Math · Probability

Variance & Standard Deviation

Variance turns a collection of values into a measure of its spread. Change the observations to see how their distance from the mean becomes a useful signal.

Concept

The mean is the balance point of a population. Variance is the average squared distance from that point; standard deviation returns that spread to the original units. This exploration uses population variance, dividing by $n$ .

Equation

$σ^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - μ)}^{2}$

Focus or select an equation to trace that part of the calculation.

Manipulate

Drag a point, or focus one and use Left and Right Arrow. Hold Shift for a larger step.

Live calculation

Observe

Points farther from $μ$ make larger squared deviations. Because distance is squared, a distant outlier contributes much more than a nearby point: greater spread from the mean means greater variance and standard deviation.

Standardization

Data science often changes a value into a z-score.

$z = \frac{x - μ}{σ}$

This puts the current population on a common scale.

Why this matters

Spread helps describe datasets, spot outliers, and reason about statistical models. Standardizing features can keep one large-scale measurement from dominating another during machine-learning preprocessing.