Statistical proof

Statistical proof is a way to show how sure we can be about an idea or a guess after looking at information and doing special checks. It helps us understand facts better and explains why we can trust what we find. The goal is to convince others and to make the idea clear for everyone to see and agree with.

The way we prove something with numbers depends on different beliefs and ways of thinking. Some people use one method, while others use another, and these choices affect how we understand the results. These ideas come from big questions in science about how we know things.

One important way scientists test ideas is by trying to show they might be wrong. This helps them learn and improve their understanding. Statistical proof is not about being 100% sure but about learning from tests and mistakes. It is also used in courts to help decide what is true.

Axioms

There are two kinds of axioms: ideas we accept as true without testing, and hypotheses. In the late 1600s, four important rules were created to help us understand probabilities, or how likely something is to happen.

These rules say that probabilities are never negative, a certainty has a probability of one, the chance of two impossible events together is the sum of their chances, and the chance of one event happening given another depends on how often both happen together compared to how often the second one happens. These rules help us understand chance and make good guesses using data and logic.

Test and proof

Main article: Statistical tests

A proof in statistics comes from testing ideas using data. Think of it like checking if a guess is right by looking at facts. We use special ways to see if what we see in our data is just by chance or if it really means something.

When we test data, we compare it to what we would expect if nothing special was happening. If the data is very different from what we expected more than just by chance, we think our guess might be right. This helps scientists and researchers make better decisions using numbers.

Bayes' theorem

Main article: Bayes' theorem

See also: Evidence under Bayes' theorem

Bayesian statistics use a special way to think about proof for inference. The main idea is shown in a math formula:

This formula helps us understand how likely something is true after we see new information. It takes the chance we thought something was true before (called the prior probability) and combines it with how well the new information matches our idea (called the likelihood). The result tells us how likely the idea is true now (called the posterior probability).

Scientists can use this to compare different ideas, or hypotheses, to see which one is more likely based on what they observed. Some people wonder if this method fits with another way of thinking used by a scientist named Karl Popper, who said that ideas are only supported, not proven right, after testing. Some believe that Bayesian methods and Popper’s ideas can work together in this way.

In legal proceedings

Main article: Legal burden of proof

Statistical proof helps show if unfair treatment is happening in legal cases. It can be used to prove three main things: that something happened, who might be responsible, and why they did it.

Statistical proof became more common in United States courts in the 1970s after a big case called Castaneda v. Partida. The court decided that big differences in statistics can show unfair treatment, changing who has to prove their point in court. Since then, this kind of proof has been used in many cases about unfairness and inequality.

In one case, McCleskey v. Kemp, a man named McCleskey was on trial for a serious crime. Experts showed statistics that suggested people charged with hurting white individuals were more likely to face very harsh punishments. But the court said the statistics alone were not enough to prove that the decision in his specific case was unfair. They said each case still needs to be looked at carefully.