A Conceptual View of ANOVA

Goals:

Conceptually, the goal of ANOVA is to

• determine the amount of variability in groups of data
• to determine where it comes from
• to see if the variability is greater between groups than within groups.

Visual Demonstration

We can demonstrate how this works visually with three hypothetical sets of data.

• In each set of data there are 3 groups sampled from 3 populations.
• The populations have means of 15, 30 and 45. We have colored the data to show the groups. We use
  1. Red for the group with population mean=15
  2. Green for the group with population mean=30
  3. Blue for the group with population mean=45
• The three sets of data differ according to the variances of the 3 populations:
  1. Dataset 1 sampled from populations with variances of 4, 4, and 4.
  2. Dataset 2 sampled from populations with variances of 4, 64, and 4.
  3. Dataset 3 sampled from populations with variances of 64, 64, and 64.
• With each visualization we present the corresponding F-Test value and its p value.

Example 1 Population Means = 15, 30, 45 Population variances = 4, 4, 4 F=854.24, p<.0001

Example 2 Population Means = 15, 30, 45 Population variances = 4, 64, 4 F=11.66, p<.0001

Example 3 Population Means = 15, 30, 45 Population variances = 64, 64, 64 F=1.42, p=.2440

Note that in these examples, the means of the three groups haven't varied, but the variances have. We see that when the groups are well separated, the F value is very significant. On the other hand, when they overlap a lot, the F is much less significant.

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