31/12/2025
📈What is One-Way ANOVA📈
One-Way Analysis of Variance (ANOVA) is a statistical method used to compare the means of three or more independent groups to see whether at least one group mean is significantly different from the others.
One-way → only one factor (independent variable)
ANOVA → compares variation (variance), not means directly
Example
Effect of 3 fertilizers on plant height
Student marks under 3 teaching methods
Yield under different growing media
🔴 Why not use multiple t-tests?
If you compare many groups using multiple t-tests:
Type-I error (false positives) increases
Results become unreliable
👉 ANOVA controls this error by testing all groups at once.
🔴 Basic Idea Behind One-Way ANOVA
ANOVA checks whether:
> Variation between group means is large compared to
Variation within the groups
If group means differ more than expected by random variation, the result is statistically significant.
🔴 Components of a One-Way ANOVA Table
Let’s explain each part using your table.
1. Between Groups
Measures variation caused by differences among group means
Large value → groups may be different
2. Within Groups
Measures variation inside each group
Represents random error or natural variability
3. Total
Total variation in the data
Between Groups + Within Groups
Explanation of Each Column
Sum of Squares (SS)
Quantifies total variability
Larger SS → more variation
From your table:
Between Groups SS = 96.976
Within Groups SS = 6757.691
Total SS = 6854.667
Degrees of Freedom (df)
Between Groups df = k − 1
Here df = 2 → number of groups = 3
Within Groups df = N − k
Here df = 433
Total df = N − 1
Here df = 435
Mean Square (MS)
Calculated as:
> Mean Square = Sum of Squares ÷ df
Between Groups MS = 48.488
Within Groups MS = 15.607
F-Value
Calculated as:
> F = MS Between ÷ MS Within
From your table:
F = 3.107
Interpretation:
Higher F → stronger evidence that group means differ
🔴 Significance Value (Sig. or p-value)
p = 0.046
Common significance level (α) = 0.05
Since:
> 0.046 < 0.05
✅ The result is statistically significant
🔴 Final Interpretation of Your ANOVA Table
There is a statistically significant difference among the group means for Total Mastery
(F(2, 433) = 3.107, p = 0.046).
This means:
At least one group mean is different
ANOVA does not tell which groups differ
🔴 What to Do After One-Way ANOVA?
Because ANOVA only tells that a difference exists, you should perform a post-hoc test, such as:
Tukey’s HSD
Bonferroni
L*D
These tests identify which specific groups differ.
🔴 Assumptions of One-Way ANOVA
1. Independence of observations
2. Normal distribution of residuals
3. Homogeneity of variances (equal variances)
📌If assumptions are violated:
Use Kruskal–Wallis test (non-parametric alternative)
