Math · Probability

Normal Distribution & Z-Scores

A normal distribution is a compact way to describe where values gather and how far they tend to spread. Move its center, spread, and one observed value.

Concept

The bell curve describes a population around a center. Its mean, $μ$ , sets the location; its standard deviation, $σ$ , sets the spread. The total probability stays at one even when the curve becomes narrower and taller, or wider and flatter.

Equation

$f (x) = \frac{1}{σ \sqrt{2 π}} \cdot e^{\frac{- {(x - μ)}^{2}}{2 σ^{2}}}$

Focus or select a term to trace its role in the curve.

Manipulate

Mean

μ

Standard deviation

σ

Observed value

x

Move $x$ directly on the curve, or use the labeled sliders. Keyboard arrows work on every control.

Z-score

68–95–99.7 rule

Choose a band to shade the probability lying within that many standard deviations of $μ$ .

Observe

$μ$ moves the whole distribution without changing its shape. $σ$ moves the $\pm 1 σ$ and $\pm 2 σ$ guides and changes the shape, while the area under the curve remains one. A z-score is simply the signed count of $σ$ steps from $μ$ .

Data & machine learning

Z-scores make measurements on different raw scales comparable. In Dataset A, $μ = 100$ , $σ = 10$ , $x = 120$ ; in Dataset B, $μ = 50$ , $σ = 5$ , $x = 60$ . Both values have $z = 2$ : each is two standard deviations above its own mean. This is useful for standardized features, unusual observations, probability models, and preprocessing.