# Analysis of variance (ANOVA)

### Analysis of variance (ANOVA)

#### Lessons

Recall:
$\sigma^2=\frac{(x_1-\mu)^2+(x_2-\mu)^2+ \cdots (x_n-\mu)^2}{n}$

$\overline{x}=\frac{x_1+x_2+ \cdots +x_n}{n}$

Degrees of Freedom
The degrees of freedom for a calculation is the number of variables that are free to vary. Think of calculating the mean of several variables.

$d.f.=n-1$

Sums of Squares:

The Sum of Squares Within Groups (SSW) is calculated by first finding the sum of squares for each individual group, and then adding them together.

The Sum of Squares Between Groups (SSB) is calculated by first finding the mean for all the groups (the Grand Mean) and then seeing what is the sum of squares from each individual group to the Grand Mean.

The Total Sum of Squares (TSS or SST) is just the sum of sqaures of the every single item from all the groups. Just imagine that all the groups come together to form one big group.

Total Sum of Squares = Sum of Squares Within + Sum of Squares Between (TSS=SSW+SSB)

Hypothesis Testing with F-Distribution
This method is just the test that the variances between groups do not vary.

$F=\frac{between\;group\;variability}{within\;group\;variability}$ $=\frac{\frac{SSB}{df}}{\frac{SSW}{df}}$

$F(df_{SSB},df_{SSW})$ is the critical value for an F-distribution
• Introduction
a)
What is Analysis of Variance (ANOVA)?

b)
Hypothesis Testing with F-Distribution

• 1.
Determining Degrees of Freedom
A test was done to study the reaction time of car drivers at different periods of the day
 Reaction Time of Drivers (seconds) Morning: Afternoon: Evening: Night: 1.32 0.25 2.34 1.54 0.71 1.48 1.75 1.98 2.27 1.24 0.64 1.76 0.57 0.89 0.98 1.15 $\overline{x}=1.2175$ $\overline{x}=0.965$ $\overline{x}=1.4275$ $\overline{x}=1.6075$ $\overline{x}=1.304375$
a)
What are the degrees of freedom for each time of day?

b)
How many degrees of freedom are there if we wanted to measure the Grand Mean (the mean of all the groups)?

• 2.
Determining the Sum of Squares
The following case study was done on what type of beverages office workers drink in the morning and their productivity.
 Juice/Milk Drinkers: Tea Drinkers: Coffee Drinkers: 3 5 8 5 5 6 3 6 7 1 4 7
a)
What is the Total Sum of Squares (TSS or SST) for this case study? Also what are the degrees of freedom for this group?

b)
What is the Sum of Squares Within Groups (SSW)? Also what is the number of degrees of freedom for all these groups?

c)
What is the Sum of Squares Between Groups (SSB)? Also what is the number of degrees of freedom for this calculation?

d)
Verify that: TSS=SSW+SSB for both the variation and the degrees of freedom.

• 3.
Hypothesis Testing with F-Distribution
The following case study was done on what type of beverages office workers drink in the morning and their productivity.
 Juice/Milk Drinkers: Tea Drinkers: Coffee Drinkers: 3 5 8 5 5 6 3 6 7 1 4 7

With a significance level of $\alpha$=0.05 test the claim that "what you drink in the morning does not affect how productive you are at work."

Use the fact that in the previous example we found that SSW=12 with 9 degrees of freedom. And we also had that SSB=32 with 2 degrees of freedom.