Statistics is one of the many branches of mathematics that is responsible for collecting, organizing, projecting, analyzing, interpreting and presenting data following laws of probability, this allows us to predict certain types of behaviors by applying them to the scientific, industrial or social field. What is student t-test?

Within statistics we can use several hypothesis tests, one of the most complete is the **Student’s T** test , it was developed by the English mathematician and chemist William Sealy Goset, better known by his pseudonym *“Student”.*

This statistical test consists of the probability distribution, due to the need to estimate what is the mean of a population with a small, normally distributed sample. That is, less than 30, which is why this test is widely used in the field of medicine.

To perform this test, a **normal distribution of the data** is needed , since this statistical test is a parametric test and is used when the population standard deviation is unknown because if this statistical data were known, instead of using this test, the normal distribution for hypothesis tests.

## Basic concepts of Student’s T

To correctly apply the **Student’s t** test, we must take into account several basic concepts of decision theory theory for large samples. What is student t-test?

### The percentile

It is the result of dividing a set of data into one hundred equal parts, each of these parts represents 1% in the representation of the Gaussian bell graph, it runs from the left part to the right part.

### Gauss’s bell

It is a graph that represents the normal distribution of a set of statistical data. The normal distribution is used for large samples, this means a statistical data greater than 30 while the Student’s t is used for small samples, less than 30.

## Characteristics of the Student’s T

- It belongs to a family of bell distributions.
- It is symmetric around a mean of zero.
- It is more flattened than the standard normal distribution.
- It has more area at the ends and less area in the center.
- As the sample size increases, it approaches a standard normal distribution.

## Scenarios where to apply the Student’s t

There are several scenarios in which we can apply this statistical test and it will always depend on the type of sample that has been collected. What is student t-test?

### A related sample

This means that there are two measures which have been obtained at two different times and which are also related, an example of this is when an intervention is carried out, under this context we can have data and information before the intervention and after the intervention, then we can observe if in each subject the result varied before and after.

### Two samples with homogeneous variances

It refers to the fact that the samples taken for our statistical test are similar in the two samples.

### Two samples with heterogeneous variances

This means that our statistical test has totally different samples, data and information.

## How to determine the stage to know?

To determine which of the two-sample scenarios is being used, it is necessary to know homoscedasticity, if the data from the two samples have this characteristic then the two-sample scenario with homogeneous variances should be used, in the case that the samples do not have homoscedasticity. The two-sample scenario with heterogeneous variances should be used.

The **Student’s t** statistical test **has several assumptions** , in this case for the scenarios that have two samples it is assumed that the data have a normal distribution, and it must be presented in each of the two samples and also these samples are totally independent, the values we have in one sample do not depend at all on the other sample. What is student t-test?

When we use the scenario of a related sample, we have only one assumption and the assumption is that the difference between the two related variables has a normal distribution and the perfect example is when an intervention is carried out, since we have data from before and after the itself, from this we can find the difference between each subject since the values of before and after are subtracted thus finding the values of the difference.

This difference must have a normal distribution, in this scenario it is not indicating that the data in each of the samples or groups have a normal distribution, it indicates that the difference is the one that has a normal distribution and not the data from each of the groups, which is what the assumption with two variables or two samples indicated. What is student t-test?

## Degrees of freedom

The **Student’s t** test depends on the **degrees of freedom** . It is the determined number that allows us to know the variability of events in a sample, in simpler words we can say that they are the number of values that we can freely choose, existing a fixed total.

There are two **formulas for degrees of freedom** , one formula when we have a sample that is related, and the other formula that is when we are working either of the two scenarios with two samples. What is student t-test?

To visualize this in a more comfortable way, we can imagine a family in which there is a mother and 4 children, the mother prepares 10 loaves with ham, the fixed total is 10 loaves with ham, the first child tells his mother that He wants to eat 3 loaves of bread, the second son asks for 2 loaves, the third son asks for 3 loaves and the fourth son, due to being late, will not be able to choose how many loaves of ham he wants, because he is conditioned by what his other 3 brothers asked for That is why the fourth child had only 2 loaves left.

The important thing is that of the 4 brothers only 3 were able to choose how many loaves they wanted, in this case the degree of freedom is 3 who were the ones who could choose and the last one was conditioned to complete the 10 loaves.