## F Test Definition

The F Test Formula can be used to run the statistical test, which aids in determining whether the standard deviation of two population sets with normal distributions of data points is the same.

Any test that applies F-distribution is an F Test Formula. A value on the F-distribution is called an F-value. A number of statistical analyses produce an F value. If the value is significant statistically, the test may be considered to be valid.

The null and alternative hypotheses must be framed before running the F Test Formula. The next step is to establish the level of relevance that the test must meet. The degrees of freedom for the numerator and denominator must then be determined. It will assist in calculating the F-table value. Then, to decide whether to reject the null hypothesis, the F-value shown in the table is contrasted with the computed F-value.

## F Test Formula

Any test that makes use of the F-distribution is referred to as an “F Test” in this context. Most of the time, when someone refers to the F Test Formula, they are really referring to the F Test Formula to Compare Two Variances. However, a number oF Tests, such as regression analysis, the Chow test, and the Scheffe Test, use the f-statistic (a post-hoc ANOVA test).

An f distribution is used as the basis for the statistical test known as the F Test. To determine if the variances of the two provided samples (or populations) are equal or not, a two-tailed F Test Formula is performed. A one-tailed hypothesis F Test Formula, on the other hand, is one that determines if one population variance is more or less than the other.

The F Test Formula statistic is used in a test called the F Test Formula to determine if the variances of two samples (or populations) are equal. An F Test needs an f distribution in the population and independent events for the samples in order to be valid. The null hypothesis can be rejected after conducting the hypothesis test if the F Test Formula results are statistically significant; otherwise, it cannot be rejected.

Using hypothesis testing, the F Test is used to verify the equivalence of variances. The following is the F Test Formula for several hypothesis tests:

Using Excel, SPSS, Minitab, or any other type of technology is recommended if students are conducting an F Test. This is because manually calculating the F Test with variances is laborious and time-consuming. As a result, students are likely to make negligent errors along the way.

The only steps students actually need to perform when using technology to run an F Test are Steps 1 and 4 (for example, when using Excel to run an F Test two samples for variances) (dealing with the null hypothesis). Steps 2 and 3 will be calculated for students by technology.

List the alternate theory and 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).

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 number 21 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

A test statistic is compared to a crucial value to determine whether to reject or not reject the null hypothesis. The crucial value, in terms of a graph, separates a distribution into acceptance and rejection zones. The null hypothesis can be rejected if the test statistic is within the rejection region; otherwise, it cannot be rejected. Following are the steps to determine the F Test Formula critical value at a particular α level (or significance level),

Find the first sample’s degrees of freedom. To do this, take 1 out of the initial sample size. Therefore, x = n1−1 .

Subtract 1 from the sample size to get the second sample’s degrees of freedom. This formula is y = n 2−1.

The significance level is if the test has a right-tailed distribution, then. The alpha level is 1 for a left-tailed test. However, the significance threshold is determined by α / 2 in the case of a two-tailed test.

The crucial value at the required alpha level is located using the F table.

The F Test critical value is determined by the junction of the x column and the y row in the f table.

### 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

Depending on the distribution that the population data follow, the F Test Formula and T-Test are two different kinds of statistical tests that are employed for hypothesis testing. The differences between the F Test Formula and the T-Test are outlined in the table below.

F Test:

• A test statistic called an F Test Formula is used to determine whether the variances of two populations are equal.
• Data exhibits an F distribution.
• The formula for the F Test statistic is F = σ2 1 /σ 2 2.
• Variances are tested using the F Test Formula.

T-Test

• When the population standard deviation is unknown and the sample size is small (n < 30), the T-test is applied.
• The data is distributed using the Student t-test.
• The formula for the T-Test statistic for one sample is t = ¯¯¯x − μ s √ n, where ¯¯¯ x, where x is the sample mean, μ is the sample standard deviation, and n is the sample size.
• It is employed to evaluate means.

An F distribution is used in the F Test, a statistical test, to determine whether the variances of two populations are equal.

F = σ2 1 /σ 2 2 is the F Test formula for the test statistic.

A cut-off value known as the f crucial value is used to determine whether or not the null hypothesis may be rejected.

An F Test Formula is used to evaluate the variability of group means and the corresponding variability in the group observations. A one-way ANOVA is an example of an F Test Formula.

### Examples on F Test

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:

• For the test to be right-tailed, the larger variance must always be in the numerator (the top number). Calculating right-tailed tests is simpler.
• Divide alpha by two before determining the appropriate critical value for two-tailed tests.
• The variances must be squared if students are given standard deviations.
• Use the higher critical value if the degrees of freedom for the students aren’t given in the F Table.
1. 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:

H 0: 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 5:

• 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.

## 1. What does the critical value for an F Test Formula mean?

The cut-off value used to determine whether or not to reject the null hypothesis in an F Test can be referred to as the F critical value.

## 2. Can a negative F statistic be used in an F Test Formula?

The F Test Formula statistic cannot be negative because it measures the ratio of variances. This is due to the fact that a number’s square is always positive.

## 3. Describe the F Test Formula.

A one-tailed or two-tailed hypothesis test is used to determine if the variances of two populations are equal or not. This is known as the F Test in statistics.

## 4. What distinguishes the F Test from the T-test?

To ascertain whether the variances of two samples are equal, one performs an F Test on an F distribution. When there are fewer samples and the population standard deviation is unknown, the T-Test is applied to a student T-distribution. It is applied to mean comparisons.