1. Chi Square

2. What is it?
A statistical test to determine if the actual results are different enough from the predicted results to suggest that the hypothesis is not correct
Use the null hypothesis:
Any difference between the observed and expected data is due to CHANCE
The goal of the Chi Square analysis is to confirm or refute this null hypothesis

3. Calculating Chi Square
Χ2= (O-E)2 E O- observed E- expected

4. If we use pennies… We expect heads ½ the time and tails ½ the time, so
The expected value for heads is 0.5 and the expected value for tails is 0.5
In genetics problems, we use a Punnett square to get our expected values…convert the fractions to decimals

5. Hypothesis
If I flip a coin 300 times, I expect 150 heads, 150 tails for a total of 300

6. Test the Hypothesis I actually flip the penny 300 times…
I observe 162 heads, 138 tails, 300 total

7. Calculate Chi Square Value
For each possible outcome, (O-E)2 E
Heads: (O-E)2 ( )2 = 144/150 = 0.96
Tails: (O-E)2 ( )2 = 144/150 = 0.96

8. Find Total Chi Square Add Chi Square values together
= 1.92
All the information is placed on a Chi Square Table
Class (of data)
Expected
Observed
(O – E)
(O – E)2
(O – E)2/E
Heads
150
162 12
144
.96
Tails
138
-12
Total
300
Sum of X2 = 1.92

9. Degrees of Freedom The number of possible outcomes, minus one.
With coins, you have two possible outcomes: heads or tails.
There is one degree of freedom

10. Compare total to a table of critical values
Use Chi-Square Distribution table
Read across appropriate degrees of freedom row
Find the range of the p value- it is unlikely you will be exactly on a p value, but somewhere between two

11. Critical Values of the X2 Distribution (1.92 with 1 degree of freedom)

12. What does this mean?
So what does that mean? A probability of 0.10 corresponds to a “chance” of 10%; a probability of 0.50 to a “chance” of 50%. This chi-square result means that, if our hypothesis is correct, and we performed exactly this experiment over and over again, 10% or 50% of the time, our results would be at least this far from what we predicted. Or, the probability that we would get results at least as bad as these, even though our hypothesis is correct is between 0.10 and 0.50.

13. So is the hypothesis supported or not?
The usual “level of discrimination” used by investigators is P(X2) = Thus, if your chi-square value has a probability of 0.05 or lower, it is very likely (but not certain) that your hypothesis is not supported.
If the calculated Chi Square is larger than the critical value, we REJECT the null hypothesis because our data is too different from the expected to be due to chance alone…there must be some other explanation

14. Practice #1

15. Hypothesis: White fruit is dominant over yellow fruit P = Ww x ww
½ White; ½ yellow 1:1
.5 white; .5 yellow W w Ww ww

16. Calculate Expected
There are 200 total seeds produced (observed) 200 * .5 = 100 (expect 1 out of 2) 200 * .5 = 100 (expect 1 out of 2) _____ 200

17. Observed:
110 white 90 yellow

18. Chi Square
(Observed – Expected)2 Expected White: ( )2 = 1 Add χ2 together 100 to get total. Yellow: (90-100)2 = 1 Total = 2 100

19. Degrees of freedom = # of possibilities - 1

20. What does that mean? 1◦ Freedom X2 value of 2 p value is 0.2-0.1
NOT a significant difference between observed and expected
Hypothesis is supported by observed results
Due to random chance

21. Practice #2

22. Hypothesis: White fruit is dominant over yellow fruit P = Ww x ww
½ White; ½ yellow 1:1
.5 white; .5 yellow W w Ww ww

23. Calculate Expected
There are 2000 total seeds produced (observed) 2000 * .5 = 1000 (expect 1 out of 2) 2000* .5 = 1000 (expect 1 out of 2) _____ 2000

24. Observed:
1100 white 900 yellow

25. Chi Square
(Observed – Expected)2 Expected White: ( )2 = Yellow: ( )2 = 10 Total = 20

26. What does that mean? 1◦ Freedom X2 value of 20 p value is ˂0.01
THERE IS a significant difference between observed and expected
Hypothesis is NOT supported by observed results
Due to some other factor than random chance

27. Practice #3 ttWw x TtWw (T= tall t= short; W= wide w= narrow)
3/8 tall, wide
1/8 tall, narrow
3/8 short, wide
1/8 short, narrow tW tw TW
TtWW
TtWw Tw
Ttww
ttWW
ttWw
ttww

28. There are 462 offspring produced
Calculate expected:
462 * = (can’t have ¼ of organism)
462 * =
462 * =
____
462

29. Observed
Of the 462 offspring observed, 158 are tall, wide 64 are tall, narrow 143 are short, wide 97 are short, narrow Calculate the Chi Square!

30. Calculate Chi Square
Tall, wide: ( )2 = Tall, narrow: (64-58)2 = Short, wide: ( )2 = 5.20 Short, narrow: (97-58)2 = Total = 33.34

31. Degrees of freedom = # of possibilities - 1

32. Analyze 3 degrees of freedom (4 possible phenotypes, less one)
Chi- Square value of 33.34
p value is ˂ 0.01
Hypothesis is NOT supported by data
Likely due to some factor other than chance