## Discover the origins, theory and uses behind the famous t-distribution

The **t-distribution****, **is a continuous probability distribution that may be very just like the **normal distribution****, **nonetheless** **has the next key differences:

**Heavier tails**:*More of its probability mass is positioned on the extremes (higher**kurtosis**). Which means it’s more more likely to produce values removed from its mean.***One parameter**:*The t-distribution has just one parameter, the**degrees of freedom**, because it’s used after we are unaware of the population’s variance.*

An interesting fact concerning the t-distribution is that it is usually known as the “Student’s t-distribution.” It’s because the inventor of the distribution, **William Sealy Gosset**, an English statistician, published it using his pseudonym “Student” to maintain his identity anonymous, thus resulting in the name “Student’s t-distribution.”

Let’s go over some theory behind the distribution to construct some mathematical intuition.

## Origin

The origin behind the t-distribution comes from the thought of modelling normally distributed data without knowing the population’s variance of that data.

For instance, say we sample ** n** data points from a standard distribution, the next shall be the mean and variance of this sample respectively:

Where:

*x̄**is the sample mean.**s**is the sample standard deviation.*

Combining the above two equations, we will construct the next random variable:

Here ** μ** is the population mean and

**is the t-statistic belongs to the t-distribution!**

*t*See here for a more thorough derivation.

## Probability Density Function

As declared above, the t-distribution is parameterised by just one value, the degrees of freedom, *ν, *and its **probability density function** looks like this: