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ANOVA: Comparing Differences Between Multiple Groups
Analysis of Variance (ANOVA): Comparing More Than Two Group Means
Previously, we used the t-test to compare the means of two groups. But what if we want to compare the means of three or more groups? Running multiple t-tests increases the risk of error. This is where Analysis of Variance (ANOVA) comes in.
1. The Core Principle: Judging ‘Means’ Through ‘Variance’
Although the name is ‘Analysis of Variance’, its purpose is to examine the ‘difference in means’. It splits the total variation in the data into two types:
- Between-group Variance: How far apart are the groups from each other? (The difference we want to identify)
- Within-group Variance: How much do data points scatter within the same group? (Random error)
F-statistic = Between-group Variance / Within-group Variance
If this F-value is large enough, we conclude that “there is a significant difference in means between the groups.”
2. Example ANOVA Table
Actual analysis results are summarized in a table format like the one below:
Analysis of Productivity Differences by Line Placement (ANOVA Table)
| Source of Variation | Sum of Squares (SS) | Degrees of Freedom (df) | F-statistic | P-value |
|---|---|---|---|---|
| Between Groups (Treatment) | 1,250 | 2 | 12.5 | 0.001 |
| Within Groups (Error) | 8,400 | 84 | - | - |
| Total | 9,650 | 86 | - | - |
3. What Happens After ANOVA? (Post-hoc Tests)
An ANOVA result showing a “significant difference” doesn’t tell us which specific groups are different from each other. To find this out, we perform an additional procedure called a Post-hoc Test.
💡 Professor’s Tip
ANOVA is the foundation of ‘Design of Experiments’. It’s like the first map you use when comparing the effectiveness of three different marketing ads or testing the durability of various factory parts.