One-Way ANOVA Calculator
Calculate One-Way Analysis of Variance ($F = \frac{\text{MSB}}{\text{MSW}}$), degrees of freedom ($df_1, df_2$), $p$-value, and $\eta^2$ effect size.
Calculate One-Way ANOVA F-Test
Enter Sum of Squares Between (SSB), Sum of Squares Within (SSW), group count (k), and total N.
Calculation Results
Calculated using One-Way ANOVA F-ratio equations: F = \frac{\text{MSB}}{\text{MSW}} = \frac{\text{SSB} / (k - 1)}{\text{SSW} / (N - k)}
Quick Summary
The One-Way ANOVA Calculator evaluates the $F$-ratio statistic ($F = \frac{\text{MSB}}{\text{MSW}}$), degrees of freedom ($df_1 = k-1, df_2 = N-k$), and $\eta^2$ effect size.
Formula Explanation
\text{MSB} = \frac{\text{SSB}}{k - 1}, \quad \text{MSW} = \frac{\text{SSW}}{N - k}
F = \frac{\text{MSB}}{\text{MSW}}, \quad \eta^2 = \frac{\text{SSB}}{\text{SS}_{\text{total}}}How It Works
One-Way ANOVA partitions total dataset variance into between-group variance (treatment effect) and within-group variance (random error). The One-Way ANOVA Calculator evaluates Mean Square Between (MSB) and Mean Square Within (MSW) to compute the $F$-ratio.
Step-by-Step Worked Example
Practical Problem: An experiment compares $k = 3$ teaching methods across total $N = 30$ students. $\text{SSB} = 120$, $\text{SSW} = 280$. Calculate $F$-statistic and $\eta^2$ effect size.
- Step 1: Calculate Between-Groups degrees of freedom ($df_1$): $df_1 = 3 - 1 = \mathbf{2}$.
- Step 2: Calculate Within-Groups degrees of freedom ($df_2$): $df_2 = 30 - 3 = \mathbf{27}$.
- Step 3: Calculate Mean Square Between (MSB): $\text{MSB} = 120 / 2 = \mathbf{60.0000}$.
- Step 4: Calculate Mean Square Within (MSW): $\text{MSW} = 280 / 27 = \mathbf{10.3704}$.
- Step 5: Calculate F-ratio & Eta Squared ($\eta^2$): $F = 60.0 / 10.3704 = \mathbf{5.7857}$; $\eta^2 = 120 / 400 = \mathbf{0.3000\text{ (30\% variance explained)}}.$
Real-World Calculation Examples
Scenario 1: 3 Teaching Methods (k = 3, N = 30)
Parameters: SSB = 120, SSW = 280
Result: F = 5.7857 (Statistically significant at α = 0.05).
Scenario 2: Fertilizer Crop Yield Trial (4 Formulas)
Parameters: k = 4 groups, N = 40 plants
Result: F-test evaluation of yield differences.
Scenario 3: Multi-Arm Clinical Drug Trial (3 Dosages)
Parameters: Low, Medium, High dose vs Placebo
Result: One-Way ANOVA F-test.
Scenario 4: Customer Satisfaction Across 4 Store Locations
Parameters: k = 4 stores, N = 100 surveys
Result: Between-store variance assessment.
Key Benefits of Using This Calculator
Multi-Group Significance Testing
Tests group mean differences across 3+ groups without inflating Type I error rates.
Eta Squared ($\eta^2$) Effect Size
Computes proportion of total variance explained ($\text{SSB} / \text{SST}$).
Mean Squares Breakdown
Outputs MSB and MSW components.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is One-Way ANOVA?
One-Way Analysis of Variance (ANOVA) tests whether means across 3 or more independent groups differ significantly.
Why not run multiple 2-sample t-tests instead of ANOVA?
Running multiple t-tests inflates familywise Type I error rate ($\alpha_{\text{overall}} = 1 - (1 - \alpha)^m$); ANOVA controls overall error at $\alpha = 0.05$.
What is post-hoc testing (Tukey's HSD)?
When ANOVA yields a significant $F$-test, post-hoc tests (Tukey HSD, Bonferroni) identify WHICH specific pair of groups differs.
How do I calculate ANOVA in Excel?
Use Excel's Data Analysis Toolpak: Anova: Single Factor.
What is null hypothesis ($H_0$) in One-Way ANOVA?
$H_0: \mu_1 = \mu_2 = \dots = \mu_k$ (all group means are equal).
What non-parametric test replaces One-Way ANOVA?
The Kruskal-Wallis test replaces One-Way ANOVA for non-normal distributions.
What is Eta Squared ($\eta^2$)?
$\eta^2 = \frac{\text{SSB}}{\text{SST}}$ measures the proportion of total variance explained by group membership.
What assumptions are required for ANOVA?
1. Normally distributed residuals. 2. Homogeneity of variances (homoscedasticity). 3. Independent observations.
What is F-ratio?
The $F$-ratio is $\frac{\text{MSB}}{\text{MSW}}$, comparing treatment variance to error variance.
Who developed ANOVA?
British statistician Sir Ronald Fisher developed ANOVA in the 1920s.