Normal Distribution: The Bell Curve and 68-95-99.7 Rule

The normal distribution (Gaussian distribution) is the most important probability distribution in statistics. It describes countless natural phenomena — heights, test scores, measurement errors, blood pressure — because of the Central Limit Theorem.

The probability density function is:

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

The 68-95-99.7 (empirical) rule

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

This means if test scores have $\mu = 75$, $\sigma = 10$: about 68% score between 65 and 85, 95% between 55 and 95, and 99.7% between 45 and 105.

Try it in the visualization

Adjust $\mu$ (shifts the center) and $\sigma$ (controls width — smaller σ = taller, narrower; larger σ = shorter, wider). Toggle the 68/95/99.7 shading to see the area fractions. The total area under the curve is always 1 (100%).