Chapter 13 – How Populations Evolve
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
What
does that mean?
hypothesis.
Let’s look at the next slide to find out…
Chapter 13 – How Populations Evolve
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)
Chapter 13 – How Populations Evolve
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
Chapter 13 – How Populations Evolve
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…
Chapter 13 – How Populations Evolve
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.
Chapter 13 – How Populations Evolve
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…
Chapter 13 – How Populations Evolve
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…?
Chapter 13 – How Populations Evolve
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?
Chapter 13 – How Populations Evolve
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.
Chapter 13 – How Populations Evolve
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?
Chapter 13 – How Populations Evolve
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.
Chapter 13 – How Populations Evolve
Χ2 (Chi-square) Test
Observed
Expected
Heads
41
50
Tails
59
50
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.
Chapter 13 – How Populations Evolve
Χ2 (Chi-square) Test
Observed
Expected
Heads
41
50
Tails
59
50
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…
Chapter 13 – How Populations Evolve
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
(5% chance that null is correct).
Therefore the null hypothesis cannot be rejected.
Chapter 13 – How Populations Evolve
Χ2 (Chi-square) Test
Example 1
Chapter 13 – How Populations Evolve
Χ2 (Chi-square) Test
Example 1 Answer
observed
404
420
400
376
expected
(obs-exp)^2/exp
400
0.04
400
1
400
0
400
1.44
2.48X2
DOF = #groups – 1 = 4 – 1 = 3
P = ~0.4 or 40%
Therefore, there is a 40% chance that the null hypothesis is
supported (that there is no difference between the groups)
and therefore, according to this data, the card machine is
fair.
Chapter 13 – How Populations Evolve
Χ2 (Chi-square) Test
Example 2
A genetics engineer was attempting to cross a tiger and a cheetah. She
predicted a phenotypic outcome of the traits she was observing to be in
the following ratio 4 stripes only: 3 spots only: 9 both stripes and
spots. When the cross was performed and she counted the individuals
she found 50 with stripes only, 41 with spots only and 85 with
both. According to the Chi-square test, did she get the predicted
outcome?
Chapter 13 – How Populations Evolve
Χ2 (Chi-square) Test
Example 2 Answer
Expected ratio Observed #
Expected #
O-E
(O-E)2
(O-E)2/E
4 stripes
50
44
6
36
0.82
3 spots
41
33
8
64
1.94
9 stripes/spots
85
99
-14
196
1.98
16 total
176 total
176 total
0 total
DOF = #groups – 1 = 3 – 1 = 2
P = ~0.18 or 18%
Therefore, there is a 18% chance that the null hypothesis is
supported (that there is no difference between the groups)
and therefore, according to this data, the null can be
accepted and the observed is not significantly different
than the expected.
Sum(X2) = 4.74