Normal (Gaussian) Distribution

🔔

The Normal distribution, often called the "bell curve," is the single most important probability distribution in statistics. It appears everywhere in nature and forms the backbone of machine learning, hypothesis testing, and inferential statistics.

Core Concepts

If a random variable $X$ follows a Normal distribution, we write:

X \sim \mathcal{N}(\mu, \sigma^2)

Where:

$\mu$ (mu) is the mean (which dictates the center of the bell).
$\sigma^2$ (sigma squared) is the variance (which dictates the spread). The square root, $\sigma$ , is the standard deviation.

Probability Density Function (PDF)

🤯

Don't panic! The equation below looks terrifying, but it perfectly defines the elegant, smooth shape of the bell curve.

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

The Anatomy of the PDF (The "Why")

Why does the bell curve have $\pi$ and $e$ ? Every part of this formula serves a specific mathematical purpose:

The Exponent ( $e^{-x^2}$ ): This is the "engine." It creates the symmetric drop-off. As $x$ gets further from the mean, the square $x^2$ grows rapidly, and $e^{-\text{huge number}}$ approaches zero. This is what creates the "bell."
The Center ( $\mu$ ): By subtracting $\mu$ from $x$ , we shift the entire curve left or right so that its peak is exactly at the mean.
The Spread ( $\sigma$ ): By dividing by $\sigma$ , we stretch or compress the curve horizontally. This is a basic Linear Transformation.
The Normalizing Constant ( $\frac{1}{\sigma\sqrt{2\pi}}$ ): This is just a number. Its only job is to ensure that the Total Area under the curve is exactly 1.0. Without this, the height of the curve would be wrong, and it wouldn't be a valid probability distribution. The $\sqrt{2\pi}$ comes from the Gaussian Integral in multivariate calculus.

The Empirical Rule (68-95-99.7)

One of the most useful heuristics in statistics applies to all Normal distributions:

$\approx 68.27\%$ of data falls within 1 standard deviation ( $\pm 1\sigma$ ) of the mean.
$\approx 95.45\%$ of data falls within 2 standard deviations ( $\pm 2\sigma$ ) of the mean.
$\approx 99.73\%$ of data falls within 3 standard deviations ( $\pm 3\sigma$ ) of the mean.

Real-World Examples

The Normal distribution is everywhere because of the Central Limit Theorem (covered later).

Human Heights: The heights of adult males in a country cluster around a mean and taper off symmetrically.
Measurement Error: The tiny errors a machine makes when precisely measuring weights or lengths.
Test Scores: Standardized tests like the SAT or GRE are explicitly designed to force scores into a normal distribution.

Interactive Visualization

Below is an interactive graph.

Drag the Mean ( $\mu$ ) to shift the entire curve left or right along the x-axis. The shape does not change.
Drag the Standard Deviation ( $\sigma$ ) to see how variance impacts the curve. A smaller $\sigma$ makes the curve tall and spiked (data is highly predictable). A larger $\sigma$ makes the curve flat and wide (data is highly spread out).

Normal Distribution

Drag μ and σ to shift the curve and change its spread.

Mean (μ)

0.0

Standard Deviation (σ)

1.0

Test Your Knowledge

Example: Normal Curve Empirical Rule

A population of adult male heights is normally distributed with a mean $\mu = 70$ inches and a standard deviation $\sigma = 3$ inches. What percentage of men are between 64 inches and 76 inches tall?

View Step-by-Step Solution

Notice that 64 inches is exactly $\mu - 2\sigma$ ( $70 - 2(3) = 64$ ). Notice that 76 inches is exactly $\mu + 2\sigma$ ( $70 + 2(3) = 76$ ).

According to the Empirical Rule (68-95-99.7), approximately 95% of normally distributed data falls within 2 standard deviations of the mean.

Continuous Uniform Standard Normal (Z)