One smooth, symmetric, bell-shaped curve describes an enormous amount of real data — and it is pinned down by just two numbers. The mean says where the bell sits; the standard deviation says how wide it spreads. Drag them and watch the curve slide and stretch.
Normal (μ = 100, σ = 15)
under the curveinside the bandsampled dataμpinned curvedrag handles
Normal (μ = 100, σ = 15)drag the sliders — or the red dots on the curve
The two parameters
100
15.0
Center μ = med = mode100
Spread σ15.0
Peak height 1 ÷ σ√(2π).0266
Total area computed1.000
Try a move
How much area sits inside ± kσ?
The empirical rule (68–95–99.7): in any normal distribution about 68% of values fall within 1 SD of the mean, about 95% within 2 SDs, and about 99.7% within 3. Move μ and σ wherever you like — the endpoints change, the percentages do not.
Two things the curve is not. The height of the curve is a density, not a probability — a single exact value on a continuous scale has probability 0, so probability lives in area, never in height. And "normal" here is only the name of a shape: it does not mean typical, healthy, or correct, and a histogram that merely looks bell-ish is not automatically normal.
Does real data pile up like that?
Each draw generates observations from this exact normal distribution and stacks them into bins. The gold histogram is what actually came out; the blue curve is the model it was drawn from.
Observations0
Sample mean—μ = 100.0
Sample SD—σ = 15.0
Draw a batch and the bars will land raggedly around the curve. Keep drawing and the ragged edges smooth out — the curve was never a promise about any one observation, only about the shape a large pile of them takes.