2. PCB 3043L - General Ecology Data Analysis

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

4. What is a variable? Variable: any defined characteristic that varies from one biological entity to another. Examples: plant height, bird weight, human eye color, no. 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.

5. 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.

6. 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)

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

8. 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

9. 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

10. Statistical analysis of data Measures of central tendency Measures of dispersion and variability Site 1Site 2 54 72 38 83 67 Heights of pine trees at 2 sites in Everglades National Park

11. 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 = (1 + 2 + 2 + 2+ 3 + 5 + 6 + 7 + 8 + 9 + 10)/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

12. Measures of dispersion and variability How widely is the data distributed? range: largest value minus smallest value variance (s 2 or σ 2 ) ………….…………. standard deviation (s or σ)………………… Large spread Small spread

13. Example data set: 0, 1, 3, 3, 5, 5, 5, 7, 7, 9, 10 Variance = 9.8 Standard Deviation = 3.13 Range = 10 Example data set: 0, 10, 30, 30, 50, 50, 50, 70, 70, 90, 100 Variance = 980 Standard Deviation = 31.30 Range = 100 Measures of dispersion and variability

14. 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

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

16. Proportions of a Normal Distribution How many bears have a weight > 80kg μ = 70kgσ = 10kgX = 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

17. 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

18. Probability distribution tables There are multiple probability tables for different types of statistical tests. e.g. Z-Table, t-Table, Χ 2 -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

19. Statistical significance 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) What does this mean? There is less than a 5% chance that the null hypothesis will be true The lower the α value the more certain we about rejecting the null Hypothesis