1. Chi-Squared Χ2 Analysis
Genetics & Math
Objective: I can calculate if the number of offspring resulting from a cross is true to a pattern of heredity and deviates from the “perfect value” only due to random chance OR some other factor

2. Example Genetics Problem
Red is dominant over white. A heterozygous red breeds with a hybrid. What are the offspring? (phenotype chances)
75% red; 25% white (so if have 1000 offspring, would expect…)
750 red and 250 white (but will this ALWAYS happen EXACTLY as expected ?)
NO! Could have 749 red and 251 white…ok?
But what if have 690 red and 310 white…(is this close enough to 75%, 25%…?)…likely?

3. Chi-Squared Χ2 Analysis
Genetics predicts a probability (a chance) of getting a certain number of offspring at a certain ratio
Does reality always PERFECTLY match?
Sometimes, doesn’t match because there is just a random “sampling” error
Sometimes, doesn’t match because there is a legitimate “force” or variable acting on the results (new pattern of heredity?)
Chi-squared = “Goodness of Fit” Test that determines which using math (statistics)

4. Example (not genetics)
Flip a coin: Probability of Heads = 50% (½)
Probability of Tails = 50% (½)
Flip 10 times: 7 heads & 3 tails…is this okay?
Flip 100 times: 70 heads & 30 tails…is this likely?
Flip 1000 times: 700 heads & 300 tails…likely?
Probability says won’t get expected values EXACTLY, but should get closer with more trials
Null hypothesis: any observed deviation from expected is only due to chance
Reject null hypothesis if show deviation from expected caused by SOMETHING…
Suspect friend’s coin is weighted (trick coin)
Null hypothesis: no statistical significance in the difference between what was observed and what was expected

5. Calculating Chi-Square X2 A Mathematical Way to Accept or Reject the Null Hypothesis (Info Sheet)
For EACH category…(i = category…)
Subtract Expected from Observed
Square and divide by expected
X2 = Σ(sum of) (Oi-Ei)2 Ei
*Don’t have to memorize formula…
Match X2 with a table (will always be given a table)
Degrees of Freedom (df) = # of categories - 1

6. How to use Table (Critical Values)
Examine correct row (df: degrees of freedom)
Match X2 value to top of column = probability
0.90 = 90% probability that the differences are simply a result of chance (random sampling error)  accept null hypothesis
0.05 = 5% probability that the differences are simply a result of chance (random sampling error)  reject null hypothesis (95% sure what to do)
0.05 = standard scientific threshold
If prob. lower (X2 ≥ value)  reject N.H.
If prob. higher (X2 <value)  accept N.H.

7. Flip 10 times: 7 heads & 3 tails…is this likely?
Flip 1000 times: 700 heads & 300 tails…likely?

8. Example (genetics)
Incomplete dominance with red flowers and white flowers making pink
Cross two pink flowers – what would you expect if there were 1000 offspring?
Expect 250 red, 500 pink, and 250 white,
BUT get 318 red, 498 pink, and 184 white
Null Hypothesis = difference due to chance
Chi-Square to calculate critical value at (p)
probability = 0.05 (5% that due to chance)
If X2 <value)  accept N.H. (just random)
If X2 ≥ value  reject N.H. (new pattern?)

9. Expect 250 red, 500 pink, and 250 white,
BUT get 318 red, 498 pink, and 184 white

10. Trends and Patterns
If have a small sample (few trials), is it reasonable for observed to deviate far from expected? (i.e. flip a coin twice, and…)
If have a large sample (many trials), is it reasonable for observed to deviate far from expected? (i.e. flip a coin 100 times…)
The more data you have (larger the sample), the observed data should be closer to the expected value…

11. If you still don’t get it, let’s practice!
Also, check out this video…
Chi-Square is one of many mathematical and statistical ways science proves that the results they get are truly because of a change in variable, as opposed to random chance