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T-test and F-test

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  • November 14, 2021

In this article, we will go through T-test and F-test in detail.


A t-test is a type of test statistic that helps to compare and determine the means of two groups. It aids in identifying whether or not there is a substantial difference in the means of two groups. In simple words, a t-test compares the average values of two data sets to determine whether or not they are from the same population. It is an inferential statistic. A t-test is like a hypothesis testing tool for testing and creating predictions for a specific population.

T-Distribution and T-Scores

T-score, also known as a T-value is the number of standard deviations from the mean in a t-distribution. The t-distribution is a form of normal distribution. We normally prefer it for smaller sample sizes. We also prefer it when the data is roughly normally distributed. T-scores are essentially normalized scores on each dimension for each kind. It is the sample mean minus the population mean over the entire sample standard deviation divided by the square root of the number of observations.

The test findings of t-tests are all dependent on t-values. It is possible to calculate t-values by comparing your sample mean(s) to the null hypothesis and taking into account both the sample size and the variability in the data. A t-value of 0 implies that the sample findings are identical to the null hypothesis. The actual value of the t-value increases as the difference between the sample data and the null hypothesis grows.
Higher t-values suggest that there is a significant difference between the two sample sets. Likewise, the lower the t-value, the greater the similarity between the two sample sets. A big t-score implies that the groups are different, whereas a small t-score shows that they are similar. Generally, any t-value larger than +2 or less than – 2 is acceptable.

Degrees of Freedom

The greatest number of completely independent values, or values with the ability to fluctuate, in a data sample is what we consider as degrees of freedom.
Subtract the number of relations from the number of observations to get calculate the degrees of freedom.

Degrees of Freedom = n – 1

where “n” refers to the number of things in your dataset.

How Does a T-Test Work?

In essence, it takes a sample from each dataset and creates the issue statement based on the assumption that the two means are equal. Once the null hypothesis has been adopted, the null hypothesis is accepted or rejected depending on the t-test results when compared to the standard values.

Different Types of T-Tests

T-tests can be classified into the following types:

  • Paired-sample
  • Unpaired/ Independent-samples t-test
  • One-sample

Let us briefly understand them one by one.

Paired T-Tests

A paired t-test is just a t-test performed on dependent samples. Tests on the same thing/person/element are known as dependent samples. For instance, on the same person, running two tests, one 10 days previously and one 10 days later.

Unpaired T-Tests

The unpaired t-test, also known as the independent samples t test, is used to compare the means of two datasets. When we don’t know the population mean or standard deviation and we have two independent samples, we apply this test.

One-sample T-Tests

A one-sample t-test compares the mean of a single group to a known mean. In this scenario, the test variable must be continuous. The purpose of this test is to identify whether or not an unknown population mean differs from a given value.


An F-test is any statistical test in which the test statistic has an F-distribution. It is conducted under the null hypothesis. The F-test compares the standard deviations of two samples to determine their variability. Its goal is to determine whether the variances of two populations are the same. It does it by calculating the ratio of two variances. As a result, if the variances are the same, the variance ratio is 1. The F-test is most typically used when comparing statistical models that have been fitted to a data set to determine which model best reflects the population from which the data were sampled.


With the assumption that the null hypothesis is true, the p-value is the probability of receiving results that are at least as severe as the observed outcomes of a statistical hypothesis test. The p-value  is a statistic for determining the likelihood that an observed difference occurred by chance. You must use the F statistic in conjunction with the p value to determine if your overall results are statistically significant or reliable.


The F value numerically depicts the F distribution. The F Statistic’s probability distribution is the F Distribution. We can obtain a F value via a variety of statistical tests. Examine the value to see if the test is statistically significant. When determining if your data are significant enough to reject the null hypothesis, the F value should always be used in combination with the p value. Basically, a high f value indicates that something is important, but a low p value indicates that all of your findings are important.

How Are T-test and F-test Different from Each Other?

We should have figured out what the T-test and F-test is all about by now. To sum up and be entirely clear on the topic, let us quickly summarize the t-test and f-test and examine its differences.

A T-test is useful for comparing and determining the means of two groups. It analyses the average values of two data sets to see if they are from the same population. The F-test examines two samples’ standard deviations to see how different they are. It determines if two populations have the same variations.
We use the T-test when the standard deviation is unknown and when the sample population is equal to or under 30.We use the F-test when the sample population is higher than 30 or huge.
There exists different types of T-tests for comparing the means of two groups such as paired (dependent), and unpaired (independent) t-tests.There exists only one type of F-test.
The null hypothesis of a T-test is that the sample mean is 0.The null hypothesis of an f-test is that the variance of the two samples is the same.
As previously stated, the degree of freedom formula for the t-test is n-1.The f-test has a degree of freedom of (n1-1,n2-1), where n1 and n2 are the number of observations in samples 1 and 2, respectively.