Normal Distribution and the Bell Curve

The normal distribution

The normal (Gaussian) distribution is the most important distribution in statistics. It's defined by two parameters: mean $\mu$ (center) and standard deviation $\sigma$ (spread).

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

The 68-95-99.7 rule (empirical rule)

For any normal distribution:

• 68.27% of data falls within $\mu \pm 1\sigma$
• 95.45% within $\mu \pm 2\sigma$
• 99.73% within $\mu \pm 3\sigma$

Step-by-step: $P(-1 < Z < 1)$ for standard normal

The standard normal has $\mu = 0$, $\sigma = 1$.

$P(-1 < Z < 1) = 0.6827 = 68.27\%$

This means if test scores follow $N(75, 10)$: about 68% of students score between 65 and 85.

Properties

• Symmetric about $\mu$ (left half mirrors right half)
• Total area under the curve = 1
• Mean = median = mode = $\mu$
• Changing $\sigma$: smaller σ → taller, narrower; larger σ → shorter, wider

Try it in the visualization

Adjust $\mu$ and $\sigma$. The 68/95/99.7 regions shade in different colors. The area under any region is computed live.