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Χ2 (Chi-square) Test
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…
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)
Χ2 (Chi-square) Test
Hypotheses
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
Χ2 (Chi-square) Test
NULL HYPOTHESIS
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…
Χ2 (Chi-square) Test
You flip the coin 100 times and you getting the following
results:
Observed
Expected
Heads
41
50
Tails
59
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.
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? 2
Χ approaches 0
Let’s determine χ2 for the coin flipping study…
Χ2 (Chi-square) Test
Observed
Expected
Heads
41
50
Tails
59
50
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…?
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?
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.
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?
Χ2 (Chi-square) Test
The P-value
P-value = ~0.07
The p-value tells us the probability that the NULL
hypothesis (observed and expected not different) is correct.
Χ2 (Chi-square) Test
Observed
Expected
Heads
41
50
Tails
59
50
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.
Χ2 (Chi-square) Test
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
(95% chance that null is correct).
Therefore the null hypothesis cannot be rejected.