Distributions

How probability spreads across a variable's values: normal, uniform, binomial.

Maths · Intermediate · ~4 min

In plain English

The shape of how values spread out. Bell-shaped, long-tailed, or piled at zero — the shape tells you which tools are safe to use.

Why it's worth your time

Most statistical methods quietly assume a shape. When the data has a different one, the method doesn't warn you, it just gives a wrong answer.

If you remember three things

Normal: symmetric, thin tails, the assumption behind most tests
Long-tailed: means are unstable, use medians and log transforms
Poisson/binomial for counts, not Gaussian

Overview

A probability distribution describes how likely each value of a random variable is, with the total probability always summing to 1. The normal (bell) curve is defined by its mean and standard deviation; uniform and binomial are other common shapes. A PDF gives density at each value, and a CDF gives the probability of falling at or below a value.

In an interview

A distribution spreads a total probability of 1 across the values a variable can take. The normal bell is set by its mean and standard deviation; the uniform makes every value equally likely; the binomial counts successes in n trials. A PDF is the height (density) at each value; the CDF accumulates it into the probability of being at or below a point.

Production defaults

Check first: plot the histogram and a Q-Q plot before applying anything that assumes normality
Skewed positive data: log1p transform is the standard first move (revenue, counts, durations)
Counts: Poisson or negative binomial. A Gaussian on counts predicts negative quantities

What breaks

A few users dominate every metric — Long tail. Report medians and percentiles; the mean is describing the tail, not the users.
Confidence intervals are far too narrow — Normality assumed on non-normal data. Bootstrap instead.

Watch it explained

The Normal Distribution, Clearly Explained!!! — StatQuest with Josh Starmer, 5:12