1. STATISTICAL ANALYSIS Frequency Distribution # Indivi duals Median Mean MedianMean Median Figure 2.Frequency distributions of three different samples. ABC negativenormalpositive

2. Descriptive Statistics: used to describe, simplify, and summarize a collection of data in a clear understandable way. Ex. Mean, standard deviation, frequency Inferential Statistics: allows you to make inferences about a population from a sample. Its used to test the Ho. Ex. t-test, ANOVA, ANCOVA Raw Data: the data you collect directly from the organisms or environment you are studying. Population (N): the total # of individuals in the population of interest. Sample (n): the number of observations or individuals measured

3. Mean: _ X = 1.2+3.0+0.5+2.3+1.5 5 = 1.7 m Median: Middle number in a data set. Order from smallest to largest. 0.5 1.2 1.5 2.3 3.0 Range: Difference between the largest and smallest data points in a sample. 3.0 - 0.5 = 2.5 m Fern Height (m) 1.2 3.0 0.5 2.3 1.5 Hand calculations

4. Sample Variance _ _ X X-X(X-X) 2 0.5-1.2 1.44 1.2-0.5 0.25 1.5-0.2 0.04 2.3 0.6 0.36 3.0 1.3 1.69 3.78 n = 5 _ X = 1.7 3.78 4 S 2 == 0.945 Standard Deviation S = 0.945 = 0.972 Standard Error

5. Confidence Interval 95% confidence interval mean Mean = 1.7 SE = 0.435 Standard Error df = n-1 df = 5-1 = 4

6. df = n-1 _ X - u s n t = Testing Ho using t-test Distribution must be normal Use when n > 30 Ho = 0 Ha > 0 t = 1.7 - 0 0.972/ 5 = 0.39 df=5-1=4 Ignore - signs t-Table two tail is more strict one-tailtwo-tail 2.776 0.39 Your t-value is less than t- critical. Fail to reject Ho.

7. Comparing two sample means All right. It is time to use Excel! X1X256473559462857X1X256473559462857 t = _ X 1 - X 2 s 1 2 + s 2 2 n 1 n 2 df = n 1 + n 2 -2 Variances must be equal Ho: X 1 =X 2 Ha: X 1 =X 2

8. Using Excel Click Tools, Data Analysis, t-test: Two-sample Assuming Unequal Variances Enter data Data Set 0

9. t-critical Value Your Value Reporting results in the literature: Sample X1 and sample X2 are significantly different, t(12) = -4.26, p < 0.05, 2-tail. df Do you accept or reject the Ho? Lets the reader know that you set alpha to 0.05. The p-value is less than 0.05, therefore the results are significant.

10. When you compare more than 2 samples, you need to do an Analysis of Variance (ANOVA). The relationship between an ANOVA and t-test: F=t 2 An ANOVA is the similar to a t-test in the sense that you compare the F value to the critical F value. You also need to look at the p-value. If you set alpha to 0.05 and your p value is less that alpha, you can reject the Ho; if greater than alpha then fail to reject the Ho. ANOVA

11. Descriptive Stats

12. Correlation & Regression Correlation: describe the strength of association between two variables. The correlation coefficient is designated as r. r can range from -1 to +1. +1 = a strong positive correlation 0 = no correlation -1 = a strong negative correlation r = +1 r = -1 r = 0

13. This relationship can be represented by a regression equation. Equation for a straight line: y = bx + a y = DV, b = slope, x = IV, a = intercept Simple Linear Regression: shows relationship between variables.