What is a p-value?

Saoirse Trought


A powerful little tool that is often overlooked by many but holds a dear place in the heart of statisticians.


Probability value. More commonly known as the beloved p-value. If you google “what is a p-value?”, the Wikipedia result is alarmingly complicated so I’d strongly recommend NOT looking at that. Unfortunately, we cannot dive straight into p-values. I need to provide more of a backstory as to why we need them and what statistical analysis methods they’re associated with.


Hypothesis testing


We make a claim and must test the validity of said claim. For example, if I was to say that one assignment takes me two hours to complete, we would test this claim to analyse if it is true or not. We analyse the null hypothesis, that we denote by HO, that claims an assignment does take me two hours to complete. We also examine the alternative hypothesis, denoted by HA, that an assignment does NOT take me two hours to complete. Essentially, checking to see if the claim I made was valid or not valid (FYI I would be doing VERY well to be getting an assignment done in two hours so this example is purely hypothetical).


Testing our claims


We scrutinise sample data that we have collected to investigate the claims and more often than not, we’ll conduct Z-tests using Z-scores. Ask yourself - what level of confidence do we want to work in? What level of accuracy are we testing the claims for? 90%? 95% (which is the most common and insightful)? 97.5%? Store this concept at the back of your head for the meantime as it is extremely important and will be discussed at a later date. For now, accept that hypothesis testing and p-values come hand in hand and are often accompanied by a lovely Z-test.


What is a p-value?


Assume that our null hypothesis is indeed true. Our bold claim that we have made is correct and now we would like to further our argument by determining the strength of our initial claim. This is where our p-value makes their debut. It is used to weigh the strength of the evidence we have provided to prove our null hypothesis to be true. The p-value is a PROBABILITY and therefore, must always adopt a value between 0 and 1. In a more Wikipedia-esque fashion: “a small p-value is statistically significant, indicating we should reject the null hypothesis. A larger p-value indicates we accept the proposed null hypothesis”. Quite simply - the smaller the p-value, the stronger the evidence to reject the claimed null hypothesis.


Where can we find this mysterious p-value?


Believe it or not, there are statistical tables designed to provide us statisticians with these answers and makes our lives a lot easier. When calculating a statistical test, such as a Z-test score, we dig out these magical tables and note the corresponding probability to the calculated test score. Multiply this probability by 2 and voilà, you have your p-value.


Always remember:


If p is low HO must go, meaning we must reject the null hypothesis. Reject the assumption that the initial claim is true. Most definitely reject the claim that it takes me only two hours to complete an assignment.



My name is Saoirse Trought and I am a Mathematical Sciences student at University College Cork, Ireland. Besides my obvious interest in all things maths, I happen to be fluent as Gaeilge. I enjoy sailing, staycation-ing and attempting to run UCC MathSoc. I'm looking forward to combining some of these interests with statistics and seeing where it takes us!

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