This One-Way ANOVA Calculator helps you compare means between three or more groups to determine if there are statistically significant differences. For example, you could compare test scores between multiple teaching methods, or analyze plant growth under different conditions. The calculator performs comprehensive statistical analysis including assumption checks (normality, homogeneity of variance), effect size calculations, and generates publication-ready APA format reports. To learn about the data format required and test this calculator, click here to populate the sample data.
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One-way ANOVA
Definition
One-way ANOVA (Analysis of Variance) tests whether there are significant differences between the means of three or more independent groups. It extends the t-test to multiple groups while controlling the Type I error rate.
Formula
Key Components:
Between-groups sum of squares, where:
- = mean of group
- = grand mean
- = sample size of group
Within-groups sum of squares, where:
- = th observation in group
- = mean of group
- = sample size of group
Final Test Statistic:
Where:
- = within-groups sum of squares
- = number of groups
- = total sample size
Key Assumptions
Practical Example
Step 1: State the Data
| Group A | Group B | Group C |
|---|---|---|
| 8 | 6 | 9 |
| 9 | 5 | 10 |
| 7 | 8 | 10 |
| 10 | 7 | 8 |
Step 2: State Hypotheses
- (all means equal)
- at least one mean is different
Step 3: Calculate Summary Statistics
- Group A:
- Group B:
- Group C:
- Grand mean:
Step 4: Calculate Sums of Squares
Step 5: Calculate Mean Squares
Step 6: Calculate F-statistic
Step 7: Draw Conclusion
The critical value at is .
The calculated F-statistic () is greater than the critical value (), and the p-value () is less than our significance level (). We reject the null hypothesis in favor of the alternative. There is statistically significant evidence to conclude that not all group means are equal. Specifically, at least one group mean differs significantly from the others.
Effect Size
Eta-squared () measures the proportion of variance explained:
Guidelines:
- Small effect:
- Medium effect:
- Large effect:
For the example above, the effect size is:which indicates a large effect.
Code Examples
library(tidyverse)
group <- factor(c(rep("A", 4), rep("B", 4), rep("C", 4)))
values <- c(8, 9, 7, 10, 6, 5, 8, 7, 9, 10, 10, 8)
data <- tibble(group, values)
anova_result <- aov(values ~ group, data = data)
summary(anova_result)import numpy as np
from scipy import stats
group_A = [8, 9, 7, 10]
group_B = [6, 5, 8, 7]
group_C = [9, 10, 10, 8]
# Perform one-way ANOVA
f_stat, p_value = stats.f_oneway(group_A, group_B, group_C)
# Print results
print(f'F-statistic: {f_stat:.4f}')
print(f'p-value: {p_value:.4f}')Alternative Tests
Consider these alternatives when assumptions are violated:
- Kruskal-Wallis Test: Non-parametric alternative when normality is violated
- Welch's ANOVA: When variances are unequal
- Brown-Forsythe Test: Robust to violations of normality