BOT3015L Data analysis and interpretation

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BOT3015LData analysis and interpretation
Presentation created by Jean Burns and Sarah
Tso All photos from Raven et al. Biology of
Plants except when otherwise noted
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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

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Types of data
1. Discrete Having categories (i.e. flowers
present/flowers absent, large/medium/small)
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Seed heteromorphism a discrete character.
Hetermorphic
Not hetermorphic
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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)
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Seed size a continuous character
Commelina benghalensis seed size variation
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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.

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

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Types of statistics
1. Descriptive Summarize a set of data. 2.
Inferential Draw conclusions from a data set.
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Types of statistics
1. Descriptive Summarize a set of data. 2.
Inferential Draw conclusions from a data set.
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Mean a type of descriptive statistic
Arithmetic mean
http//www.steve.gb.com/science/statistics.html
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Mean a type of descriptive statistic

• Measure of the central tendency of a data set.

Mean 2.9
Frequency
Value
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Standard deviation a type of descriptive
statistic
Standard deviation
http//www.steve.gb.com/science/statistics.html
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Standard deviation a type of descriptive
statistic.

• Measure of spread of variability in a data set.

Standard deviation 0.25
Frequency
Value
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Standard deviation a type of descriptive
statistic.

• Measure of spread of variability in a data set.

Standard deviation 0.58
Standard deviation 0.41
Frequency
Value
Value
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Types of statistics
1. Descriptive Summarize a set of data. 2.
Inferential Draw conclusions from a data set.
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Pearsons ?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 CO2
concentration.
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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 CO2 concentration.
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Regression a type of inferential statistic
Used on continuous response variable, when you
have continuous treatments (independent
variables). Example Stomatal aperture response
to varied CO2 concentration (when the CO2
concentration is actually measured). Talk to
your TA if you want to know how to do this
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Observation both internal and external factors
affect stomatal aperture
Question What is the effect of CO2
concentration on stomatal aperture or the number
of open and closed stomata?
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Experimental Design

• Question What is the effect of reducing CO2
  concentration on the number of open stomata?
• Treatment Reduce CO2 concentration using sodium
  hydroxide
• CO2 NaOH gt NaHCO3 (sodium bicarbonate)
• Control Ambient atmospheric CO2 concentration
• Data Count the number of open and closed stomata
  (are these data discrete or continuous?)

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Hypothesis testing for discrete data
Pearsons Chi Square (?2) a test of association
between to categorical variables. Ho Both
treatments yield an equal number of open and
closed stomata. HA1 NaOH treatment results in
fewer open stomata than the control. HA2 NaOH
treatment results in more open stomata than the
control.
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Step 1 Make a contingency table
open stomata closed stomata
NaOH 5 15
Ambient CO2 15 5
This is a 2 x 2 contingency table, having two
columns and two rows, but it can have other
dimensions.
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Step 2 Make a contingency table
open stomata closed stomata Row Totals
NaOH 5 15 20
Ambient CO2 15 5 20
Column Totals 20 20 N 40
Add the row and column totals and the grand
total, N.
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Step 3 Calculate expected values based on null
hypothesis
open stomata closed stomata Row Totals
NaOH 5 (10) 15 (10) 20
Ambient CO2 15 (10) 5 (10) 20
Column Totals 20 20 N 40
Ho Both treatments yield an equal number of open
and closed stomata. For each cell, the expected
value is Row total x column total/ N.
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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
NaOH 5 (10) 15 (10) 20
Ambient CO2 15 (10) 5 (10) 20
Column Totals 20 20 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
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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 Ho

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Hypothesis testing for continuous data
Ho Both treatments yield the same stomatal
aperture.
HA1 NaOH treatment results in narrower stomatal
aperture.
HA2 NaOH treatment results in larger stomatal
aperture.
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Hypothesis testing for continuous data
Ho Both treatments yield the same stomatal
aperture.
A t-test will distinguish between Ho and HA, then
you must look at the direction of the difference
to interpret the results.
HA1 Water treatment results in larger stomatal
aperture.
HA2 NaOH treatment results in larger stomatal
aperture.
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We will use a t-test for this example
http//www.steve.gb.com/science/statistics.html
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Question is there a difference in the means
between two treatments?
Large overlap not different.
http//www.steve.gb.com/science/statistics.html
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Question is there a difference in the means
between two treatments?
small
t lt 2
large
Large overlap not different.
http//www.steve.gb.com/science/statistics.html
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Question is there a difference in the means
between two treatments?
Large overlap not different.
http//www.steve.gb.com/science/statistics.html
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Question is there a difference in the means
between two treatments?
larger
t gt 2
large
Little overlap different.
http//www.steve.gb.com/science/statistics.html
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Question is there a difference in the means
between two treatments?
Little overlap different.
http//www.steve.gb.com/science/statistics.html
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Question is there a difference in the means
between two treatments?
large
t gt 2
small
Little overlap different.
http//www.steve.gb.com/science/statistics.html
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What if the answer is not so obvious?
This is why we need statistics.
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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 tests power. Larger sample
sizes (and DF) result in more power to detect
differences between the means.
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t-value distribution
frequency
t-value
1. Get tcrit from a table of t-values, for P
0.05 and the correct DF. 2. If tobserved gt tcrit,
then the test is significant. 3. If P lt 0.05, the
means are different.
http//www.psychstat.missouristate.edu/introbook/s
bk25m.htm
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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).

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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

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Creating graphs in excel

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

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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

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Creating graphs in excel

• Open excel (Start/Applications/Microsoft Excel)
• Enter the data in table format
• Calculate the mean and standard deviation
• Mean enter formula
• average(cells to calculate the mean from)
• Example
• AVERAGE(A2A11)

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Creating graphs in excel

• Open excel (Start/Applications/Microsoft Excel)
• Enter the data in table format
• Calculate the mean and standard deviation
• Standard deviation enter formula
• stdev(cells to calculate the mean from)
• Example
• STDEV(A2A11)

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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
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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
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Creating graphs in excel

• 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

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Now your chart should look like this
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Creating graphs in excel

• 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
• Note you should only have error bars if the
  data are continuous.

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Now your chart should look like this
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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

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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.
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t-test with Excel

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

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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