# Theoretical probability distributions

Interactive applications using R package Shiny

statistics
Author

Pedro J. Aphalo

Published

2023-11-21

Modified

2023-11-21

Abstract
Shiny Apps or dashboards for interactive exploration on the dependence of the t and F distributions on the degrees of freedom.
Keywords

IPS003, design of experiments

Three Apps or dashboards created with R package Shiny. It takes a few moments to activate the Apps in the server as they are stopped after 10 min without user interaction.

# 1t-distribution compared to the Normal distribution

As the number of degrees of freedom increases the t distribution approaches the Normal distribution.

The observed t-value is calculated as $${\bar x} / s_{\bar x}$$, or more generaly the estimate of the value of a parameter divided by its standard deviation.

t-distribution App

# 2F-distribution

The observed F-value is the ratio between two estimates of variance $$s^2_1 / s^2_2$$. If the variance as numerator is larger than the denominator $$F > 1$$. The degrees of freedom of the two variances determine the shape of the distribution.

F-distribution App

If we know the observed F-value, and degrees of freedom we can look up the P-value from the theoretical distribution.

F-distribution P-value look-up App