StatisticsANOVA

What is ANOVA?

Analysis of Variance

Test if a measured parameter is different between groups

Literally comparing the sample means by analyzing the variances

Differing Variances – same mean

Remember Variance?

= _____

n – 1

The mean of the squared deviations

Sum of Squares

= _____

n – 1

Partitioning the Variance

SSTotal = SSbetween + SSwithin

SSWithin = mean of squared deviations within each group summed

SSbetween = SStotal – SSwithin

The goal is to create the tableSource SS df MS F p

Between (g)roups – 1 SSbetween/df MSbetween/MSwithin

Within n – g SSwithin/df blank blank

Total n – 1 blank blank blank

SS = Sum of Squaresdf = degrees of freedomMS = 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

F distribution

F table – differs by alpha

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.

Theoretical example

Point number Group 1 Group 2

1 90 150

2 130 100

3 120 30

4 180 40

5 180 60

6 140 50

7 130 40

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?

Point Locations

1 4

2 5 7 9 11

3 6 8 10 12 13 15

14

Hypothesis

Walnut woods has the same BA/acre in the west half and the east half

Use F statistic to infer significance

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.

Result

Anova: Single Factor

SUMMARY

Groups Count Sum Average Variance

Column 1 7 970 138.5714 1047.619

Column 2 7 470 67.14286 1857.143

ANOVA

Source of Variation SS df MS F P-value F crit

Between Groups 17857.14 1 17857.14 12.29508 0.00433 4.747225

Within Groups 17428.57 12 1452.381

Total 35285.71 13       

F table – differs by alpha

Result

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

Questions?