Normal distribution

The bell curve

The normal distribution (also called the Gaussian distribution) is the most important distribution in statistics. It describes many naturally occurring measurements — heights, test scores, measurement errors — and plays a central role in statistical theory.

Its shape is a symmetric, bell-shaped curve centered at its mean $\mu$, with spread controlled by its standard deviation $\sigma$.

The parameters

A normal distribution is fully described by two parameters:

• $\mu$ (mu): the mean — the center of the distribution.
• $\sigma$ (sigma): the standard deviation — how spread out the values are. Larger $\sigma$ means a wider, flatter bell.

We write $X \sim \mathcal{N}(\mu, \sigma^2)$ to say "$X$ follows a normal distribution with mean $\mu$ and variance $\sigma^2$."

The probability density function is:

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

You do not need to memorize this — but notice that it is symmetric around $\mu$ and falls off as $x$ moves away from the mean.

The 68-95-99.7 rule

A practical rule for any normal distribution:

• ~68% of values fall within 1 standard deviation of the mean ($\mu \pm \sigma$)
• ~95% fall within 2 standard deviations ($\mu \pm 2\sigma$)
• ~99.7% fall within 3 standard deviations ($\mu \pm 3\sigma$)

Example: adult male heights in the US are approximately $\mathcal{N}(177\text{ cm},\, 7^2)$. So about 95% of men are between $177 - 14 = 163$ cm and $177 + 14 = 191$ cm.

The standard normal

The standard normal distribution is the special case $\mathcal{N}(0, 1)$ — mean 0, standard deviation 1. Any normal distribution can be converted to standard normal via standardization:

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

The value $Z$ (called a z-score) tells you how many standard deviations $X$ is above or below the mean. A z-score of $2$ means the value is 2 standard deviations above average.

Why the normal distribution is everywhere

The central limit theorem (CLT) explains its ubiquity: the sum (or average) of many independent random variables — regardless of their individual distributions — tends toward a normal distribution as the number of variables grows. Since many real-world quantities are the result of many small independent influences adding together, normality arises naturally.

Key properties

• Symmetric around $\mu$: mean = median = mode.
• Fully described by just $\mu$ and $\sigma$.
• The sum of independent normal random variables is also normal.
• Tails extend to $\pm\infty$ but become negligibly small beyond $3\sigma$.