The Normal Distribution

The King of Statistics

The Normal (Gaussian) Distribution appears everywhere: the heights of people, the errors in scientific measurements, even the scores on standardized tests. It is the most important continuous distribution in statistics.

f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

Parameters and Properties

$\mu$ (Mean): The center of the distribution (determines location).
$\sigma$ (Standard Deviation): The spread of the distribution (determines width).
Variance ( $Var(X)$ ): $\sigma^2$
Support: $x \in (-\infty, \infty)$

✦Intuition

Why is it everywhere?

The reason the Normal distribution is so ubiquitous is the Central Limit Theorem. It states that if you add up a large number of independent random variables (even if they aren't normally distributed themselves!), their sum will approximate a Normal distribution. Since many natural phenomena (like human height) are the result of adding up countless tiny independent factors (thousands of genetic and environmental variables), the final result follows a perfect bell curve.

The Empirical Rule (68-95-99.7)

For any normal distribution, you can predict exactly where the data lives:

68% of outcomes are within $\mu \pm 1\sigma$ .
95% are within $\mu \pm 2\sigma$ .
99.7% are within $\mu \pm 3\sigma$ . Anything beyond 3 standard deviations is extremely rare, often called a "six-sigma" event in industrial quality control.

The Standard Normal Distribution

Every Normal distribution is just a shifted and scaled version of the Standard Normal Distribution, denoted as $Z \sim N(0, 1)$ . This fundamental curve has a mean of 0 ( $\mu = 0$ ) and a standard deviation of 1 ( $\sigma = 1$ ).

We can convert any normal variable $X$ into a standard $Z$ -score using the following transformation:

Z = \frac{X - \mu}{\sigma}

This Z-score tells you exactly how many standard deviations away from the mean your value is. A Z-score of +1.5 means the value is 1.5 standard deviations above average.

The Standard Normal Curve (Z)

The 'Z-distribution' is the standard benchmark where mean is 0 and variance is 1. Any normal variable X can be converted to Z using Z = (X-μ)/σ.

Standard Normal PDF (μ=0, σ=1)

Advanced Practice

Example 1: IQ Scores

medium

IQ scores are normally distributed with a mean of 100 ( $\mu = 100$ ) and a standard deviation of 15 ( $\sigma = 15$ ). What percentage of the population has an IQ between 85 and 130?

Example 2: Quality Control Rejects

hard

A machine fills bottles with exactly 500ml of liquid, but with a normally distributed error. The standard deviation is 4ml ( $\sigma = 4$ ). Bottles with less than 490ml are rejected. What must the machine's target mean setting ( $\mu$ ) be so that exactly 1% of bottles are rejected? (Assume a Z-score of -2.33 corresponds to the bottom 1%).

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The Uniform Distribution

Course Progression

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The Exponential Distribution