← All topicsInteractive Guide

Statistics & Probability Explained

From averages and spreads to bell curves, probability rules, and confidence intervals — visual and step-by-step.

Measures of Center & Spread

Understand what each metric tells you, why outliers skew some but not others, and how standard deviation is calculated.

Choose a dataset:

Sorted dataset:(Data count (n): 0)(Total sum (∑x): 0)

(Data count (n) 0)

x̄

0.00

x̄ = (∑x) / n

The balance point of the data. Sum of all items divided by the count.

Sensitive to extreme values (outliers). A single huge number pulls the mean up.

0.00

Middle value at position (n + 1) / 2

The physical middle of the sorted data. If count is even, it is the average of the two middle numbers.

Robust against outliers. Income and house prices usually use median instead of mean.

—

No single mode

The value that appears most often in the dataset.

A dataset can have one mode, multiple modes (bimodal), or no mode at all.

0.00

0 − 0

The difference between the maximum and minimum observed values.

Quick to calculate, but only depends on the two most extreme points.

—

Sample Variance (s²): needs at least 2 values

Sample standard deviation needs at least 2 values (divides by n − 1).

s = √( ∑(x − x̄)² / (n − 1) )

Sample formula divides by (n − 1) to account for estimation bias from a sample.

The typical distance of data points from the mean. The foundation of dispersion.

We square deviations so negatives don't cancel, sum them, divide, and take the square root to return to original units.

Value (x)	Deviation (x − x̄)	Squared Deviation (x − x̄)²
(0 values)