Statistics

1. You will go through the process of science, learning how statistics is applied to a study of saguaros. Statistics

2. OBSERVATION • Saguaros seem to occur in different numbers on the north versus south slope of Gates Pass. Gates Pass www.gamineral.org/t04-gates_pass.html

3. RESEARCH QUESTION • Based on the observation that saguaros seem to occur in different numbers on the north and south slopes of Gates Pass:What descriptive question (versus causal question) could you ask? • Descriptive: Is saguaro density affected by whether the slope faces north versus south? Only requires a count to answer. • Causal: Why is saguaro density affected by whether the slope faces north versus south? Requires controlled studies of many factors to answer.

4. LITERATURE REVIEW • Not needed to come up with the multiple hypotheses for this question because it is a descriptive question – there either is not an effect or there is an effect of slope direction on saguaro density.

5. MULTIPLE HYPOTHESES First, what is the null hypothesis (H0) – the one that states that there is not a cause and effect relationship? • H0: Saguaro density is not affected by whether the slope faces north versus south. Second, what are the alternative hypotheses (H1 and H2 in this case)? • H1: Saguaro density is greater on the north-facing slope. • H2: Saguaro density is greater on the south-facing slope.

6. DEDUCTIONS • What evidence would you need to be convinced each hypothesis is correct or incorrect? • In other words, By how much would the densities have to differ for you to be convinced that the direction of the slope affects saguaro density? • We will come back to this later…but in “real life” you are supposed to come up with deductions before collecting any data.

7. TESTS: Three Data Sets Imagine that these are three possible sets of data that you could have collected by counting saguaros on a north and south slope. Note that I have kept the total number of saguaros the same (200) for each data set.

8. TESTS: THREE EXAMPLES Here are the same data displayed in a bar graph

9. TENTATIVE CONCLUSION • For each data set (A-C) did the evidence convince you that the differences in densities were significant enough to warrant ruling out the null hypothesis (that the distribution of saguaros on the two slopes was just random) and tentatively concluding that slope did affect saguaro density?

10. TENTATIVE CONCLUSION p=probability this would happen randomly? Perhaps it would help to know what the chance is that the distribution is random? p=?% p=?% p=?%

11. TENTATIVE CONCLUSION But how do we determine the probability that the distribution is random? STATISTICS

12. STATISTICS • If you are comparing counts, then use the chi (pronounced kie) square test. • Example: count of saguaros on two slopes. • If you are comparing averages, then use the t-test. • Example: comparing average height of saguaros on two slopes.

13. Chi-Square Test • You compare the actual counts to what the expected count would be if the distribution was random. • In our case, with a total of 200 saguaros counted on both slopes, what would the expected distribution be if they were distributed perfectly random?

14. Chi-Square Test • The chi-squared test tells you the probability that the difference between observed and expected occurred by chance.

15. Chi-Square Test Type in the numbers in the gray boxes and then hit enter • Use my Excel file online

16. TENTATIVE CONCLUSION p=probability this would happen randomly Using my Excel file online, you would come up with these probabilities. p=0.89=89% p=0.16=16% p=0.005=<1%

17. DEDUCTIONS • Which brings us back to deductions. • What probability of being wrong are we willing to risk? • The worse mistake you can make in science (Type 1 error) is to conclude that the difference is due to cause and effect when it was really random. • Are we willing to be wrong 89% of the time? 16% of the time? Less than 1% of the time? • Scientists most often use 5% = p<0.05

18. DEDUCTIONS • If slope direction does not affect saguaro density, then p > or equal to 0.05 (5%). • If saguaro density is greater on the north slope, then p will be less than 0.05 and saguaro density will be greater on the north slope. • If saguaro density is greater on the south slope, then p will be less than 0.05 and saguaro density will be greater on the south slope.

19. TENTATIVE CONCLUSION • For data sets A&B, because p>0.05 there is no significant difference in saguaro density so slope direction unlikely to affect saguaro density. • For data set C, because p<0.05 saguaro density is significantly greater on the south slope so slope direction likely affects saguaro density.

20. North slope South slope Significant? Data set A Data set B Data set C No; p=0.89 No; p=0.16 Yes; p=0.005 99 90 80 101 110 120 This could be your table Table 1. Number of saguaros per hectare on the north and south slope of Gates Pass near Tucson, AZ as counted September 1, 2009.

21. This could be your graph Figure 1. Number of saguaros per hectare on the north and south slope of Gates Pass near Tucson, AZ as counted September 1, 2009.

22. T-Test Example • The t-test tells you the probability that the difference between two averages is random, and considers variability within the data. • For example • H0: Slope direction does not affect saguaro height. • H1: Slope direction does affect saguaro height. Gates Pass www.gamineral.org/t04-gates_pass.html

23. Sample Data Table 1. Saguaro height (in meters) on the north and south slope of Gates Pass near Tucson, AZ as counted September 1, 2009.

24. Sample Data Table 2. Average saguaro height (in meters) for 7 saguaros measured on the north and south slope of Gates Pass near Tucson, AZ on September 1, 2009. Are these significantly different? It depends on sample size and amount of variability in the data.

25. Number saguaros Number saguaros 1.0 1.0 2.0 2.0 3.0 3.0 4.0 4.0 Height Height (m) Lots of variability:Probably not significantly different South slope average North slope average x x x x x x x x x x x x x x North slope average Little variability:Probably significantly different South slope average x x x x x x x x x x x x x x

26. Group 1 Group2 3.5 1.0 0.2 3.2 Etc. Etc. P value 0.272 T-Test • Use my Excel file online • Click on the t-test tab at the bottom. TENTATIVE CONCLUSION?

27. Remember, this is the Magic number: P=0.05

28. If P<0.05then we tentatively conclude that there is a significant difference because there is less than 5% chance that it could have happened randomly.

29. If P>0.05 then we tentatively conclude that there is NOT a significant difference because there is more than 5% chance that it could have happened randomly.

30. Group 1 Group2 3.5 1.0 0.2 3.2 Etc. Etc. P value 0.272 T-Test • Average saguaro height on the north slope (1.5 m) is not significantly different (p=0.27) from average saguaro height on the south slope (2.4 m). TENTATIVE CONCLUSION?

31. REVIEW • Before you collect data (i.e., in your proposal) you have to decide how much difference is enough to convince you that there is cause and effect going on versus just random chance. • Statistics (e.g., chi-square and t-test) can be used to calculate the probability that the difference is random. • Use p<0.05 to rule out the null hypothesis and tentatively conclude there are significant differences that suggest cause and effect.