Normal Distribution Rule

Brian Lee

The normal distribution rule, also called the 68-95-99.7 rule, is a handy way to estimate probabilities without looking up values in a statistics table. It applies to any variable

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

where most observations cluster around the mean \(\mu\) and the spread is measured by the standard deviation \(\sigma\).

For such a variable:

• about 68% of observations fall within \(\mu \pm \sigma\)
• about 95% fall within \(\mu \pm 2\sigma\)
• about 99.7% fall within \(\mu \pm 3\sigma\)

Equivalently, using the standard score

\[ z = \frac{X - \mu}{\sigma}, \]

these probabilities are

\[ P(|z| \le 1) \approx 0.68 \] \[ P(|z| \le 2) \approx 0.95 \] \[ P(|z| \le 3) \approx 0.997 \]

Summary Table

This heuristic helps traders gauge likely price moves and tail risks quickly, setting expectations for how far a normally distributed variable might wander from its mean.