ANOVA Calculator | One-Way & Two-Way Analysis of Variance

ANOVA (ANALYSIS OF VARIANCE) ENGINE

One-Way ANOVA $\cdot$ Variance Breakdown $\cdot$ F-Test

Parametric Test

Enter numeric values for each group (comma or space-separated). You can dynamicly add or remove groups.

Group 1 Data:

Group 2 Data:

Group 3 Data:

Significance Level ($\alpha$): 0.01 (1%) 0.05 (5%) 0.10 (10%)

ANOVA Summary Table

Source	SS	df	MS	F-Stat	p-Value
Between (Treatment)	0.0000	0	0.0000	0.0000	0.0000
Within (Error)	0.0000	0	0.0000
Total	0.0000	0	-	-	-

Decision ($\alpha = 0.05$) -

Eta-Squared ($\eta^2$) Effect Size -

ANOVA (Analysis of Variance) Overview

Analysis of Variance (ANOVA) is a statistical methodology used to compare the means of three or more independent groups. Instead of conducting multiple two-sample t-tests (which elevates the overall Type I error rate), ANOVA partitioning measures total variation into components attributable to between-group variance and within-group variance.

1. One-Way vs. Two-Way ANOVA

Feature	One-Way ANOVA	Two-Way ANOVA
Independent Variables (Factors)	1 Factor (e.g., Treatment Type)	2 Factors (e.g., Treatment Type $\times$ Dosage)
Hypotheses Tested	Group means equality ($H_0: \mu_1 = \mu_2 = \dots = \mu_k$)	Main Effect A, Main Effect B, and Interaction Effect ($A \times B$)
F-Statistics Produced	1 Test Statistic ($F$)	3 Test Statistics ($F_A, F_B, F_{AB}$)

2. Formulas & Variance Partitioning

• Total Sum of Squares ($SS_{Total}$): Measures overall variation across all observations relative to the grand mean ($\bar{X}_{..}$). $$SS_{Total} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_{..})^2$$
• Between-Group Variation ($SS_{Between}$): Measures variation among group means ($\bar{X}_i$) relative to the grand mean. $$SS_{Between} = \sum_{i=1}^{k} n_i (\bar{X}_i - \bar{X}_{..})^2$$
• Within-Group Variation ($SS_{Within}$): Measures experimental error or noise within individual groups. $$SS_{Within} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2$$
• Test Statistic ($F$): Ratio of Mean Square Between to Mean Square Within. $$F = \frac{MS_{Between}}{MS_{Within}} = \frac{SS_{Between} / (k - 1)}{SS_{Within} / (N - k)}$$

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