1. Chi square analysis Just when you thought statistics was over!!

2. More statistics…  Chi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis.  For example, if, according to Mendel's laws, you expected 10 of 20 offspring from a cross to be male and the actual observed number was 8 males, then you might want to know about the "goodness to fit" between the observed and expected.

3. Hmmmmm…  Were the deviations (differences between observed and expected) the result of chance, or were they due to other factors?  How much deviation can occur before you, the investigator, must conclude that something other than chance is at work, causing the observed to differ from the expected?

4. Null hypothesis  The chi-square test is always testing what scientists call the null hypothesis, which states that there is no significant difference between the expected and observed result.

5. Chi Square x 2 =  ( O - E ) 2 E The formula…. Just get it over with already!!

6. Sample problem  Suppose that a cross between two pea plants yields a population of 880 plants, 639 with green seeds 241 with yellow seeds.  You are asked to propose the genotypes of the parents.  Your hypothesis is that the allele for green is dominant to the allele for yellow and that the parent plants were both heterozygous for this trait.  If your hypothesis is true, then the predicted ratio of offspring from this cross would be 3:1 (based on Mendel's laws) as predicted from the results of the Punnett square

7. GreenYellow Observed (o)639241 Expected (e)660220 Deviation (o - e)-2121 Deviation 2 (o - e) 2 441 d 2 /e0.6682 x 2 = d 2 /e = 2.668.. Chi Square x 2 =  ( O - E ) 2 E

8. So what does 2.688 mean?  Figure out your Degree of freedom (dF)  Degrees of freedom can be calculated as the number of categories in the problem minus 1.  In our example, there are two categories (green and yellow); therefore, there is 1 degree of freedom.

9. Now that you know your dF…  Determine a relative standard to serve as the basis for accepting or rejecting the hypothesis.  The relative standard commonly used in biological research is p > 0.05.  The p value is the probability that the deviation of the observed from that expected is due to chance alone (no other forces acting).  In this case, using p > 0.05, you would expect any deviation to be due to chance alone 5% of the time or less.

10. Conclusion  Refer to a chi-square distribution table  Using the appropriate degrees of 'freedom, locate the value closest to your calculated chi-square in the table.  Determine the closest p (probability) value associated with your chi-square and degrees of freedom.  In this case ( X 2 =2.668), the p value is about 0.10, which means that there is a 10% probability that any deviation from expected results is due to chance only.

11. Degrees of Freedom (df) Probability (p) 0.950.900.800.700.500.300.200.100.050.010.001 10.0040.020.060.150.461.071.642.713.846.6410.83 20.100.210.450.711.392.413.224.605.999.2113.82 30.350.581.011.422.373.664.646.257.8211.3416.27 40.711.061.652.203.364.885.997.789.4913.2818.47 51.141.612.343.004.356.067.299.2411.0715.0920.52 61.632.203.073.835.357.238.5610.6412.5916.8122.46 72.172.833.824.676.358.389.8012.0214.0718.4824.32 82.733.494.595.537.349.5211.0313.3615.5120.0926.12 93.324.175.386.398.3410.6612.2414.6816.9221.6727.88 103.944.866.187.279.3411.7813.4415.9918.3123.2129.59 NonsignificantSignificant

12. Step-by-Step Procedure for Chi-Square  1. State the hypothesis being tested and the predicted results.  2. Determine the expected numbers (not %) for each observational class.  3. Calculate X 2 using the formula.  4. Determine degrees of freedom and locate the value in the appropriate column.  5. Locate the value closest to your calculated X 2 on that degrees of freedom (df) row.  6. Move up the column to determine the p value.  7. State your conclusion in terms of your hypothesis.

13. Analysis  If the p value for the calculated X 2 is p > 0.05, accept your hypothesis. 'The deviation is small enough that chance alone accounts for it. A p value of 0.6, for example, means that there is a 60% probability that any deviation from expected is due to chance only. This is within the range of acceptable deviation.

14.  If the p value for the calculated X 2 is p < 0.05, reject your hypothesis, and conclude that some factor other than chance is operating for the deviation to be so great. For example, a p value of 0.01 means that there is only a 1% chance that this deviation is due to chance alone. Therefore, other factors must be involved.

15. Chi Square x 2 =  ( O - E ) 2 E 100 Flips of a coin Contingency table 40 60 50 100 Heads Tails O E ( 40 - 50 ) 2 50 ( 60 - 50 ) 2 50 + ( 10 ) 2 50 ( 10 ) 2 50 + = = 100 50 100 50 + = = 2 + 2 = 4.00 df = 1

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17. Distribution of colors….or so they say..… hmmmmmmm