Transcript Standard deviation - Department of Biological Science

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
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
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
independent variable. Effect.
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.
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.
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 CO2 concentration.
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.
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
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?
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 => NaHCO3 (sodium bicarbonate)
Control: Ambient atmospheric CO2 concentration
Data: Count the number of open and closed stomata (are
these data discrete or continuous?)
Hypothesis testing for discrete data
Pearson’s 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.
Step 1: Make a contingency table
# open # closed
stomata 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.
Step 2: Make a contingency table
# open # closed
stomata 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.
Step 3: Calculate expected values based
on null hypothesis
# open # closed
stomata 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.
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
Step 4: Compare calculated 2 with the critical
value from a Chi Square distribution table
df
1
2
3
4
5
6
7
8
9
10
P = 0.05
3.84
5.99
7.82
9.49
11.07
12.59
14.07
15.51
16.92
18.31
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
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.
Hypothesis testing for continuous data
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 for this
example:
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
Click the chart button or “Insert” “Chart…”
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
*Note: you should only have error bars if the data
are continuous.
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
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.
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.
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