# A guide on how to read statistical tables

Below a Shiny app to help you read the main statistical tables:

This Shiny app helps you to compute probabilities for the main probability distributions.

# How to use this app?

- Open the app via this link
- Choose the distribution
- Set the parameter(s) of the distribution (the parameters depend of course on the chosen distribution)
- Select whether you want to find the lower tail, upper tail or an interval
- Choose the value of x

On the right panel (or below depending on the size of your screen) you will see:

- a recap of the data you just entered
- the numerical solution (i.e., the probability)
- a visualization of the solution
- the probability density function together with the mean, the standard deviation and the variance

# Example

Here is an example with the most common distribution: the **normal distribution**.

Suppose the following problem: The cost of weekly maintenance and repair of a business has been observed over a long period of time and turns out to be distributed according to a normal distribution with an average of 402€ and a standard deviation of 22€. Having set a budget of 439€ for next week, what is the probability that the cost exceeds this budget?

To solve this problem, follow these steps in the app:

- Choose the normal distribution, as it is said that the costs follow a normal distribution
- Set the mean \(\mu\) equal to 402, as it is said that the average cost is 402€
- In the statement, the standard deviation is given (and not the variance) so select “Standard deviation \(\sigma\)” and set it equal to 22
- We are asked what is the probability the the cost
**exceeds**the budget. Therefore, we look for the probability**above**a certain x, so select upper tail \(P(X > x)\) - We are now asked to find the probability that the cost exceeds 439€, so set x equal to 439

The solution panel gives a recap of the data:

\[X ∼ \mathcal{N}(\mu = 402, \sigma^2 = 484)\] where \(484 = 22^2\), and the solution: \[P(X > 439) = P(Z > 1.68) = 0.0463\] where \(Z = \frac{X - \mu}{\sigma} = \frac{439 - 402}{22} = 1.68\) and \(Z ∼ \mathcal{N}(\mu = 0, \sigma^2 = 1)\) (known as the standard normal distribution). Thus, the probability that the cost next week exceeds the budget of 439€ is 0.0463, or 4.63%.

It also shows the normal distribution (with \(\mu = 402\) and \(\sigma^2 = 484\)) with the shaded area corresponding to the probability we are looking for. It then gives some details about the density function, the mean, the standard deviation and the variance.

# Code

Here is the entire code (or see the last version on GitHub) in case you would like to enhance it.

*Note that the link may not work if the app has hit the monthly usage limit. Try again later if that is the case.*

Thanks for reading. I hope you will find this app useful to compute probabilities for the main distributions.

As always, if you have a question or a suggestion related to the topic covered in this article, please add it as a comment so other readers can benefit from the discussion.

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