---
tag: XZSA2NCATS
kind: concept
name: Mathematical statistics
msc: [62A01]
sources:
  - url: https://en.wikipedia.org/wiki/Mathematical_statistics
same_as:
  wikidata: Q745328
  wikipedia: Mathematical statistics
---

## Summary

Mathematical statistics is the part of mathematics that studies how to learn about an unknown probability distribution from data.
It models data as the values of random variables whose law belongs to a [[statistical model|JZ6R7J36K5]].
It then studies procedures such as estimators, tests and confidence sets through their behaviour under each law of the model.
It gives the rigorous basis for [[statistical inference|FXCQWFY5RV]].

## Explanations

### L1 human

Data are noisy, so a conclusion drawn from data can be wrong.
Mathematical statistics uses probability theory to measure how often a method of drawing conclusions goes wrong.
Picture a coin with an unknown chance $p$ of heads, tossed $100$ times.
The proportion of heads is a natural guess for $p$, and mathematical statistics tells how far this guess typically lies from $p$: about $\sqrt{p(1-p)/100}$, at most $0.05$.