One-Way ANOVA

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.

Calculator

1. Load Your Data

2. Select Columns & Options

Select group column:

Select value column:

Significance Level:

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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:

SS_{between} = \sum_{i=1}^{k} n_i(\bar{x}_i - \bar{x}_g)^2

Between-groups sum of squares, where:

$\bar{x}_i$ = mean of group $i$
$\bar{x}_g$ = grand mean
$n_i$ = sample size of group $i$

SS_{within} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (x_{ij} - \bar{x}_i)^2

Within-groups sum of squares, where:

$x_{ij}$ = $j$ th observation in group $i$
$\bar{x}_i$ = mean of group $i$
$n_i$ = sample size of group $i$

Final Test Statistic:

F = \frac{MS_{between}}{MS_{within}} = \frac{SS_{between}/(k-1)}{SS_{within}/(N-k)}

Where:

$SS_{within}$ = within-groups sum of squares
$k$ = number of groups
$N$ = total sample size

Key Assumptions

Independence: Observations must be independent

Normality: Data within each group should be normally distributed

Homogeneity of Variance: Groups should have equal variances

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

$H_0: \mu_1 = \mu_2 = \mu_3$ (all means equal)
$H_a:$ at least one mean is different
$\alpha = 0.05$

Step 3: Calculate Summary Statistics

Group A: $\bar{x}_A = 8.50, s_A = 1.29$
Group B: $\bar{x}_B = 6.50, s_B = 1.29$
Group C: $\bar{x}_C = 9.25, s_C = 0.96$
Grand mean: $\bar{x} = 8.08$

Step 4: Calculate Sums of Squares

SS_{between} = 16.17

SS_{within} = 12.75

SS_{total} = 28.92

Step 5: Calculate Mean Squares

MS_{between} = \frac{SS_{between}}{k-1} = 8.08

MS_{within} = \frac{SS_{within}}{N-k} = 1.42

Step 6: Calculate F-statistic

$F = \frac{MS_{between}}{MS_{within}} = 5.71$

Step 7: Draw Conclusion

The critical value $F_{(2,9)}$ at $\alpha = 0.05$ is $4.26$ .

The calculated F-statistic ( $F = 5.71$ ) is greater than the critical value ( $4.26$ ), and the p-value ( $p = 0.025$ ) is less than our significance level ( $\\alpha = 0.05$ ). 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 ( $\eta^2$ ) measures the proportion of variance explained:

\eta^2 = \frac{SS_{between}}{SS_{total}}

Guidelines:

Small effect: $\eta^2 \approx 0.01$
Medium effect: $\eta^2 \approx 0.06$
Large effect: $\eta^2 \approx 0.14$

For the example above, the effect size is: $\eta^2 = \frac{16.17}{28.92} = 0.56$ 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)


# calculate SS between manually
mean_values <- tapply(values, group, mean)
grand_mean <- mean(values)
ss_between <- sum(table(group) * (mean_values - grand_mean)^2)
print(str_glue("SS between: {ss_between}"))

# calculate everything manually
ss_within <- sum((values - ave(values, group, FUN = mean))^2)
print(str_glue("SS within: {ss_within}"))

ss_total = sum((values - grand_mean)^2)
print(str_glue("SS total: {ss_total}"))

ms_between = ss_between / (length(unique(group)) - 1)
print(str_glue("MS between: {ms_between}"))

ms_within = ss_within / (length(values) - length(unique(group)))
print(str_glue("MS within: {ms_within}"))

f = ms_between / ms_within
print(str_glue("F: {f}"))

p = 1 - pf(f, length(unique(group)) - 1, length(values) - length(unique(group)))
print(str_glue("p value: {p}"))


# Perform One-Way ANOVA using aov()
data <- tibble(group, values)
anova_result <- aov(values ~ group, data = data)
summary(anova_result)

Python

import numpy as np
from scipy import stats

# Data
group_A = [8, 9, 7, 10]
group_B = [6, 5, 8, 7]
group_C = [9, 10, 10, 8]


# calculate ss for each group
ss_A = np.sum((group_A - np.mean(group_A)) ** 2)
ss_B = np.sum((group_B - np.mean(group_B)) ** 2)
ss_C = np.sum((group_C - np.mean(group_C)) ** 2)

ss_whithin = ss_A + ss_B + ss_C
print(f'SS within: {ss_whithin:.4f}')

# calculate ss between
grand_mean = np.mean([np.mean(group_A), np.mean(group_B), np.mean(group_C)])
ss_between = len(group_A) * (np.mean(group_A) - grand_mean) ** 2 + len(group_B) * (np.mean(group_B) - grand_mean) ** 2 + len(group_C) * (np.mean(group_C) - grand_mean) ** 2
print(f'SS between: {ss_between:.4f}')

print(f'SS total: {ss_whithin + ss_between:.4f}')
ms_within = ss_whithin / (len(group_A) + len(group_B) + len(group_C) - 3)
print(f'MS within: {ss_whithin:.4f}')
ms_between = ss_between / 2
print(f'MS between: {ss_between:.4f}')

f = ms_between / ms_within
print(f'F-statistic: {f:.4f}')



# Perform one-way ANOVA
print('Perform one-way ANOVA using scipy.stats')
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}')

One-Way ANOVA

Calculator

1. Load Your Data

2. Select Columns & Options

Learn More

One-way ANOVA

Definition

Formula

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: Calculate Summary Statistics

Step 4: Calculate Sums of Squares

Step 5: Calculate Mean Squares

Step 6: Calculate F-statistic

Step 7: Draw Conclusion

Effect Size

Code Examples

Alternative Tests

Verification

Related Calculators

Independent T-Test Calculator

Paired T-Test Calculator

Two-Way ANOVA Calculator

Kruskal-Wallis Test Calculator

Help us improve

One-Way ANOVA

Calculator

1. Load Your Data

2. Select Columns & Options

Learn More

One-way ANOVA

Definition

Formula

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: Calculate Summary Statistics

Step 4: Calculate Sums of Squares

Step 5: Calculate Mean Squares

Step 6: Calculate F-statistic

Step 7: Draw Conclusion

Effect Size

Code Examples

Alternative Tests

Verification

View Verification Details

Related Calculators

Independent T-Test Calculator

Paired T-Test Calculator

Two-Way ANOVA Calculator

Kruskal-Wallis Test Calculator

Help us improve