1. PCB 3043L - General Ecology
Data Analysis

2. OUTLINE Organizing an ecological study Basic sampling terminology
Statistical analysis of data
Why use statistics?
Describing data
Measures of central tendency
Measures of spread
Normal distributions
Using Excel
Producing tables
Producing graphs
Analyzing data
Statistical tests
T-Tests
ANOVA
Regression

3. Organizing an ecological study
What is the aim of the study?
What is the main question being asked?
What are your hypotheses?
Collect data
Summarize data in tables
Present data graphically
Statistically test your hypotheses
Analyze the statistical results
Present a conclusion to the proposed question

5. Basic sampling terminology
Variables
Populations
Samples
Parameters
Statistics

6. What is a variable?
Variable: any defined characteristic that varies from one biological entity to another.
Examples: plant height, bird weight, human eye color, N° of tree species
If an individual is selected randomly from a population, it may display a particular height, weight, etc.
If several individuals are selected, their characteristics may be very similar or very different.

7. What is a population?
Population: the entire collection of measurements of a variable of interest.
Example: if we are interested in the heights of pine trees in Everglades National Park (Plant height is our variable) then our population would consist of all the pine trees in Everglades National Park .

8. What is a sample?
Sample: smaller groups or subsets of the population which are measured and used to estimate the distribution of the variable within the true population
Example: the heights of 100 pine trees in Everglades National Park may be used to estimate the heights of trees within the entire population (which actually consists of thousands of trees)
How do you think you can increase your accuracy on your study?

9. What is a parameter?
Parameter: any calculated measure used to describe or characterize a population
Example: the average height of pine trees in Everglades National Park

10. What is a statistic?
Statistic: an estimate of any population parameter
Example: the average height of a sample of 100 pine trees in Everglades National Park

11. Why use statistics?
It is not always possible to obtain measures and calculate parameters of variables for the entire population of interest.
Statistics allow us to estimate these values for the entire population based on multiple, random samples of the variable of interest.
The larger the number of samples, the closer the estimated measure is to the true population measure.
Statistics also allow us to efficiently compare populations to determine differences among them.
Statistics allow us to determine relationships between variables.

12. Statistical analysis of data
Heights of pine trees at 2 sites in Everglades National Park
Site 1
Site 2 5 4 7 2 3 8 6
Measures of central tendency
Measures of dispersion and variability

13. Measures of central tendency
Where is the center of the distribution?
mean ( or μ): arithmetic mean……
median: the value in the middle of the ordered data set
mode: the most commonly occurring value
Example data set : 1, 2, 2, 2, 3, 5, 6, 7, 8, 9, 10
Mean = ( )/11 = 55/11 = 5
Median = 1, 2, 2, 2, 3, 5, 6, 7, 8, 9,10 = 5
1, 2, 2, 2, 3, 5, 6, 7, 8, 9,10,11 = (5+6)/2 = 5.5
Mode = 1, 2, 2, 2, 3, 5, 6, 7, 8, 9, 10 = 2

15. Measures of dispersion and variability
How widely is the data distributed?
range: largest value minus smallest value
variance (s2 or σ2) ………….………….
standard deviation (s or σ)…………………
Small spread=more clustered data
Large spread Small spread

16. Same mean but variance are different

17. Measures of dispersion and variability
Example data set: 0, 1, 3, 3, 5, 5, 5, 7, 7, 9, 10 Variance = 9.8 Standard Deviation = 3.13 Range = Example data set: 0, 10, 30, 30, 50, 50, 50, 70, 70, 90, 100 Variance = 980 Standard Deviation = Range = 100

18. Normal distribution of data
A data set in which most values are around the mean, with fewer observations towards the extremes of the range of values
The distribution is symmetrical about the mean
gaussian

19. Proportions of a Normal Distribution
A normal population of 1000 body weights
μ = 70kg σ = 10kg
500 weights are > 70kg
500 weights are < 70 kg

20. Proportions of a Normal Distribution
How many bears have a weight > 80kg
μ = 70kg σ = 10kg X = 80kg
We use an equation to tell us how many standard deviations from the mean the X value is located:
= =
We then use a special table to tell us what proportion of a normal distribution lies beyond this Z value
This proportion is equal to the probability of drawing at random a measurement (X) greater than 80kg
Z = X – μ σ
Z = 80 – 70 10 1

21. Z table Look for Z value on table (1.0)
Find associated P value (0.1587)
P value states there is a 15.87% ((0.1587/1)x100) chance that a bear selected from the population of 1000 bears measured will have a weight greater than 80kg

22. Probability distribution tables
There are multiple probability tables for different types of statistical tests.
e.g. Z-Table, t-Table, Χ2-Table
Each allows you to associate a “critical value” with a “P value”
This P value is used to determine the significance of statistical results

23. Using Excel Program used to organize data Produce tables
Perform calculations
Make graphs
Perform statistical tests

24. Organizing data in tables
Allows you to arrange data in a format that is best for analysis
The following are the steps you would use:

28. Performing calculations
Allows you to perform several calculations
Sum, Average, Variance, Standard deviation
Basic subtraction, addition, multiplication
More complex formulas

43. Making graphs Bar Charts……. Scatter Plots…………………. Bar Charts-
adding error bars
Frequency Histograms
Using the ‘countif’ function
Scatter Plots
Adding trendlines

58. Making graphs Bar Charts……. Scatter Plots…………………. Bar Charts-
adding error bars
Frequency Histograms
Using the ‘countif’ function
Scatter Plots
Adding trendlines

67. Analyzing Data in Excel
Statistical tests can be done to determine:
Whether or not there is a significant difference between two data sets (Student’s t-test)
Whether or not there is a significant difference between more than two data sets (ANOVA)
Whether or not there is a significant relationship between two variables (Regression analysis)

68. Analyzing Data in Excel
The following steps must be followed:
Choose an appropriate statistical test
State H0 and HA
Run test to produce Test Statistic
Examine P-value
Decide to accept or reject H0

69. Analyzing Data in Excel
Normally, you would have to calculate the critical value and look up the P value on a table
All tests done in Excel provide the P value for you
This P value is used to determine the significance of statistical results
This P value must be compared to an α value
α value is usually 0.05 or less (e.g. 0.01)
Less than 5% chance that the null hypothesis is true
The lower the α value the more certain we about rejecting the null Hypothesis
First thing you must do is select which statistical test you want to perform
This is how it is done……..

74. t-Tests
Used to compare the means of two populations and answer the question:
Is there a significant difference between the two populations?
Example: Is there a significant difference between the average height of pine trees from 2 sites in Everglades National Park?
You cannot use this test to compare two different types of data (e.g. water depth data and soil depth data).
It can only compare two sets of data based on the same data type (e.g. water depth data from two different sites)
The two data sets that are being compared must be presented in the same units. (e.g. you can compare two sets of data if both are recorded in days. You cannot compare data recorded in units of days with data recorded in units of months)

75. Your Null Hypothesis is always:
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
Your Null Hypothesis is always:
There is no significant difference between the two compared populations (μ1= μ2)
Your Alternative Hypothesis is always:
There is a difference between the two compared populations (μ1 ≠ μ2)

76. 1. Choose an appropriate statistical test 2. State H0 and HA 3
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0

82. t-Tests When you run the test, look for the p-value
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
When you run the test, look for the p-value
If p > 0.05 then fail to reject your Null Hypothesis and state that “there is no significant difference between the two compared populations”
If p < 0.05 then reject your Null Hypothesis and state that “there is a significant difference between the two compared populations”

83. t-Tests When you run the test, look for the p-value
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
When you run the test, look for the p-value
Our results show P =
Therefore P > 0.05 (This means that there is greater than a 5% chance that our null hypothesis is true)
So we must fail to reject the Null Hypothesis and state that “there is no significant difference between the two compared populations”

84. ANOVA
Used to compare the means of more than two populations and answer the question:
Is there a significant difference between the populations?
Example: Is there a significant difference between the average height of pine trees from 4 sites in Everglades National Park?
For comparing a particular feature of two or more populations, use a Single Factor ANOVA
For comparing a particular feature of two or more populations, subdivided into two groups, use a Two Factor ANOVA

85. Your Null Hypothesis is always:
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
Your Null Hypothesis is always:
There is no significant difference between the compared populations (μ1 = μ2 = μ3 = μ4 …..)
Your Alternative Hypothesis is always:
There is a difference between the compared populations (μ1 ≠ μ2 ≠ μ3 ≠ μ4 …..)

86. 1. Choose an appropriate statistical test 2. State H0 and HA 3
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0

90. ANOVA When you run the test, look for the p-value
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
When you run the test, look for the p-value
If p > 0.05 then fail to reject your Null Hypothesis and state that “there is no significant difference between the compared populations”
If p < 0.05 then reject your Null Hypothesis and state that “there is a significant difference between at least two of the compared populations”

91. ANOVA When you run the test, look for the p-value
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
When you run the test, look for the p-value
Our results show P =
Therefore P < 0.05 (This means that there is less than a 5% chance that our null hypothesis is true)
So we must reject your Null Hypothesis and state that “there is a significant difference between at least two of the compared populations”

92. ANOVA Remember: The ANOVA result will only tell you that
None of the data sets are significantly different from each other OR
At least two of the data sets among the data sets being compared are significantly different
If there is a significant difference between at least two data sets, it will not tell you which two.

93. Two-way ANOVA
Used to compare the means of more than two populations that are subdivided into two or more groups and answer the question:
Is there a significant difference between the populations?
Example: Is there a significant difference between the average height of pine trees from 4 sites in Everglades National Park, during the wet and dry season?

94. 1. Choose an appropriate statistical test 2. State H0 and HA 3
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0

100. Two-way ANOVA When you run the test, look for the interaction p-value
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
When you run the test, look for the interaction p-value
If p > 0.05 then fail to reject your Null Hypothesis and state that “there is no significant difference between the compared populations”
If p < 0.05 then reject your Null Hypothesis and state that “there is a significant difference between at least two of the compared populations”

101. Two-way ANOVA Our results show P = 0.2888
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
Our results show P =
Therefore P > 0.05 (This means that there is a greater than a 5% chance that our null hypothesis is true)
So we must fail to reject the Null Hypothesis and state that “there is no significant difference between the compared populations”

102. Regression analysis
Used to determine whether or not there is a linear relationship between two variables and answer the question:
Is there a significant linear relationship between two variables?
Example: Is there a significant relationship between the average height of pine trees and soil depth in Everglades National Park?
It basically creates an equation (or line) that best predicts Y values based on X values.
You cannot use this test to compare populations. It only compares variables.
You are looking at two different variables (e.g. water depth (cm) and plant abundance (no. of individuals), so the data sets do not have to be presented in the same units

103. Your Null Hypothesis is always:
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
Your Null Hypothesis is always:
There is no significant linear relationship between the two variables
Your Alternative Hypothesis is always:
There is a significant linear relationship between the two variables

104. The closer R square is to 0, the less well it fits the data.
R squared: how well “y” can be predicted by “x”, i.e. how strong the linear relationship is between the two variables.
The closer R square is to 0, the less well it fits the data.
The closer R square is to 1, more it fits the data.
Example: R square value of 0.04
The regression line does not fit the data well
Many of the points lie far from the line, so there is not a defined linear relationship between the two variables
“x” cannot be used to predict “y”
Example: R square value of 0.94
The regression line fits the data well
The points all lie fairly close to the line, so there is a defined linear relationship between the two variables
“x” can be used to predict “y”

105. 1. Choose an appropriate statistical test 2. State H0 and HA 3
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0

112. Regression analysis
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
When you run the test, look for the Significance F or Sample p-value
If p > 0.05 then fail to reject your Null Hypothesis and state that “There is no significant linear relationship between the two variables”
If p < 0.05 then reject your Null Hypothesis and state that “There is a significant linear relationship between the two variables”

113. Regression analysis When you run the test, look for the p-value
1. Choose an appropriate statistical test 2. State H0 and HA 3. Run test to produce Test Statistic 4. Examine P-value 5. Decide to accept or reject H0
When you run the test, look for the p-value
Our results show Significance F or Sample p-value = 1.65E08 =
Therefore P < 0.05 (This means that there is less than a 5% chance that our null hypothesis is true)
So we must reject your Null Hypothesis and state that “There is a significant linear relationship between the two variables”
Next look at the R squared value
Our results show R squared = 0.975
Therefore the line fits the data well
“x” can be used to predict “y”

114. Ecological study What is the aim of the study?
What is the main question being asked?
What are your hypotheses?
Collect data
Summarize data in tables
Present data graphically
Statistically test your hypotheses
Analyze the statistical results
Present a conclusion to the proposed question

115. Distance from trail (m)
Aim: To determine whether or not there are changes in heights of Pine trees with distance from the edge of a forest trail in Everglades National Park.
Hypotheses:
HO: There is no significant relationship between distance from the edge of the trail and Pine tree height
HA: There is a significant relationship between distance from the edge of the trail and Pine tree height
Results:
Discussion/Conclusion:
The gap created by the trail may be adversely affecting Pine trees, such that they are shorter near the trail and become taller with distance from the trail.
Average tree height of pine trees along transect from forest trail to interior forest at ENP
Distance from trail (m)
Plant heights (m)
2.1 5
2.7 10
2.9 15
3.1 20
3.4 25
3.7 30
3.8 35
4.5 40
4.6 45
4.8 50
5.6
SUM
41.2
AVERAGE
3.74
STANDARD DEVIATION
1.04
P = 1.65E Since P < 0.05, reject Ho
Therefore, there is a significant relationship between distance from the edge of the trail and Pine tree height
R Square = 0.97, so there is a strong positive linear relationship between distance from the trail and plant height

116. Assignment – Worksheet 1
Three questions:
T-test
Single factor ANOVA and Two-way ANOVA
Regression analysis