Standard Normal Converter

Maurizio Mario Murino
Oggi

The Standard Normal Distribution (Z)

The Standard Normal Distribution, or Z-distribution (Z) is a particular form of normal distribution. It is mainly used to find probabilities, and has always a mean of zero and a standard deviation of one. A value on the z-distribution represents the number of standard deviation the data is under or above the mean: these are called z-scores.

Commonly, it is one of the first subjects of a statistics class. Since using the z-table to find probabilities is annoying, I developed a really simple Shiny app to help in grasping the basic calculations required from transalting a normal distribution in a standard normal.

The z-score

The z-score is the number of standard deviation that a data is under or below the mean. Every normal distribution can be translated in a z-distribution.

Let's consider a school. The average height is 175cm, with a standard deviation of 10. The z-score for a student 163cm is given by the formula:

\[ \LARGE z = (x - \mu)/\sigma \]

where \( \LARGE x \) is the student of our interest, \( \LARGE \mu \) is the distribution mean of the school and \( \LARGE \sigma \).

[1] -1.2

Graphical rapresentation

Here is our 163cm tall student on Standard Normal Distribution, -1.2 standard deviatons from the mean zero.

plot of chunk plot

How to use the program

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Insert:

  • a mean;
  • a standard deviation;
  • a score x.

It will return:

  • the plot;
  • the z-score;
  • the P(X < x).