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

The **t-distribution****, **is a continuous probability distribution that is very similar to the **normal distribution****, **however** **has the following key differences:

**Heavier tails**:*More of its probability mass is located at the extremes (higher**kurtosis**). This means that it is more likely to produce values far from its mean.***One parameter**:*The t-distribution has only one parameter, the**degrees of freedom**, as it’s used when we are unaware of the population’s variance.*

An interesting fact about the t-distribution is that it is sometimes referred to as the “Student’s t-distribution.” This is because the inventor of the distribution, **William Sealy Gosset**, an English statistician, published it using his pseudonym “Student” to keep his identity anonymous, thus leading to the name “Student’s t-distribution.”

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

## Origin

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

For example, say we sample ** n** data points from a normal distribution, the following will 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 can construct the following 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 only one value, the degrees of freedom, *ν, *and its **probability density function** looks like this: