# Difference Between T Test and F Test with Proper Definition and Brief Explanation

Hypothesis testing begins with the setting of the premises, which is followed by selecting a level of significance. Next, we have to choose the test statistic, that is, 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 an independent variable and a dependent variable. It is capable of being tested and verified for validity by impartial examination. Testing a hypothesis attempts to clarify whether the assumption is valid or not.

For an investigator, it is imperative to choose the correct test for his hypothesis, since the entire decision to validate or reject the null hypothesis is based on it. Read the given article to understand the difference between t-test and f-test.

## Content: T-test Vs F-test

- Comparative graph
- Definition
- Key differences
- conclusion

### Comparative graph

Basis for comparison | T test | Test F |
---|---|---|

Sense |
The T test is a univariate hypothesis test, applied when the standard deviation is unknown and the sample size is small. | The F test is a statistical test that determines the equality of the variances of the two normal populations. |

Test statistic |
The T statistic follows the Student’s t distribution, under the null hypothesis. | The F statistic follows the Senseco f distribution, under the 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 the Student’s t-statistic and the t-distribution to find the p-value (probability) that can be used to accept or reject the null hypothesis.

The T test looks at whether the means of two data sets are very different from each other, that is, whether the population mean is equal to or different from the standard mean. It can also be used to determine if the regression line has a nonzero slope. The test is based on a series of assumptions, which are:

- The population is infinite and normal.
- The population variance is unknown and estimated from the sample.
- The mean is known.
- The observations in the sample are random and independent.
- The sample size is small.
- H
_{0}can be one-sided or two-sided.

The mean and standard deviation of the two samples are used to make a comparison between them, so that:

where,

X _{1} = Mean of the first data set

x̄2 = Mean of the second data set

S _{1} = Standard deviation of the first data set

S _{2} = Standard deviation of the second data set

North _{1} = Size of the first data set

North _{2} = Size from the second data set

### Definition of F test

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

The F test can also be used to check if the data fit a regression model, which is acquired by least squares analysis. When there is a multiple linear regression analysis, examine the overall validity of the model or determine if any of the independent variables have a linear relationship with the dependent variable. Various predictions can be made by comparing the two data sets. The expression of the value of f-test is found in the relation of variances of the two observations, shown below:

Where, σ ^{2} = variance

The assumptions on which f-test is based are:

- The population is normally distributed.
- The samples have been drawn at random.
- The observations are independent.
- H
_{0}can be one-sided or two-sided.

## Key differences between T-test and F-test

The difference between the t-test and the f-test can be clearly drawn for the following reasons:

- A univariate hypothesis test that is applied when the standard deviation is unknown and the sample size is small is the t test. On the other hand, a statistical test, which determines the equality of the variations of the two normal data sets, is known as an f test.
- The t test is based on the T statistic following the Student’s t distribution, under the null hypothesis. On the contrary, the basis of the f test is the F statistic that follows the Senseco 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 tests used for hypothesis testing and decide whether we are going to accept the null hypothesis or reject it. Hypothesis testing does not make decisions on its own, but rather helps the researcher make decisions.