Introduction of F Test Formula
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.
Sample 1 with variance equal to 109.63 and sample size equal to 41.
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.
FAQs on F Test Formula
1. What is the difference between F and t-tests?
The difference between the F value definition and the t Statistic is that in the case of the F Test, both variances are not required to be equal. However, for a t-test, the population variance should be considered to be equal. The F Statistic also takes into account the degrees of freedom, whereas, in the probability calculations, it is not taken into consideration because it is a constant. F Test and T-test are used to compare the variance between two groups, whereas t-value definition is used to show how much two sample means that are drawn from the same population differ from each other. Students can practice t-test examples to test their understanding before attempting F Test examples.
2. What is the Significance Level for the F Test?
The significance level of an F Test is denoted by alpha value, and it is usually the pre-specified probability. A t Statistic can be used to reject or fail to reject the null hypothesis when the calculated t value is less than the t table critical values. The standard deviation does not need to be estimated in the case of the F Test if both samples are of equal size. The degrees of freedom for an F Test is equal to the number of samples minus one. The degrees of freedom can be defined as the number of observations that are free to vary in the calculation of Statistics. One can use the F table pdf to look up the critical values.
3. What are Some Applications of F-test?
Some of the applications of the F-test include comparing variance between two groups, testing difference between means of two independent groups, analysis of variance, and tests for homogeneity of variance. In all these cases, it is essential to ensure that the population variances are equal. If they are not equal, then an appropriate correction needs to be made to the obtained F value. You can download the F table pdf to get the relevant values.
An F-test is used to compare the variance between two groups. The larger variance is divided by the smaller variance to obtain the F Statistic. If this value is greater than the F critical value, then it is concluded that the difference between variances in the two groups is significant.
4. What is a Null in an F-test?
The null hypothesis usually States that the values being tested are not Statistically different from each other, whereas the alternative hypothesis States otherwise. In the case of an F Test, the null hypothesis States that there is no difference between population variances, while the alternative hypothesis States that there is a difference between population variances. The F Statistic is a calculation that is used along with the t-test. It is used to compare the variances between two groups. If the variance in one group is larger than the variance in another group, the F Statistic will be larger than the F critical value. This difference is Statistically significant if the F Statistic is greater than the F critical value. Answers to the frequently asked questions can be found in the F Test example and in other related topics for quick revision. The t-test examples are also a good resource for practising this concept.
5. How to Interpret the Results of the F-test?
In the case of an F-test, you can interpret the results by looking at the p-value. If the p-value is greater than 0.05, then it is concluded that there is no significant difference between population variances, and if it falls below 0.05, then it is concluded that there is a significant difference between population variances. It should be kept in mind that all F Test example problems are solved with step by step calculations.
To compare the variance between two groups, one should set up the null hypothesis as Ho: the population variances are equal and alternative hypothesis as Ha: the population variances are not equal. One uses an F Statistic to compare whether the variances in the two populations are Statistically different. If the p-value is less than 0.05, then it can be concluded that the variances are indeed significantly different.
6. 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.
7. 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.