Difference Between T-test and F-test

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Hypothesis testing starts with setting up the premises, which is followed by selecting a significance level. Next, we have to choose the test statistic, i.e. t-test or f-test. While t-test is used to compare two related samples, f-test is used to test the equality of two populations.

The hypothesis is a simple proposition that can be proved or disproved through various scientific techniques and establishes the relationship between independent and some dependent variable. It is capable of being tested and verified to ascertain its validity, by an unbiased examination. Testing of a hypothesis attempts to make clear, whether or not the supposition is valid.

For a researcher, it is imperative to choose the right test for his/her hypothesis as the entire decision of validating or refusing the null hypothesis is based on it. Take a read of the given article to understand the difference between t-test and f-test.

Comparison Chart

Basis for Comparison	T-test	F-test
Meaning	T-test is a univariate hypothesis test, that is applied when standard deviation is not known and the sample size is small.	F-test is statistical test, that determines the equality of the variances of the two normal populations.
Test statistic	T-statistic follows Student t-distribution, under null hypothesis.	F-statistic follows Snedecor f-distribution, under null hypothesis.
Application	Comparing the means of two populations.	Comparing two population variances.

Definition of T-test

A t-test is a form of the statistical hypothesis test, based on Student’s t-statistic and t-distribution to find out the p-value (probability) which can be used to accept or reject the null hypothesis.

T-test analyses if the means of two data sets are greatly different from each other, i.e. whether the population mean is equal to or different from the standard mean. It can also be used to ascertain whether the regression line has a slope different from zero. The test relies on a number of assumptions, which are:

The population is infinite and normal.
Population variance is unknown and estimated from the sample.
The mean is known.
Sample observations are random and independent.
The sample size is small.
H₀ may be one sided or two sided.

Mean and standard deviation of the two sample are used to make comparison between them, such that:

t-test where,

x̄₁ = Mean of the first dataset
x̄2 = Mean of the second dataset
S₁ = Standard deviation of the first dataset
S₂ = Standard deviation of the second dataset
n₁ = Size of first dataset
n₂ = Size of second dataset

Definition of F-test

F-test is described as a type of hypothesis test, that is based on Snedecor f-distribution, under the null hypothesis. The test is performed when it is not known whether the two populations have the same variance.

F-test can also be used to check if the data conforms to a regression model, which is acquired through least square analysis. When there is multiple linear regression analysis, it examines the overall validity of the model or determines whether any of the independent variables is having a linear relationship with the dependent variable. A number of predictions can be made through, the comparison of the two datasets. The expression of the f-test value is in the ratio of variances of the two observations, which is shown as under:

f-test Where, σ² = variance

The assumptions on which f-test relies are:

The population is normally distributed.
Samples have been drawn randomly.
Observations are independent.
H₀ may be one sided or two sided.

Key Differences Between T-test and F-test

The difference between t-test and f-test can be drawn clearly on the following grounds:

A univariate hypothesis test that is applied when the standard deviation is not known and the sample size is small is t-test. On the other hand, a statistical test, which determines the equality of the variances of the two normal datasets, is known as f-test.
The t-test is based on T-statistic follows Student t-distribution, under the null hypothesis. Conversely, the basis of the f-test is F-statistic follows Snedecor f-distribution, under the null hypothesis.
The t-test is used to compare the means of two populations. In contrast, f-test is used to compare two population variances.

Conclusion

T-test and f-test are the two, of the number of different types of statistical test used for hypothesis testing and decides whether we are going to accept the null hypothesis or reject it. The hypothesis test does not take decisions itself, rather it assists the researcher in decision making.