1. Statistics ANOVA

2. What is ANOVA?

3. Analysis of Variance  Test if a measured parameter is different between groups  Literally comparing the sample means by analyzing the variances

4. Differing Variances – same mean

5. Remember Variance?

6. Sum of Squares

7. Partitioning the Variance  SSTotal = SSbetween + SSwithin  SSWithin = mean of squared deviations within each group summed  SSbetween = SStotal – SSwithin

8. The goal is to create the table SourceSSdfMSFp Between(g)roups – 1SS between /dfMS between /MS within Withinn – gSS within /dfblank Totaln – 1blank SS = Sum of Squares df = degrees of freedom MS = Mean Squares (like variance since you divide by the portion of the sample size) F = the F test statistic, need to choose a alpha level i.e. 5% p = probability

9. F distribution

10. F table – differs by alpha

11. For Instance  If we had split the class into 2 groups and had each group cruise the entire walnut woods tract the same way. We could compare the values obtained by group 1 to those from group 2.

12. Theoretical example Point numberGroup 1Group 2 190150 2130100 312030 418040 518060 614050 713040 These are BA per acre estimates from point sampling Walnut Woods 2013. 14 total points divided in half to create 2 groups. Group 2 basically equals points 8-14. Did the higher numbered points have fewer stems?

13. Point Locations 1 4 2 5 7 911 3 6 8 10121315 14

14. Hypothesis  Walnut woods has the same BA/acre in the west half and the east half  Use F statistic to infer significance

15. Single factor ANOVA in Excel  1. On the Data tab, click Data Analysis.  2. Select Anova: Single Factor and click OK.  3. Click in the Input Range box and select the range B2:C8.  4. Click in the Output Range box and select an empty cell such as A10.  5. Click OK.

16. Result Anova: Single Factor SUMMARY GroupsCountSumAverageVariance Column 17970138.57141047.619 Column 2747067.142861857.143 ANOVA Source of VariationSSdfMSFP-valueF crit Between Groups17857.141 12.295080.004334.747225 Within Groups17428.57121452.381 Total35285.7113

17. F table – differs by alpha

18. Result  Since F > F crit we reject the null hypothesis that the 2 samples have the same BA/acre.

19. Questions?