Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a statistical technique used to compare the means of three or more groups. It tests whether there are any statistically significant differences between the means of independent (unrelated) groups.

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1. When to Use ANOVA

• Comparing mean blood pressure across three treatment groups

• Testing whether multiple teaching methods affect exam scores differently

• Analyzing differences in plant growth under different fertilizer types

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2. Types of ANOVA

a. One-way ANOVA

Used when comparing means across groups based on one independent variable (factor).

• Example: Comparing average test scores across four different schools

b. Two-way ANOVA

Used when examining the effect of two independent variables and their interaction.

• Example: Testing the effects of diet (A/B) and exercise (yes/no) on weight loss

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3. ANOVA Terminology

• Between-group variance: Variability due to differences between group means

• Within-group variance: Variability due to differences within each group

• F-statistic: Ratio of between-group variance to within-group variance

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4. Assumptions of ANOVA

• Independence of observations

• Normally distributed residuals within each group

• Homogeneity of variances (equal variances across groups)

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5. Interpreting ANOVA

• Null hypothesis (H0): All group means are equal

• Alternative hypothesis (Ha): At least one group mean is different

• If the p-value from the F-test < 0.05, reject H0

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6. Post-hoc Tests

If ANOVA is significant, use post-hoc tests to identify which group means differ:

• Tukey’s HSD

• Bonferroni correction

• Scheffé’s method

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7. Example (R Code)

# One-way ANOVA
result <- aov(score ~ group, data = mydata)
summary(result)

# Post-hoc test
TukeyHSD(result)

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8. Summary

ANOVA is a powerful method for testing differences among group means. It relies on specific assumptions and, when significant, should be followed by post-hoc comparisons to identify which groups differ.