Standard Normal (Z-Distribution)

Core Concepts

A Standard Normal distribution is defined simply as a Normal distribution where the mean is exactly $0$ and the standard deviation is exactly $1$.

We denote this special random variable as $Z$:

Probability Density Function (PDF)

Because $\mu = 0$ and $\sigma = 1$, the terrifying PDF of the general normal distribution simplifies dramatically into a much cleaner equation:

The Z-Score

Why do we care about a distribution locked at $0$ and $1$? Because any Normal distribution in the universe can be transformed into the Standard Normal distribution using a Z-score.

A Z-score tells you exactly how many standard deviations a specific data point ($x$) is away from its mean ($\mu$).

Why is this powerful? (Apples to Oranges)

Imagine you score an 85 on a Math test (where the class mean was 70 and standard dev was 10) and a 90 on an English test (where the class mean was 85 and standard dev was 5). Which test did you actually perform better on relative to your peers?

• Math Z-score: $z = \frac{85 - 70}{10} = +1.5$
• English Z-score: $z = \frac{90 - 85}{5} = +1.0$

Interactive Visualization

Below is the standard normal distribution ($\mu=0, \sigma=1$). Use the slider to select a Z-Score and instantly calculate the probability (the area under the curve) to the left or right of that score.

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