Here's a detailed explanation of One-Way ANOVA (Analysis of Variance), including its meaning, assumptions, procedure, example, and interpretation, presented in paragraph form for academic or research use:
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One-Way ANOVA (Analysis of Variance)
Meaning and Purpose
One-Way ANOVA (Analysis of Variance) is a parametric statistical test used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups. Unlike a t-test, which compares only two groups, ANOVA can test multiple group means simultaneously, thus reducing the chance of Type I error (false positives).
For example, a researcher may want to compare the effectiveness of three different diets on weight loss. One-Way ANOVA can test whether the average weight loss differs significantly across the three diet groups.
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Key Assumptions of One-Way ANOVA
To apply One-Way ANOVA, the following assumptions must be met:
1. Independence: Observations must be independent of each other.
2. Normality: The dependent variable should be approximately normally distributed within each group.
3. Homogeneity of variances: The variance among the groups should be approximately equal (tested using Levene’s Test).
If these assumptions are violated, a non-parametric alternative like Kruskal-Wallis test may be used.
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Hypotheses in One-Way ANOVA
Null Hypothesis (H₀): All group means are equal.
Alternative Hypothesis (H₁): At least one group mean is different.
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ANOVA Table Structure
Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) = SS/df F-Ratio = MS Between / MS Within
Between Groups SS<sub>Between</sub> k - 1 MS<sub>Between</sub> F
Within Groups SS<sub>Within</sub> N - k MS<sub>Within</sub>
Total SS<sub>Total</sub> N - 1
Where:
= number of groups
= total number of observations
= sum of squares (variation)
= ratio of variance between groups to variance within groups
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Procedure to Perform One-Way ANOVA
1. State the Hypotheses: Set up null and alternative hypotheses.
2. Calculate Group Means and Variance: Compute the overall mean, group means, and variances.
3. Compute the F-Ratio: Using the ANOVA table, calculate the F statistic.
4. Compare F with Critical Value: Use F-distribution table at a given significance level (e.g., α = 0.05).
5. Decision: If calculated F > critical F, reject the null hypothesis.
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Example
Scenario: A professor wants to test whether students from three different departments (A, B, C) have different average test scores.
Department Scores
A 85, 90, 78
B 88, 92, 85
C 72, 75, 70
Using One-Way ANOVA, we calculate the group means and variances, compute the F-ratio, and check if the variation between the groups is statistically significant.
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Interpretation of Results
If the F-ratio is statistically significant (p-value < 0.05), it indicates that at least one group mean is different from the others.
To determine which group(s) differ, a post-hoc test like Tukey’s HSD can be applied.
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Applications of One-Way ANOVA
Comparing effectiveness of different treatments
Evaluating customer satisfaction across regions
Testing employee performance across departments
Studying mean productivity in different machines or techniques
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Limitations
Assumes normality and equal variances, which may not always hold.
Only tells whether a difference exists, not which group is different.
Cannot handle more than one independent variable (use Two-Way ANOVA in that case).
