
CBSE Important Questions›

CBSE Previous Year Question Papers›
 CBSE Previous Year Question Papers
 CBSE Previous Year Question Papers Class 12
 CBSE Previous Year Question Papers Class 10

CBSE Revision Notes›

CBSE Syllabus›

CBSE Extra Questions›

CBSE Sample Papers›
 CBSE Sample Papers
 CBSE Sample Question Papers For Class 5
 CBSE Sample Question Papers For Class 4
 CBSE Sample Question Papers For Class 3
 CBSE Sample Question Papers For Class 2
 CBSE Sample Question Papers For Class 1
 CBSE Sample Question Papers For Class 12
 CBSE Sample Question Papers For Class 11
 CBSE Sample Question Papers For Class 10
 CBSE Sample Question Papers For Class 9
 CBSE Sample Question Papers For Class 8
 CBSE Sample Question Papers For Class 7
 CBSE Sample Question Papers For Class 6

ISC & ICSE Syllabus›

ICSE Question Paper›
 ICSE Question Paper
 ISC Class 12 Question Paper
 ICSE Class 10 Question Paper

ICSE Sample Question Papers›
 ICSE Sample Question Papers
 ISC Sample Question Papers For Class 12
 ISC Sample Question Papers For Class 11
 ICSE Sample Question Papers For Class 10
 ICSE Sample Question Papers For Class 9
 ICSE Sample Question Papers For Class 8
 ICSE Sample Question Papers For Class 7
 ICSE Sample Question Papers For Class 6

ICSE Revision Notes›
 ICSE Revision Notes
 ICSE Class 9 Revision Notes
 ICSE Class 10 Revision Notes

ICSE Important Questions›

Maharashtra board›

RajasthanBoard›
 RajasthanBoard

Andhrapradesh Board›
 Andhrapradesh Board
 AP Board Sample Question Paper
 AP Board syllabus
 AP Board Previous Year Question Paper

Telangana Board›

Tamilnadu Board›

NCERT Solutions Class 12›
 NCERT Solutions Class 12
 NCERT Solutions Class 12 Economics
 NCERT Solutions Class 12 English
 NCERT Solutions Class 12 Hindi
 NCERT Solutions Class 12 Maths
 NCERT Solutions Class 12 Physics
 NCERT Solutions Class 12 Accountancy
 NCERT Solutions Class 12 Biology
 NCERT Solutions Class 12 Chemistry
 NCERT Solutions Class 12 Commerce

NCERT Solutions Class 10›

NCERT Solutions Class 11›
 NCERT Solutions Class 11
 NCERT Solutions Class 11 Statistics
 NCERT Solutions Class 11 Accountancy
 NCERT Solutions Class 11 Biology
 NCERT Solutions Class 11 Chemistry
 NCERT Solutions Class 11 Commerce
 NCERT Solutions Class 11 English
 NCERT Solutions Class 11 Hindi
 NCERT Solutions Class 11 Maths
 NCERT Solutions Class 11 Physics

NCERT Solutions Class 9›

NCERT Solutions Class 8›

NCERT Solutions Class 7›

NCERT Solutions Class 6›

NCERT Solutions Class 5›
 NCERT Solutions Class 5
 NCERT Solutions Class 5 EVS
 NCERT Solutions Class 5 English
 NCERT Solutions Class 5 Maths

NCERT Solutions Class 4›

NCERT Solutions Class 3›

NCERT Solutions Class 2›
 NCERT Solutions Class 2
 NCERT Solutions Class 2 Hindi
 NCERT Solutions Class 2 Maths
 NCERT Solutions Class 2 English

NCERT Solutions Class 1›
 NCERT Solutions Class 1
 NCERT Solutions Class 1 English
 NCERT Solutions Class 1 Hindi
 NCERT Solutions Class 1 Maths

JEE Main Question Papers›

JEE Main Syllabus›
 JEE Main Syllabus
 JEE Main Chemistry Syllabus
 JEE Main Maths Syllabus
 JEE Main Physics Syllabus

JEE Main Questions›
 JEE Main Questions
 JEE Main Maths Questions
 JEE Main Physics Questions
 JEE Main Chemistry Questions

JEE Main Mock Test›
 JEE Main Mock Test

JEE Main Revision Notes›
 JEE Main Revision Notes

JEE Main Sample Papers›
 JEE Main Sample Papers

JEE Advanced Question Papers›

JEE Advanced Syllabus›
 JEE Advanced Syllabus

JEE Advanced Mock Test›
 JEE Advanced Mock Test

JEE Advanced Questions›
 JEE Advanced Questions
 JEE Advanced Chemistry Questions
 JEE Advanced Maths Questions
 JEE Advanced Physics Questions

JEE Advanced Sample Papers›
 JEE Advanced Sample Papers

NEET Eligibility Criteria›
 NEET Eligibility Criteria

NEET Question Papers›

NEET Sample Papers›
 NEET Sample Papers

NEET Syllabus›

NEET Mock Test›
 NEET Mock Test

NCERT Books Class 9›
 NCERT Books Class 9

NCERT Books Class 8›
 NCERT Books Class 8

NCERT Books Class 7›
 NCERT Books Class 7

NCERT Books Class 6›
 NCERT Books Class 6

NCERT Books Class 5›
 NCERT Books Class 5

NCERT Books Class 4›
 NCERT Books Class 4

NCERT Books Class 3›
 NCERT Books Class 3

NCERT Books Class 2›
 NCERT Books Class 2

NCERT Books Class 1›
 NCERT Books Class 1

NCERT Books Class 12›
 NCERT Books Class 12

NCERT Books Class 11›
 NCERT Books Class 11

NCERT Books Class 10›
 NCERT Books Class 10

Chemistry Full Forms›
 Chemistry Full Forms

Biology Full Forms›
 Biology Full Forms

Physics Full Forms›
 Physics Full Forms

Educational Full Form›
 Educational Full Form

Examination Full Forms›
 Examination Full Forms

Algebra Formulas›
 Algebra Formulas

Chemistry Formulas›
 Chemistry Formulas

Geometry Formulas›
 Geometry Formulas

Math Formulas›
 Math Formulas

Physics Formulas›
 Physics Formulas

Trigonometry Formulas›
 Trigonometry Formulas

CUET Admit Card›
 CUET Admit Card

CUET Application Form›
 CUET Application Form

CUET Counselling›
 CUET Counselling

CUET Cutoff›
 CUET Cutoff

CUET Previous Year Question Papers›
 CUET Previous Year Question Papers

CUET Results›
 CUET Results

CUET Sample Papers›
 CUET Sample Papers

CUET Syllabus›
 CUET Syllabus

CUET Eligibility Criteria›
 CUET Eligibility Criteria

CUET Exam Centers›
 CUET Exam Centers

CUET Exam Dates›
 CUET Exam Dates

CUET Exam Pattern›
 CUET Exam Pattern
F Test Formula
Ftest is a statistical method used in hypothesis testing to determine if the variances of two populations or samples are equal. The data in an Ftest follows an Fdistribution, and the test compares two variances by calculating the ratio between them using the F statistic. Depending on the problem’s parameters, the Ftest can be either onetailed or twotailed.
Quick Links
ToggleThe F value derived from conducting an Ftest is essential for performing a oneway ANOVA (Analysis of Variance) test. In this article, we will explore the Ftest in detail, including the F statistic, its critical value, the formula, and the steps for conducting an Ftest for hypothesis testing.
What is F Test in Statistics?
An Ftest is any statistical test that utilizes the Fdistribution. The Fvalue is a specific point on the Fdistribution, generated by various statistical tests to determine whether the test results are statistically significant. In statistics, an Ftest refers to any test statistic that follows an Fdistribution under the null hypothesis. This test is used to compare statistical models based on the available data set. The Ftest formula was named by George W. Snedecor in honor of Sir Ronald A. Fisher.
Definition of F Test Formula
The Ftest 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 Ftest 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 Ftest is employed to assess the equality of variances through hypothesis testing. The Ftest formula varies depending on the type of hypothesis test being conducted, as detailed below:
LeftTailed Test:
 Null Hypothesis (H₀): σ_{12} =σ_{22}
 Alternate Hypothesis (H₁): σ_{12}<σ_{22 }
Decision Criteria: Reject the null hypothesis if the F statistic is less than the critical F value.
RightTailed Test:
 Null Hypothesis (H₀): σ_{12} =σ_{22}
 Alternate Hypothesis (H₁): σ_{12}>σ_{22 }
Decision Criteria: Reject the null hypothesis if the F statistic is greater than the critical F value.
TwoTailed Test:
 Null Hypothesis (H₀): σ_{12} =σ_{22}
 Alternate Hypothesis (H₁): σ_{12}≠σ_{22}
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 FTest 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 = σ_{12}/σ_{22}
Where:
 σ_{12} is the larger sample variance.
 σ_{22} is the smaller sample variance.
To compare the variances of two different data sets using the Ftest, you need to apply the Ftest 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:
σ^{2} = ∑(x − \(\bar{x}\))^{2}/n−1
Where:
 σ^{2} 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 Ftest with technology, follow these steps:
 Formulate the null hypothesis and the alternative hypothesis.
 Calculate the Fvalue using the appropriate formula.
 Determine the Fstatistic, which is the critical value for this test. This Fstatistic is calculated as the ratio of the variance between group means to the variance within groups.
 Based on the Fstatistic, 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 righttailed and a twotailed F Test Formula, the numerator will contain the variance with the larger value. As a result, the sample that corresponds to the σ_{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 lefttailed test (sample 2).
F Test Critical Value
The critical value for an Ftest is determined based on the Fdistribution, 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.
 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).
 Choose a Significance Level ($α$):
 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.
 Use an FDistribution Table:
 FDistribution 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 oneway 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:
 H_{0}: All groups’ means are equal.
 H_{1}: 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 df_{1}=K – 1 and df_{1}=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 TTest
The Ftest and ttest 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 Ftest and the ttest:
F Test  T Test 
Compares the variances of two populations or samples  Compares the means of two populations or samples 
Ratio of two sample variances (larger/smaller)  Difference between sample means divided by the standard error 
Hypothesis=Tests if variances are equal  Hypothesis=Tests if means are equal 
Null Hypothesis (H₀)= The variances are equal (σ_{12} =σ_{22})  Null Hypothesis (H₀)= The means are equal (μ_{1 }= μ_{2} ) 
Used when comparing variability  Used when comparing average values 
Examples on F Test
Example 1: Given the Fstatistic 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 onetailed test, we consult the Ftable.
Solution:
Referring to the Ftable with degrees of freedom 8 and 3, we find the critical Fvalue to be 8.845 for a onetailed test at a 5% significance level.
Since the calculated Fstatistic (2.38) is less than the critical Fvalue (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 Ftest 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 Ftest 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 bellshaped distribution. Also required are independent events for the samples. Students should also keep in mind the following crucial information:
Example 4: Perform an FTest 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 Fcritical value in step two. Here, use the denominator with the lowest variance and the numerator with the highest variance:
F Value= σ_{21}/σ_{22}
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 FTable, students will determine the important FValue. Students will take advantage of the 0.025 tables. CriticalF 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.
FAQs (Frequently Asked Questions)
1. What does the critical value for an F Test Formula mean?
The cutoff 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 onetailed or twotailed 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 Ttest?
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 TTest is applied to a student Tdistribution. It is applied to mean comparisons.
5. How do we determine the significance of the Fvalue?
The significance of the Fvalue is assessed by comparing it to a critical value obtained from statistical tables. If the calculated Fvalue exceeds the critical value at a chosen level of significance, typically 0.05, then we reject the null hypothesis, indicating significant differences in variances.