Statistics
ANOVA
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
𝑥𝑖 − 𝑥
2
𝑖=1
_____
=
n–1
 The mean of the squared deviations
Sum of Squares
𝑛
𝑥𝑖 − 𝑥
𝑖=1
2
_____
=
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 table
Source
SS
df
MS
F
Between
(g)roups – 1
SSbetween/df
MSbetween/MSwithin
Within
n–g
SSwithin/df
blank
blank
Total
n–1
blank
blank
blank
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
p
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
14
15
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
Between
Groups
17857.14
1
17857.14
Within
Groups
17428.57
12
1452.381
Total
35285.71
13
F
12.29508
P-value
0.00433
F crit
4.747225
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?