The Normal Distribution

The bell curve

The normal distribution (a.k.a. Gaussian) is the famous bell-shaped curve. It's defined by two numbers:

• Mean $\mu$ — where the peak is.
• Standard deviation $\sigma$ — how wide the bell is.

It shows up everywhere: heights, test scores, measurement errors, sums of many small random effects.

The 68-95-99.7 rule

For any normal distribution, the percentage of data within $k$ standard deviations of the mean is:

Range	Percent of data
$\mu \pm 1\sigma$	~68%
$\mu \pm 2\sigma$	~95%
$\mu \pm 3\sigma$	~99.7%

Example. SAT scores are roughly normal with $\mu = 1050$, $\sigma = 200$.

• About 68% of test-takers score between $850$ and $1250$.
• About 95% score between $650$ and $1450$.
• About 99.7% score between $450$ and $1650$.

Z-scores

A z-score tells you how many standard deviations a value is from the mean:

$z = \frac{x - \mu}{\sigma}$

A positive $z$ means above the mean; negative means below.

Example. SAT $\mu = 1050$, $\sigma = 200$. If you score $1350$:

$z = \frac{1350 - 1050}{200} = \frac{300}{200} = 1.5$

You're 1.5 standard deviations above the mean.

Why z-scores matter

Z-scores let you compare values across different normal distributions on a common scale. A score of "$z = 2$" on the SAT and "$z = 2$" on the ACT both mean "roughly the top 2.5%" regardless of the original units.

Standard normal table (rough values)

For the standard normal (mean 0, std dev 1), the proportion of data below a given $z$:

$z$	$P(Z < z)$
$-2$	2.3%
$-1$	15.9%
$0$	50%
$1$	84.1%
$2$	97.7%

So if your $z = 1.5$, you scored higher than about 93% of test-takers.

Key takeaways

• The normal distribution is fully described by mean and standard deviation.
• 68% within 1 SD, 95% within 2 SDs, 99.7% within 3 SDs.
• Z-scores rescale anything to "how many SDs from average" — letting you compare across distributions.