[Maths Class Notes] on F Test Formula Pdf for Exam

The F Test Formula is a Statistical Formula used to test the significance of differences between two groups of Data. It is often used in research studies to determine whether the difference in the means of two populations is Statistically significant.

It is based on the F Statistic, which is a measure of how much variation exists in one group of Data compared to another. Students who are studying for their Statistics course will need to be familiar with this Formula. Our article will provide a detailed explanation of how to use the F Test Formula. It will also provide examples of how to use it in practice.

The use of the F Test Formula is a critical step in any research study, and it is important to understand how to use it correctly. You will be able to find the F Test Formula in most Statistics textbooks.

What is the Definition of F-Test Statistic Formula?

It is a known fact that Statistics is a branch of Mathematics that deals with the collection, classification and representation of Data. The tests that use F – distribution are represented by a single word in Statistics called the F Test. F Test is usually used as a generalized Statement for comparing two variances. F Test Statistic Formula is used in various other tests such as regression analysis, the chow test and Scheffe test. F Tests can be conducted by using several technological aids. However, the manual calculation is a little complex and time-consuming. This article gives an in-detail description of the F Test Formula and its usage.

Definition of F-Test Formula

F Test is a test Statistic that has an F distribution under the null hypothesis. It is used in comparing the Statistical model with respect to the available Data set. The name for the test is given in honour of Sir. Ronald A Fisher by George W Snedecor. To perform an F Test using technology, the following aspects are to be taken care of.

• State the null hypothesis along with the alternative hypothesis.

• Compute the value of ‘F’ with the help of the standard Formula.

• Determine the value of the F Statistic. The ratio of the variance of the group of means to the mean of the within-group variances. 

• As the last step, support or reject the Null hypothesis.

F-Test Equation to Compare Two Variances:

In Statistics, the F-test Formula is used to compare two variances, say σ1 and σ2, by dividing them. As the variances are always positive, the result will also always be positive. Hence, the F Test equation used to compare two variances is given as:

F_value =[frac{variance1}{variance2}]

i.e. F_value = [frac{sigma_{1}^{2}}{sigma_{2}^{2}}]

F Test Formula helps us to compare the variances of two different sets of values. To use F distribution under the null hypothesis, it is important to determine the mean of the two given observations at first and then calculate the variance. 

[sigma ^{2}=frac{sum (x-bar{x})^{2}}{n-1}]

In the above formula, 

σ2 is the variance

x is the values given in a set of data

x is the mean of the given Data set 

n is the total number of values in the Data set

While running an F Test, it is very important to note that the population variances are equal. In more simple words, it is always assumed that the variances are equal to unity or 1. Therefore, the variances are always equal in the case of the null hypothesis.

F Test Statistic Formula Assumptions

F Test equation involves several assumptions. In order to use the F – test Formula, the population should be distributed normally. The samples considered for the test should be independent events. In addition to these, it is also important to consider the following points.

• Calculation of right-tailed tests is easier. To force the test into a right-tailed test, the larger variance is pushed in the numerator.

• In the case of two-tailed tests, alpha is divided by two prior to the determination of critical value. 

• Variances are the squares of the standard deviations.

If the obtained degree of freedom is not listed in the F table, it is always better to use a larger critical value to decrease the probability of type 1 errors. 

F-Value Definition: Example Problems

Example 1: 

Perform an F Test for the following samples.

1. Sample 1 with variance equal to 109.63 and sample size equal to 41.

2. Sample 2 with variance equal to 65.99 and sample size equal to 21.

Solution: 

Step 1:

The hypothesis Statements are written as:

H_0: No difference in variances 

H_a: Difference invariances 

Step 2: 

Calculate the value of F critical. In this case, the highest variance is taken as the numerator and the lowest variance in the denominator.

F_value = [frac{sigma_{1}^{2}}{sigma_{2}^{2}}]

F_value = [frac{109.63}{65.99}]

F_value = 1.66

Step 3:

The next step is the calculation of degrees of freedom.

The degrees of freedom is calculated as Sample size – 1

The degree of freedom for sample 1 is 41 -1 = 40.

The degree of freedom for sample 2 is 21 – 1 = 20.

Step 4:

There is no alpha level described in the question, and hence a standard alpha level of 0.05 is chosen. During the test, the alpha level should be reduced to half the initial value, and hence it becomes 0.025.

Step 5:

Using the F table, the critical F value is determined with alpha at 0.025. The critical value for (40, 20) at alpha equal to 0.025 is 2.287.

Step 6:

It is now the time for comparing the calculated value with the standard value in the table. Generally, the null hypothesis is rejected if the calculated value is greater than the table value. In this F value definition example, the calculated value is 1.66, and the table value is 2.287. 

It is clear from the values that 1.66 < 2.287. Hence, the null hypothesis cannot be rejected.

Fun Facts About F-Value Definition:

• In the case of Statistical calculations where the null hypothesis can be rejected, the F value can be less than 1; however, not exactly equal to zero.

• The F critical value cannot be exactly equal to zero. If the F value is exactly zero, it indicates that the mean of every sample is exactly the same, and the variance is zero. 

• One of the key points to remember while working with the F Statistic is that the population variances are always considered to be equal. If this condition is not met, the obtained F value might not be correct.

• The degrees of freedom is taken as the number of samples minus one. In the case of a two-sample problem, there are two samples, and hence it becomes 2 – 1 = 1.

• When the alpha level is not mentioned in the F Test, the standard value used in most of the cases is equal to 0.05.

Conclusion

In case of a problem with two sample Data sets, the F value can be obtained by dividing the larger variance by the smaller one. In order to perform a test at a pre-specified alpha level, it is always better to use standard values from the F table rather than using calculated values. The F value definition example has demonstrated how to calculate the F Statistic along with the relevant steps and interpretation of results. Students can use the F Statistic Formula to understand how it is used for t-test calculations. t-value definition examples are also available on this website. You can download the F table pdf to perform your own calculations.