Statistics 303. Chapter 12 ANalysis Of VAriance. ANOVA: Comparing Several Means. The statistical methodology for comparing several means is called analysis of variance, or ANOVA. In this case one variable is categorical. This variable forms the groups to be compared.
ANalysis Of VAriance
1. Each of the I population or group distributions is normal. -check with a Normal Quantile Plot (or boxplot) of each group
2. These distributions have identical variances (standard deviations).
-check if largest sd is > 2 times smallest sd
3. Each of the I samples is a random sample.
4. Each of the I samples is selected independently of one another.
The null hypothesis (step 1) for comparing several means is
where I is the number of populations to be compared
The alternative hypothesis (step 2) is
This compares the variation between groups (group mean to group mean) to the variation within groups (individual values to group means).
This is what gives it the name “Analysis of Variance.”
P-valueANOVA: Comparing Several Means
where df1 = I – 1 (number of groups minus 1) and
df2 = N – I (total sample size minus number of groups).
Note: MSE is the pooled sample variance and SSG + SSE = SSTot
is the proportion of the total variation explained by the difference in means
Variation between groupsANOVA: Comparing Several Means
Stddev1 = 0.89316
Stddev2 = 0.86603
Stddev3 = 0.64507
Stddev4 = 0.85206
Sample size: N = 32
Variation Within Groups
Average Within Group Variation (MSE)
Average Between Group Variation (MSG)
Variation Between Groups
Step 1: The null hypothesis is
Step 2: The alternative hypothesis is
Step 3: The significance level is a = 0.05
MSGANOVA: Comparing Several Means
MSG and MSE are found in the ANOVA table when the analysis is run on the computer:
where df1 = I – 1 (number of groups minus 1) = 4 – 1 = 3 and df2 = N – I (total sample size minus I) = 32 – 4 = 28
An additional test will tell us which means are different from the others.