# P-Values in Statistics: Significance, Definition & Explanation

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*The p-value is the measure of whether the outcome of endeavor is due to an actual effect or mere random chance. It is used to compare the world we encounter to a world that is dominated by chance.*

We also recommend watching Descriptive & Inferential Statistics: Definition, Differences & Examples and What is Bivariate Data? - Definition & Examples

## Introduction

We have a bet going: I am going to flip a fair coin, and every time it comes up heads, you owe me $2, but every time it comes up tails, I owe you $2. By all probability, this game should come out even if we play forever. But, suppose that after 9 flips, the coin came up heads all 9 times. What are the odds the the coin will come up heads on the 10th time?

The correct answer is 50%. Any time a fair coin is flipped, the odds of any outcome are 50-50, because any given flip is *independent* of all of the other flips. The real question is, *what are the odds that this is a fair coin?* In other words, we are asking whether the results we are seeing differ from what we would expect due to random chance, and the probability that the outcome differs from random chance is called the **p-value**.

## Statistics and the P-Value

We can think of statistics as the comparison between the world as it is and the world we expect if everything were totally random and ruled by probability, and the p-value is our guide to the realm of statistics. When the p-value is high, it means that it is very likely that what we are seeing is due to random chance. A low p-value, however, means that the probability of the results coming from random chance are unlikely.

## Returning to the Coin Flip Example

In the case of a fair coin, probability dictates that every flip has an equal probability of being heads or tails. If the coin keeps coming up heads, however, we have to ask how likely it is for that to happen with a fair coin. In the case above, with 9 heads in a row, we can compute the probability of getting 9 heads using a fair coin, like this:

- If you flip a fair coin 1 time, there are 2 (2^1, or 2 to the 1st power) possible outcomes: H (heads), T (tails), so the probability of any given outcome is 1 out of 2, or 1/2.
- If you flip a fair coin 2 times, there are 4 (2^2, or 2 to the 2nd power) possible outcomes: HH, HT, TH, and TT, so the probability of any given outcome is 1 out of 4, or 1/4.
- If you flip a fair coin 3 times, there are 8 (2^3) possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT, so the probability of any given outcome is 1 out of 8, or 1/8.

Extending this pattern to 9 flips, the probability of any given outcome (in this case, HHHHHHHHH), is 1 in 2^9, or 1/512, which is about .002 (.2%).

In other words, if you flip a fair coin 9 times, the odds of it coming up heads all nine times (*the p-value*) is .002. As such, it is not likely that a coin comes up heads nine times due to chance, so you are going to want a good, hard look at the coin I am using before you do any more betting!

## Where Else Can We Use P-Values?

We can use p-values any time we want to find out if the result of an endeavor is due to chance or some specific effect.

For example, suppose that an organization claiming to be elite has invited you to join them, promising you the opportunity to enjoy the company of similar elites and other benefits for an annual fee. You are not sure if the organization is really elite, and wants you to contribute to its specialness, or if they just want your money and are marketing to people's egos.

To find out, you can find a measure of 'eliteness' and compare a random sample of existing members of the organization to a random sample of people not in the organization to see the two groups differ. The random sample of people, if it is large enough, reflects what would happen if the organization selected people by chance. The random sample of members of the organization, if it is large enough, reflects the range of the organization's systematic choosing.

If the p-value is low enough (FYI: this involves a *t-test*, a statistical hypothesis test that won't be discussed in detail here), it means that the organization's choice of members is not random, and you are really something special. Otherwise, feel free to report the scam to the Better Business Bureau!

## Summary

The p-value is a measure of the difference between the world that we experience and the world that we *would* experience if everything happened randomly or due to chance. In statistics, we use the p-value to assess whether the result of some endeavor is due to a real effect or just due to random chance.

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