Download 08-Statistics

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.
Hetermorphic
Not 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
http://www.steve.gb.com/science/statistics.html
Mean: a type of descriptive statistic
Frequency
Measure of the central tendency of a data set.
Mean = 2.9
Value
Standard deviation: a type of
descriptive statistic
Standard deviation
http://www.steve.gb.com/science/statistics.html
Standard deviation: a type of
descriptive statistic.
Measure of spread of variability in a data set.
Frequency
Standard deviation = 0.25
Value
Standard deviation: a type of
descriptive statistic.
Measure of spread of variability in a data set.
Standard deviation = 0.41
Frequency
Standard deviation = 0.58
Value
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 CO2
concentration.
What internal and external factors
likely affect stomatal aperture?
What are the effects of CO2 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 CO2 on stomatal aperture?
Ambient CO2 x lowered CO2
CO2 + NaOH => NaHCO3 (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
Ho: Both treatments
yield the same
stomatal aperture.
A t-test will
distinguish
between Ho and
HA, then you
HA1: Water treatment
must look at the
results in larger
stomatal aperture.
direction of the
difference to
interpret the
results.
H : NaOH treatment
A2
results in larger
stomatal aperture.
We will use a t-test to interpret
the gas exchange experiment
http://www.steve.gb.com/science/statistics.html
Question: is there a difference in the
means between two treatments?
Large overlap = not different.
http://www.steve.gb.com/science/statistics.html
Question: is there a difference in the
means between two treatments?
small
large
t < ~2
Large overlap = not different.
http://www.steve.gb.com/science/statistics.html
Question: is there a difference in the
means between two treatments?
Large overlap = not different.
http://www.steve.gb.com/science/statistics.html
Question: is there a difference in the
means between two treatments?
larger
t > ~2
large
Little overlap = different.
http://www.steve.gb.com/science/statistics.html
Question: is there a difference in the
means between two treatments?
Little overlap = different.
http://www.steve.gb.com/science/statistics.html
Question: is there a difference in the
means between two treatments?
large
small
t > ~2
Little overlap = different.
http://www.steve.gb.com/science/statistics.html
What if the answer is not so obvious?
This is why we need statistics.
Degrees of freedom
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 = n1 + n2 - 2
DF is a measure of a test’s power. Larger sample
sizes (and DF) result in more power to detect
differences between the means.
frequency
t-value distribution
t-value
1. Get tcrit from a table of t-values, for P = 0.05 and
the correct DF.
2. If tobserved > tcrit, then the test is significant.
3. If P < 0.05, the means are different.
http://www.psychstat.missouristate.edu/introbook/sbk25m.htm
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.
2.
3.
4.
Open excel (Start/Applications/Microsoft Excel)
Enter the data in table format
Calculate the mean and standard deviation
Select the data you wish to graph
Select these cells
Creating graphs in excel
1.
2.
3.
4.
5.
Open excel (Start/Applications/Microsoft Excel)
Enter the data in table format
Calculate the mean and standard deviation
Select the data you wish to graph
Chart Button
Click the chart button
Creating graphs in excel
1.
2.
3.
4.
5.
6.
Open excel (Start/Applications/Microsoft Excel)
Enter the data in table format
Calculate the mean and standard deviation
Select the data you wish to graph
Click the chart button
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.
2.
3.
4.
5.
6.
7.
Open excel (Start/Applications/Microsoft Excel)
Enter the data in table format
Calculate the mean and standard deviation
Select the data you wish to graph
Click the chart button
Chose your chart options
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
Double click
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.
2.
3.
4.
Import the data into JMP
Look at data distribution
Is the distribution skewed?
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.
2.
3.
4.
5.
Import the data into JMP
Look at data distribution
Is the distribution skewed?
Transform the data
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.
2.
3.
4.
5.
Import the data into JMP
Look at data distribution
Is the distribution skewed?
Transform the data
Do the t-test on transformed data:
• Analysis
• Click arrow
Doing a t-test
1.
2.
3.
4.
5.
Import the data into JMP
Look at data distribution
Is the distribution skewed?
Transform the data
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