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

Instructor: Orin Davis
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.

## 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 that 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 is 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.

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