F Test Formula

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 is represented by a single word in Statistics called 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, manual calculation of 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 the honor 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 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 be positive always. Hence, the F test equation used to compare two variances is given as:

F_value = \[\frac{variance 1}{variance 2}\]

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

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

σ\[^{2}\] = \[\frac{Σ(x - \bar{x})^{2}}{n - 1}\]

In the above formula, 

σ2 is the variance

x is the values given in a set of data

\[\bar{x}\] 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 case of 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 case of two tailed tests, alpha is divided by two prior to 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 in variances 


Step 2: 

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

F_value = \[\frac{σ_{1}^{2}}{σ_{2}^{2}}\]

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, null hypothesis cannot be rejected.


Fun Facts About F-Value Definition:

  • In case of statistical calculations where 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. 

FAQ (Frequently Asked Questions)

1. What Does the F Test Tell You?

Any statistical test which has the test statistic of F distribution under the null hypothesis is called the F test. It is usually employed to compare the statistical models. The statistical models which are fitted to a data set are compared with each other to find the best model that fits the population from which the data samples were derived. The F test of a particular model gives an idea whether the designed linear regression model provides a proper space to the data than the model that does not contain independent variables or not.

2. What are the Characteristics of F Distribution?

A few important characteristics of F distribution are:

  • The curve is skewed to the right and hence is not symmetrical.

  • Each data set has a different curve.

  • The value of the F statistic may be greater than or equal to zero.

  • With the increase in the degrees of freedom in the numerator and the denominator, the curve attains normal approximation.