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F Test
F test is a statistical test that is used in hypothesis testing to check whether the variances of two populations or two samples are equal or not. In an f test, the data follows an f distribution. This test uses the f statistic to compare two variances by dividing them. An f test can either be onetailed or twotailed depending upon the parameters of the problem.
The f value obtained after conducting an f test is used to perform the oneway ANOVA (analysis of variance) test. In this article, we will learn more about an f test, the f statistic, its critical value, formula and how to conduct an f test for hypothesis testing.
1.  What is F Test in Statistics? 
2.  F Test Formula 
3.  F Test Critical Value 
4.  ANOVA F Test 
5.  F Test vs TTest 
6.  FAQs on F Test 
What is F Test in Statistics?
F test is statistics is a test that is performed on an f distribution. A twotailed f test is used to check whether the variances of the two given samples (or populations) are equal or not. However, if an f test checks whether one population variance is either greater than or lesser than the other, it becomes a onetailed hypothesis f test.
F Test Definition
F test can be defined as a test that uses the f test statistic to check whether the variances of two samples (or populations) are equal to the same value. To conduct an f test, the population should follow an f distribution and the samples must be independent events. On conducting the hypothesis test, if the results of the f test are statistically significant then the null hypothesis can be rejected otherwise it cannot be rejected.
F Test Formula
The f test is used to check the equality of variances using hypothesis testing. The f test formula for different hypothesis tests is given as follows:
Left Tailed Test:
Null Hypothesis: \(H_{0}\) : \(\sigma_{1}^{2} = \sigma_{2}^{2}\)
Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} < \sigma_{2}^{2}\)
Decision Criteria: If the f statistic < f critical value then reject the null hypothesis
Right Tailed test:
Null Hypothesis: \(H_{0}\) : \(\sigma_{1}^{2} = \sigma_{2}^{2}\)
Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} > \sigma_{2}^{2}\)
Decision Criteria: If the f test statistic > f test critical value then reject the null hypothesis
Two Tailed test:
Null Hypothesis: \(H_{0}\) : \(\sigma_{1}^{2} = \sigma_{2}^{2}\)
Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} ≠ \sigma_{2}^{2}\)
Decision Criteria: If the f test statistic > f test critical value then the null hypothesis is rejected
F Statistic
The f test statistic or simply the f statistic is a value that is compared with the critical value to check if the null hypothesis should be rejected or not. The f test statistic formula is given below:
F statistic for large samples: F = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\), where \(\sigma_{1}^{2}\) is the variance of the first population and \(\sigma_{2}^{2}\) is the variance of the second population.
F statistic for small samples: F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\), where \(s_{1}^{2}\) is the variance of the first sample and \(s_{2}^{2}\) is the variance of the second sample.
The selection criteria for the \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\) for an f statistic is given below:
 For a righttailed and a twotailed f test, the variance with the greater value will be in the numerator. Thus, the sample corresponding to \(\sigma_{1}^{2}\) will become the first sample. The smaller value variance will be the denominator and belongs to the second sample.
 For a lefttailed test, the smallest variance becomes the numerator (sample 1) and the highest variance goes in the denominator (sample 2).
F Test Critical Value
A critical value is a point that a test statistic is compared to in order to decide whether to reject or not to reject the null hypothesis. Graphically, the critical value divides a distribution into the acceptance and rejection regions. If the test statistic falls in the rejection region then the null hypothesis can be rejected otherwise it cannot be rejected. The steps to find the f test critical value at a specific alpha level (or significance level), \(\alpha\), are as follows:
 Find the degrees of freedom of the first sample. This is done by subtracting 1 from the first sample size. Thus, x = \(n_{1}  1\).
 Determine the degrees of freedom of the second sample by subtracting 1 from the sample size. This given y = \(n_{2}  1\).
 If it is a righttailed test then \(\alpha\) is the significance level. For a lefttailed test 1  \(\alpha\) is the alpha level. However, if it is a twotailed test then the significance level is given by \(\alpha\) / 2.
 The F table is used to find the critical value at the required alpha level.
 The intersection of the x column and the y row in the f table will give the f test critical value.
ANOVA F Test
The oneway ANOVA is an example of an f test. ANOVA stands for analysis of variance. It is used to check the variability of group means and the associated variability in observations within that group. The F test statistic is used to conduct the ANOVA test. The hypothesis is given as follows:
\(H_{0}\): The means of all groups are equal.
\(H_{1}\): The means of all groups are not equal.
Test Statistic: F = explained variance / unexplained variance
Decision rule: If F > F critical value then reject the null hypothesis.
To determine the critical value of an ANOVA f test the degrees of freedom are given by \(df_{1}\) = K  1 and \(df_{1}\) = N  K, where N is the overall sample size and K is the number of groups.
F Test vs TTest
F test and ttest are different types of statistical tests used for hypothesis testing depending on the distribution followed by the population data. The table given below outlines the differences between the F test and the ttest.
F Test  TTest 

An F test is a test statistic used to check the equality of variances of two populations  The Ttest is used when the sample size is small (n < 30) and the population standard deviation is not known. 
The data follows an F distribution  The data follows a Student tdistribution 
The F test statistic is given as F = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\)  The ttest statistic for 1 sample is given by t = \(\frac{\overline{x}\mu}{\frac{s}{\sqrt{n}}}\), where \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, s is the sample standard deviation and n is the sample size. 
The f test is used for variances.  It is used for testing means. 
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Important Notes on F Test
 The f test is a statistical test that is conducted on an F distribution in order to check the equality of variances of two populations.
 The f test formula for the test statistic is given by F = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\).
 The f critical value is a cutoff value that is used to check whether the null hypothesis can be rejected or not.
 A oneway ANOVA is an example of an f test that is used to check the variability of group means and the associated variability in the group observations.
Examples on F Test

Example 1: A research team wants to study the effects of a new drug on insomnia. 8 tests were conducted with a variance of 600 initially. After 7 months 6 tests were conducted with a variance of 400. At a significance level of 0.05 was there any improvement in the results after 7 months?
Solution: As the variance needs to be compared, the f test needs to be used.
\(H_{0}\) : \(s_{1}^{2} = s_{2}^{2}\)
\(H_{1}\) : \(s_{1}^{2} > s_{2}^{2}\)
\(n_{1}\) = 8, \(n_{2}\) = 6
\(df_{1}\) = 8  1 = 7
\(df_{2}\) = 6  1 = 5
\(s_{1}^{2}\) = 600, \(s_{2}^{2}\) = 400
The f test formula is given as follows:
F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\) = 600 / 400
F = 1.5
Now from the F table the critical value F(0.05, 7, 5) = 4.88
As 1.5 < 4.88, thus, the null hypothesis cannot be rejected and there is not enough evidence to conclude that there was an improvement in insomnia after using the new drug.
Answer: Fail to reject the null hypothesis. 
Example 2: Pizza delivery times of two cities are given below
City 1: Number of delivery times observed = 28, Variance = 38
City 2: Number of delivery times observed = 25, Variance = 83
Check if the delivery times of city 1 are lesser than city 2 at a 0.05 alpha level.
Solution: This is an example of a lefttailed F test. Thus, the alpha level is 1  0.05 = 0.95
\(H_{0}\) : \(s_{1}^{2} = s_{2}^{2}\)
\(H_{1}\) : \(s_{1}^{2} < s_{2}^{2}\)
As 38 < 83 thus, city 1 will be sample 1 and city 2 is sample 2.
\(n_{1}\) = 28, \(n_{2}\) = 25
\(df_{1}\) = 28  1 = 27
\(df_{2}\) = 25  1 = 24
\(s_{1}^{2}\) = 38, \(s_{2}^{2}\) = 83
F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\) = 38 / 83
F = 0.4578
As an F table for 0.95 alpha level is not available, the critical value is determined as follows:
F(0.95, 27, 24) = 1 / F(0.05, 24, 27)
F(0.05, 24, 27) = 1.93
F(0.95, 27, 24) = 1 / 1.93 = 0.5181
As 0.4578 < 0.5181, thus, the null hypothesis can be rejected and it can be concluded that there is enough evidence to support the claim that the delivery times in city 1 are less than in city 2.
Answer: Reject the null hypothesis 
Example 3: A toy manufacturer wants to get batteries for toys. A team collected 41 samples from supplier A and the variance was 110 hours. The team also collected 21 samples from supplier B with a variance of 65 hours. At a 0.05 alpha level determine if there is a difference in the variances.
Solution: This is an example of a twotailed F test. Thus, the alpha level is 0.05 / 2 = 0.025
\(H_{0}\) : \(s_{1}^{2} = s_{2}^{2}\)
\(H_{1}\) : \(s_{1}^{2} \neq s_{2}^{2}\)
\(n_{1}\) = 41, \(n_{2}\) = 21
\(df_{1}\) = 41  1 = 40
\(df_{2}\) = 21  1 = 20
\(s_{1}^{2}\) = 110, \(s_{2}^{2}\) = 65
F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\) = 110 / 65
F = 1.69
Using the F table F(0.025, 40, 20) = 2.287
As 1.69 < 2.287 thus, the null hypothesis cannot be rejected,
Answer: Fail to reject the null hypothesis.
FAQs on F Test
What is the F Test?
The f test in statistics is used to find whether the variances of two populations are equal or not by using a onetailed or twotailed hypothesis test.
What is the F Test Formula?
The f test formula can be used to find the f statistic. The f test formula is given as follows:
 F statistic for large samples: F = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\)
 F statistic for small samples: F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\)
What is the Decision Criterion for a Right Tailed F Test?
The algorithm to set up an right tailed f test hypothesis along with the decision criteria are given as follows:
 Null Hypothesis: \(H_{0}\) : \(\sigma_{1}^{2} = \sigma_{2}^{2}\)
 Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} > \sigma_{2}^{2}\)
 Decision Criteria: Reject \(H_{0}\) if the f test statistic > f test critical value.
What is the Critical Value for an F Test?
The F critical value for an f test can be defined as the cutoff value that is compared with the test statistic to decide if the null hypothesis should be rejected or not.
Why is an F Test Used in ANOVA?
A oneway ANOVA test uses the f test to compare if there is a difference between the variability of group means and the associated variability of observations of those groups.
Can the F statistic in an F Test be Negative?
As the f test statistic is the ratio of variances thus, it cannot be negative. This is because the square of a number will always be positive.
What is the Difference Between F Test and TTest?
An F test is conducted on an f distribution to determine the equality of variances of two samples. The ttest is performed on a student t distribution when the number of samples is less and the population standard deviation is not known. It is used to compare means.
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