The Bell Curve

The normal distribution — the bell curve — appears everywhere in statistics: test scores, measurement errors, biological traits. Its shape is defined by two parameters: the mean (center) and the standard deviation (spread).

The Formula

The probability density function for a normal distribution:

f(x) = (1 / (σ√2π)) · e^(-½((x-μ)/σ)²)

Where μ is the mean and σ is the standard deviation. As σ grows, the curve flattens. As μ shifts, the peak moves left or right without changing shape.

Interactive Explorer

Adjust the mean and standard deviation below to see how the curve responds. Notice that narrowing σ concentrates probability near the center — this is why small samples from a wide distribution can look surprisingly clustered.

Why It Matters

Many natural processes produce approximately normal distributions through the central limit theorem: the sum of many independent random variables tends toward normality. Understanding the bell curve helps you reason about outliers, confidence intervals, and the difference between typical and exceptional cases.