1. What is a χ2 (Chi-square) test used for?
Statistical test used to compare observed data
with expected data according to a hypothesis.
Let’s look at the next slide to find out…

2. 50 heads and 50 tails Χ2 (Chi-square) Test
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)

3. Hypotheses Χ2 (Chi-square) Test
1. 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

4. NULL HYPOTHESIS Χ2 (Chi-square) Test
This 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…

5. Heads Tails Observed 41 59 Expected 50 50 Χ2 (Chi-square) Test
You flip the coin 100 times and you getting the following results:
Heads
Tails
Observed 41 59
Expected 50 50
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.

6. Χ2 (Chi-square) Test 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…

7. Heads Tails Observed 41 59 Expected 50 50 Χ2 (Chi-square) Test
Χ2 = (41-50)2/ (59-50)2/50
Χ2 = (-9)2/ (9)2/50
Χ2 = 81/ /50
Χ2 = 3.24
So what does this number mean…?

8. Χ2 (Chi-square) Test Converting Χ2 to a P(probability)-value
Statisticians have devised a table to do this:
Great, but how do you use this?

9. Χ2 (Chi-square) Test Converting Χ2 to a P(probability)-value
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.

10. Χ2 (Chi-square) Test Converting Χ2 to a P(probability)-value
Then scan across and find your X2 value (3.24)
Lastly go up and estimate the p-value…
P-value = ~0.07
What does this value tell us?

11. The P-value Χ2 (Chi-square) Test P-value = ~0.07
The p-value tells us the probability that the NULL hypothesis (observed and expected not different) is correct.

12. Heads Tails Observed 41 59 Expected 50 50 Χ2 (Chi-square) Test
P-value = ~0.07
Therefore, there is a 93% chance that the null hypothesis (there is no real difference between observed and expected) is correct.

13. 50 50 Χ2 (Chi-square) Test 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 (95% chance that null is correct).
Therefore the null hypothesis cannot be rejected.