P-value

Key takeaways

P-value belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect P-value to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of P-value from memory before moving on to harder problems.

Reference excerpt

In null-hypothesis significance testing, the p-value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis. Even though reporting p-values of statistical tests is common practice in academic publications of many quantitative fields, misinterpretation and misuse of p-values is widespread and has been a major topic in mathematics and metascience. In 2016, the American Statistical Association (ASA) made a formal statement that "p-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone" and that "a p-value, or statistical significance, does not measure the size of an effect or the importance of a result", and "does not provide a good measure of evidence regarding a model or hypothesis" without "context or other evidence". That said, a 2019 task force by ASA has issued a statement on statistical significance and replicability, concluding with: "p-values and significance tests, when properly applied and interpreted, increase the rigor of the conclusions drawn from data".

Basic concepts In statistics, every conjecture concerning the unknown probability distribution of a collection of random variables representing the observed data X {\displaystyle X} in some study is called a statistical hypothesis. If we state one hypothesis only and the aim of the statistical test is to see whether this hypothesis is tenable, but not to investigate other specific hypotheses, then such a test is called a null hypothesis test. As our statistical hypothesis will, by definition, state some property of the distribution, the null hypothesis is the default hypothesis under which that property does not exist. The null hypothesis is typically that some parameter (such as a correlation or a difference between means) in the populations of interest is zero. Our hypothesis might specify the probability distribution of X {\displaystyle X} precisely, or it might only specify that it belongs to some class of distributions. Often, we reduce the data to a single numerical statistic, e.g., T {\displaystyle T} , whose marginal probability distribution is closely connected to a main question of interest in the study. The p-value is used in the context of null hypothesis testing in order to quantify the statistical significance of a result, the result being the observed value of the chosen statistic T {\displaystyle T} . The lower the p-value is, the lower the probability of getting that result if the null hypothesis were true. A result is said to be statistically significant if it allows us to reject the null hypothesis. All other things being equal, smaller p-values are taken as stronger evidence against the null hypothesis. Loosely speaking, rejection of the null hypothesis implies that there is sufficient evidence against it. As a particular example, if a null hypothesis states that a certain summary statistic T {\displaystyle T} follows the standard normal distribution N ( 0 , 1 ) , {\displaystyle {\mathcal {N}}(0,1),} then the rejection of this null hypothesis could mean that (i) the mean of T {\displaystyle T} is not 0, or (ii) the variance of T {\displaystyle T} is not 1, or (iii) T {\displaystyle T} is not normally distributed. Different tests of the same null hypothesis would be more or less sensitive to different alternatives. However, even if we do manage to reject the null hypothesis for all 3 alternatives, and even if we know that the distribution is normal and variance is 1, the null hypothesis test does not tell us which non-zero values of the mean are now most plausible. The more independent observations from the same probability distribution one has, the more accurate the test will be, and the higher the precision with which one will be able to determine the mean value and show that it is not equal to zero; but this will also increase the importance of evaluating the real-world or scientific relevance of this deviation.

Definition and interpretation

Definition The p-value is the probability under the null hypothesis of obtaining a real-valued test statistic at least as extreme as the one obtained. Consider an observed test-statistic t {\displaystyle t} from unknown distribution T {\displaystyle T} . Then the p-value p {\displaystyle p} is what the prior probability would be of observing a test-statistic value at least as "extreme" as t {\displaystyle t} if null hypothesis H 0 {\displaystyle H_{0}} were true. That is:

p = Pr ( T ≥ t ∣ H 0 ) {\displaystyle p=\Pr(T\geq t\mid H_{0})} for a one-sided right-tail test-statistic distribution.

p = Pr ( T ≤ t ∣ H 0 ) {\displaystyle p=\Pr(T\leq t\mid H_{0})} for a one-sided left-tail test-statistic distribution.

… excerpt ends here. Continue reading the full article.

Illustrations

P-value: Pierre-Simon Laplace — Pierre-Simon Laplace

Worked examples

Example 1 — a first encounter with P-value

Start with the simplest possible case. Write down what P-value claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to P-value before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about P-value ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of P-value

In research: P-value appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses P-value in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: P-value is common in secondary-school and first-year university syllabi. It links to neighbouring topics Statistical hypothesis testing, so understanding it makes those chapters shorter.
In everyday life: Look for P-value outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study P-value in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what P-value means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain P-value out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is P-value in simple terms?

In null-hypothesis significance testing, the p-value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small p-value means that such an extreme observed outcome would be very unlikely un…

Why does P-value matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study P-value?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on P-value.