Chapter 8 BOT3015L Data analysis and interpretation Presentation created by Jean Burns All photos from Raven et al. Biology of Plants except when otherwise noted

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

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

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

Seed heteromorphism: a discrete character. Not hetermorphic Hetermorphic

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)

Seed size: a continuous character Commelina benghalensis seed size variation

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 dependent variable. Effect.

Independent and dependent variables Independent: Treatment (CO2 concentration) Dependent: Stomatal aperture Assumption: Changes in CO2 concentration will alter stomatal aperture.

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

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

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

Mean: a type of descriptive statistic Arithmetic mean

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

Standard deviation: a type of descriptive statistic Standard deviation

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

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

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

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

What internal and external factors likely affect stomatal aperture? What are the effects of CO 2 on stomatal aperture? Why do we want to know? How is this important? About 1700 gallons of water are required to grow food for one adult in the US per day! (From 1993 National Geographic)

Experimental Design The question: What are the effects of CO 2 on stomatal aperture? Ambient CO 2 x lowered CO 2 CO 2 + NaOH => NaHCO 3 (sodium bicarbonate)

Hypothesis testing Ho: Both treatments yield the same stomatal aperture. HA1: NaOH treatment results in narrower stomatal aperture. HA2: NaOH treatment results in larger stomatal aperture.

Hypothesis testing 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.

We will use a t-test to interpret the gas exchange experiment

Question: is there a difference in the means between two treatments? Large overlap = not different.

Question: is there a difference in the means between two treatments? Large overlap = not different. small large t < ~2

Question: is there a difference in the means between two treatments? Large overlap = not different.

Question: is there a difference in the means between two treatments? Little overlap = different. larger large t > ~2

Question: is there a difference in the means between two treatments? Little overlap = different.

Question: is there a difference in the means between two treatments? Little overlap = different. large small t > ~2

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

Degrees of freedom DF = n 1 + n 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.

t-value distribution 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.

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).

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

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

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:

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)

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)

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

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 Chart Button

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

Now your chart should look like this:

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

Now your chart should look like this:

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

Doing a t-test 1.Import the data into JMP Open JMP Create two columns: Independent and dependent variables (double click on column heading area) Create 50 rows (double click on row heading area) Copy and paste data from JMP (select column heading and rows to paste into) Double click

Doing a t-test 1.Import the data into JMP Open JMP Create two columns: Independent and dependent variables Copy and paste data from JMP Make Treatment a nominal variable (double click on column heading, change data type to character) Or, use dummy variable, shown here

Doing a t-test 1.Import the data into JMP 2.Look at data distribution Analysis Distribution of Y Add Stomatal aperture (ok)

Doing a t-test 1.Import the data into JMP 2.Look at data distribution 3.Is the distribution skewed? Yes, data is skewed:

Doing a t-test 1.Import the data into JMP 2.Look at data distribution 3.Is the distribution skewed? 4.Transform the data Create a new column Double click on the heading of the column Add a formula Select OK Formula: ln(stomatal aperture) Evaluate Close dialog box

Doing a t-test 1.Import the data into JMP 2.Look at data distribution 3.Is the distribution skewed? 4.Transform the data 5.Do the t-test on transformed data: Analysis Fit Y by X Select and add Treatment = X Aperture = Y OK

Doing a t-test 1.Import the data into JMP 2.Look at data distribution 3.Is the distribution skewed? 4.Transform the data 5.Do the t-test on transformed data: Analysis Click arrow

Doing a t-test 1.Import the data into JMP 2.Look at data distribution 3.Is the distribution skewed? 4.Transform the data 5.To the t-test on transformed data: Analysis Click arrow Select Means, ANOVA, t-test

Interpret the results of your t-test 1.T-test T-value, larger values indicate stronger effect

Interpret the results of your t-test 1.T-test T-value, larger values indicate stronger effect 2.DF Degrees of freedom

Interpret the results of your t-test 1.T-test T-value, larger values indicate stronger effect 2.DF Degrees of freedom 3.Prob > t P-value, smaller values indicate stronger effect P < 0.05, significant difference between means.

Reminders 1.Submit Guard cell report next week: refer to “organization of a short report” (pages 9-10 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