Χ 2 (Chi-square) Test

Χ2 (Chi-square) TestChapter 13 – How Populations Evolve

What is a χ2 (Chi-square) test used for?

Statistical test used to compare observed datawith expected data according to a hypothesis.

Let’s look at the next slide to find out…

What does that mean?


Ex. Say you have a coin and you want to determine if it is fair (50/50 chance of gets heads/tails). You decide to flip the coin 100 times. If the coin is fair what do you expect/predict to observe?

50 heads and 50 tails

Now come up with a hypothesis (two possibilities)


Hypotheses1. The coin is fair and there will be no real difference between what we will observe and what we expect.

2. The coin is not fair and the observed results will be significantly different from the expected results.

The first hypothesis that states no difference between the observed and expected has a special name… NULL HYPOTHESIS


NULL HYPOTHESISThis is the hypothesis that states there will be no difference between the observed and the expected data or that there is no difference between the two groups you are observing.

Ex. You wonder if world class musicians have quicker reaction times than world class athletes. What would the null hypothesis be?

That there is no difference between these two groups.

Let’s get back to flipping coins…


You flip the coin 100 times and you getting the following results:

Heads TailsObserved 41 59

50 50Expected

Is the coin fair or not?

It’s not easy to say. It looks like it might, but maybe not…This is where statistics, in particular the χ2 test, comes in.


The formula for calculating χ2 is:

Where O is the observed value and E is the expected.

What happens to the value of χ2 as your observed data gets closer to the expected?

Χ2 approaches 0

Let’s determine χ2 for the coin flipping study…



50 50Expected

Χ2 = (41-50)2/50 + (59-50)2/50

Χ2 = (-9)2/50 + (9)2/50

Χ2 = 81/50 + 81/50

Χ2 = 3.24

So what does this number mean…?


Statisticians have devised a table to do this:

Converting Χ2 to a P(probability)-value

Great, but how do you use this?



First we need to determine Degrees of Freedom (DoF):

DoF = # of groups minus 1

We have two groups, heads group and tails group. Therefore our DoF = 1.



Then scan across and find your X2 value (3.24)

P-value = ~0.07Lastly go up and estimate the p-value…

What does this value tell us?


The P-value

The p-value tells us the probability that the NULL hypothesis (observed and expected not different) is correct.

P-value = ~0.07



50 50Expected

P-value = ~0.07

Therefore, there is a 7% chance that the null hypothesis (there is no real difference between observed and expected) is correct.



50 50Expected

P-value = ~0.07

You might say then that the other hypothesis must be correct as there is a 93% likelihood that there is a different between observed and expected…


50 50

P-value = ~0.07

However, statisticians have a p-value = 0.05 cutoff. In order for the hypothesis to be supported, p must be less than 0.05 (5% chance that null is correct).

Therefore the null hypothesis cannot be rejected.


For example, suppose that a cross between two pea plants yields a population of 880 plants, 639 with green seeds and 241 with yellow seeds. You are asked to propose the genotypes of the parents. Your hypothesis is that the allele for green is dominant to the allele for yellow and that the parent plants were both heterozygous for this trait. If your hypothesis is true, then the predicted ratio of offspring from this cross would be 3:1 (based on Mendel's laws) as predicted from the results of the Punnett square. Is it?

Χ 2 (Chi-square) Test

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