1. BOT3015L Data analysis and interpretation Presentation created by Jean Burns and Sarah Tso All photos from Raven et al. Biology of Plants except when otherwise noted

2. Today Types of data Discrete, Continuous Independent, dependent Types of statistics Descriptive, Inferential Creating graphs in excel Doing a t-test or Chi Square Lab: create graphs and do statistics for the gas exchange experiment

3. Today Types of data Discrete, Continuous Independent, dependent Types of statistics Descriptive, Inferential Creating graphs in excel Doing a t-test Lab: create graphs and do statistics for the gas exchange experiment

4. Types of data 1. Discrete: Having categories (i.e. flowers present/flowers absent, large/medium/small)

5. Seed heteromorphism: a discrete character. Not hetermorphic Hetermorphic

6. Types of data 1. Discrete: Having categories (i.e. flowers present/flowers absent, large/medium/small) 2. Continuous: Having infinite possible values (i.e. age, growth rate)

7. Seed size: a continuous character Commelina benghalensis seed size variation

8. Types of data 1.Independent: Manipulated or selected with the hypothesis that it is causally linked to the dependent variable. Cause. 2.Dependent: Measured as a response to the independent variable. Effect.

9. Independent and dependent variables Independent: Treatment (CO2 concentration) Dependent: Number of open and closed stomata, or stomatal aperture Assumption: Changes in CO2 concentration will affect stomatal aperture.

10. Today Types of data Discrete, Continuous Independent, dependent Types of statistics Descriptive, Inferential Creating graphs in excel Doing a t-test Lab: create graphs and do statistics for the gas exchange experiment

11. Types of statistics 1. Descriptive: Summarize a set of data. 2. Inferential: Draw conclusions from a data set.

12. Types of statistics 1. Descriptive: Summarize a set of data. 2. Inferential: Draw conclusions from a data set.

13. Mean: a type of descriptive statistic Arithmetic mean http://www.steve.gb.com/science/statistics.html

14. Mean: a type of descriptive statistic Measure of the central tendency of a data set. Frequency Value Mean = 2.9

15. Standard deviation: a type of descriptive statistic Standard deviation http://www.steve.gb.com/science/statistics.html

16. Standard deviation: a type of descriptive statistic. Measure of spread of variability in a data set. Frequency Value Standard deviation = 0.25

17. Standard deviation: a type of descriptive statistic. Measure of spread of variability in a data set. Frequency Value Standard deviation = 0.58Standard deviation = 0.41 Value

18. Types of statistics 1. Descriptive: Summarize a set of data. 2. Inferential: Draw conclusions from a data set.

19. Pearson’s  2 : a type of inferential statistic Used on discrete response variable, when you have discrete treatments (independent variables). Example: The number of open and closed stomata in response to lower CO 2 concentration.

20. t-test: a type of inferential statistic Used on continuous response variable, when you have discrete treatments (independent variables). Example: Stomatal aperture response to lower CO 2 concentration.

21. Regression: a type of inferential statistic Used on continuous response variable, when you have continuous treatments (independent variables). Example: Stomatal aperture response to varied CO 2 concentration (when the CO 2 concentration is actually measured). *Talk to your TA if you want to know how to do this

22. Observation: both internal and external factors affect stomatal aperture Question: What is the effect of CO 2 concentration on stomatal aperture or the number of open and closed stomata?

23. Experimental Design Question: What is the effect of reducing CO 2 concentration on the number of open stomata? Treatment: Reduce CO 2 concentration using sodium hydroxide: CO 2 + NaOH => NaHCO 3 (sodium bicarbonate) Control: Ambient atmospheric CO 2 concentration Data: Count the number of open and closed stomata (are these data discrete or continuous?)

24. Hypothesis testing for discrete data Pearson’s Chi Square (  2 ): a test of association between to categorical variables. H o : Both treatments yield an equal number of open and closed stomata. H A1 : NaOH treatment results in fewer open stomata than the control. H A2 : NaOH treatment results in more open stomata than the control.

25. Step 1: Make a contingency table # open stomata # closed stomata NaOH515 Ambient CO 2 155 This is a 2 x 2 contingency table, having two columns and two rows, but it can have other dimensions.

26. Step 2: Make a contingency table # open stomata # closed stomata Row Totals NaOH51520 Ambient CO 2 15520 Column Totals20 N = 40 Add the row and column totals and the grand total, N.

27. Step 3: Calculate expected values based on null hypothesis # open stomata # closed stomata Row Totals NaOH5 (10)15 (10)20 Ambient CO 2 15 (10)5 (10)20 Column Totals20 N = 40 H o : Both treatments yield an equal number of open and closed stomata. For each cell, the expected value is: Row total x column total/ N.

28. Step 4: Calculate the  2 and degrees of freedom  2 =  {(observed - expected) 2 / expected} d.f. = (# of columns - 1) x (# of rows - 1) # open stomata # closed stomata Row Totals NaOH5 (10)15 (10)20 Ambient CO 2 15 (10)5 (10)20 Column Totals20 N = 40  2 = (5 - 10) 2 / 10 + (15 - 10) 2 /10 + (15 - 10) 2 /10 + (5 - 10) 2 / 10 = 10 d.f. = (2 - 1) x (2 - 1) = 1

29. Step 4: Compare calculated  2 with the critical value from a Chi Square distribution table The critical value can be obtained from a table based on the degrees of freedom and the level of confidence, which is set at P = 0.05.  2 calc = 10  2 crit = 3.84, d.f. = 1 If the calculated value exceeds the critical value, you can reject your H o

30. Hypothesis testing for continuous data H o : Both treatments yield the same stomatal aperture. H A1 : NaOH treatment results in narrower stomatal aperture. H A2 : NaOH treatment results in larger stomatal aperture.

31. Hypothesis testing for continuous data H o : Both treatments yield the same stomatal aperture. H A1 : Water treatment results in larger stomatal aperture. H A2 : NaOH treatment results in larger stomatal aperture. A t-test will distinguish between H o and H A, then you must look at the direction of the difference to interpret the results.

32. We will use a t-test for this example: http://www.steve.gb.com/science/statistics.html

33. Question: is there a difference in the means between two treatments? Large overlap = not different. http://www.steve.gb.com/science/statistics.html

34. Question: is there a difference in the means between two treatments? Large overlap = not different. http://www.steve.gb.com/science/statistics.html small large t < ~2

35. Question: is there a difference in the means between two treatments? Large overlap = not different. http://www.steve.gb.com/science/statistics.html

36. Question: is there a difference in the means between two treatments? Little overlap = different. http://www.steve.gb.com/science/statistics.html larger large t > ~2

37. Question: is there a difference in the means between two treatments? Little overlap = different. http://www.steve.gb.com/science/statistics.html

38. Question: is there a difference in the means between two treatments? Little overlap = different. http://www.steve.gb.com/science/statistics.html large small t > ~2

39. What if the answer is not so obvious? This is why we need statistics.

40. Degrees of freedom DF = n 1 + n 2 - 2 DF = number of independent categories in a statistical test. For example, in a t-test, we are estimating 2 parameters the mean and the variance. Thus we subtract 2 from the degrees of freedom, because 2 elements are no longer independent. DF is a measure of a test’s power. Larger sample sizes (and DF) result in more power to detect differences between the means.

41. t-value distribution http://www.psychstat.missouristate.edu/introbook/sbk25m.htm t-value frequency 1. Get t crit from a table of t-values, for P = 0.05 and the correct DF. 2. If t observed > t crit, then the test is significant. 3. If P < 0.05, the means are different.

42. Factors influencing a difference between means Distance between means Variance in each sample (Standard Deviation, SD) T-value (means and SD) Number of samples (DF) Level of error we are willing to accept to consider two means different (P-value).

43. Today Types of data Discrete, Continuous Independent, dependent Types of statistics Descriptive, Inferential Creating graphs in excel Doing a t-test Lab: create graphs and do statistics for the gas exchange experiment

44. Creating graphs in excel 1.Open excel (Start/Applications/Microsoft Excel) 2.Enter the data in table format

45. Creating graphs in excel 1.Open excel (Start/Applications/Microsoft Excel) 2.Enter the data in table format 3.In the cells directly under treatment data:

46. Creating graphs in excel 1.Open excel (Start/Applications/Microsoft Excel) 2.Enter the data in table format 3.Calculate the mean and standard deviation Mean: enter formula =average(cells to calculate the mean from) Example: =AVERAGE(A2:A11)

47. Creating graphs in excel 1.Open excel (Start/Applications/Microsoft Excel) 2.Enter the data in table format 3.Calculate the mean and standard deviation Standard deviation: enter formula =stdev(cells to calculate the mean from) Example: =STDEV(A2:A11)

48. Creating graphs in excel 1.Open excel (Start/Applications/Microsoft Excel) 2.Enter the data in table format 3.Calculate the mean and standard deviation 4.Select the data you wish to graph Select these cells

49. Creating graphs in excel 1.Open excel (Start/Applications/Microsoft Excel) 2.Enter the data in table format 3.Calculate the mean and standard deviation 4.Select the data you wish to graph 5.Click the chart button or “Insert” “Chart…” Chart Button

50. Creating graphs in excel 1.Open excel (Start/Applications/Microsoft Excel) 2.Enter the data in table format 3.Calculate the mean and standard deviation 4.Select the data you wish to graph 5.Click the chart button 6.Chose your chart options: Column (next) Series/Category x-axis labels/highlight treatment labels (next) Titles/label axes including Units (next) Finish

51. Now your chart should look like this:

52. Creating graphs in excel 1.Open excel (Start/Applications/Microsoft Excel) 2.Enter the data in table format 3.Calculate the mean and standard deviation 4.Select the data you wish to graph 5.Click the chart button 6.Chose your chart options 7.Add error bars to your chart: Double click on the bar Y-error bars (at the top) Go to Custom Select the cells with the standard deviation *Note: you should only have error bars if the data are continuous.

54. Now your chart should look like this:

55. Today Types of data Discrete, Continuous Independent, dependent Types of statistics Descriptive, Inferential Creating graphs in excel Doing a t-test Lab: create graphs and do statistics for the gas exchange experiment

56. Performing a t-test In this course, we will demonstrate the use of Excel for statistics; however, more advanced software, designed specifically for statistical analyses, offer more detailed analyses. Use the software of your choice, being sure to indicate the software that is used.

57. t-test with Excel In excel: 1.In an empty cell, “Insert” a “Function” 2.Find “T-TEST” 3.“Array 1” is one set of values. Include each value (e.g. each aperture size under one condition) 4.“Array 2 is the other set of values (e.g. each aperture size under the other condition. 5.We will be performing a two-tailed distribution t-test. Enter “2” in “tails.” 6. We are assuming there is equal variance for the two samples, so enter “2” in “type.” 7.“OK” will return the probability (p) value. This is the probability that the difference between the sets of values is random.

58. Reminders Report submissions (paper and turnitin) refer to “organization of a lab report” in the beginning of your lab manual. Titles must be descriptive Methods must be complete Results should include descriptions (in your own words) not just graphs and tables (although those are also necessary). Discussion must demonstrate thought Submit copies of your references with your reports