One-Way ANOVA

Created:September 8, 2024

Last Updated:September 10, 2025

This calculator helps you compare the means of three or more groups. Unlike t-tests that can only compare two groups, One-Way ANOVA efficiently analyzes multiple groups while controlling for the risk of false discoveries that comes from multiple comparisons.

What You'll Get:

Complete ANOVA Table: F-statistics, p-values, and effect sizes
Assumption Testing: Normality and homogeneity of variance checks
Visual Analysis: Group comparison charts and distribution plots
Effect Size Interpretation: Know if differences are practically meaningful
Next Steps Guidance: Recommendations for post-hoc tests when needed
APA-Ready Report: Publication-quality tables and results you can copy directly

💡 Pro Tip: If you only have two groups to compare, use ourTwo-Sample T-Test Calculatorinstead for more appropriate analysis.

Ready to analyze your groups? to see how it works, or upload your own data to discover if your groups truly differ.

Calculator

1. Load Your Data

2. Select Columns & Options

Select group column:

Select value column:

Significance Level:

Exclude Outliers

Related Calculators

Independent T-Test Calculator

Tukey's HSD Calculator

Two-Way ANOVA Calculator

Kruskal-Wallis Test Calculator

Learn More

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.

Why Do We Need ANOVA?

When comparing multiple groups, you might be tempted to perform multiple t-tests between all possible pairs of groups. However, this approach leads to a serious problem: an increased risk of Type I errors (false positives) .

P(\text{at least one Type I error}) = 1 - (1 - \alpha)^k

where k is the number of tests and α is the significance level. Probability of Type I Error with Multiple Comparisons

For example:

With 3 groups (3 pairwise tests): 14.3% chance
With 4 groups (6 pairwise tests): 26.5% chance
With 5 groups (10 pairwise tests): 40.1% chance

These probabilities assume a significance level (α) of 0.05 for each test.

How Does ANOVA Work?

To compare means, ANOVA cleverly compares variances. If group means are truly different, then the variation between groups should be much larger than the variation within groups.

Between-group variance: How much do group means differ from the overall mean?

Within-group variance: How much do individual observations vary within each group?

F = \frac{\text{Between-group variance}}{\text{Within-group variance}}

Key insight: If group means are the same, F ≈ 1. If group means differ significantly, F ≫ 1. The p-value tells us how likely we'd see an F this large if there were actually no group differences.

Interactive ANOVA Explorer

Configure Groups

Group 1

Mean8

Standard Deviation0.5

Group 2

Mean11

Standard Deviation0.5

Group 1
Group 2

Understanding ANOVA Scenarios

Large mean difference (8 vs 11) + Small spread (SD=0.5) = Strong evidence of group differences

When group means are far apart and individual measurements have minimal variation, differences become clearly distinguishable.

Small mean difference (8 vs 9) + Small spread (SD=0.5) = Moderate evidence of group differences

When group means are relatively close but individual measurements show little variation, meaningful differences may still be detectable.

Large mean difference (8 vs 11) + Large spread (SD=2) = Weak evidence of group differences

When individual measurements vary widely within groups, even substantial differences between group means can be difficult to detect reliably.

Note: This is a visual demonstration only.

The "evidence strength" indications are simplified visual examples to illustrate ANOVA concepts. No actual statistical test is being performed here. In practice, an ANOVA test would calculate specific statistics (F-ratio, p-value) to formally evaluate the evidence for group differences.

Current scenario: Strong evidence of group differences

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

ANOVA Table Structure

Source of Variation	Sum of Squares (SS)	Degrees of Freedom (df)	Mean Square (MS)	F-statistic
Between Groups	$SS_{between}$	$k - 1$	$\frac{SS_{between}}{k-1}$	$\frac{MS_{between}}{MS_{within}}$
Within Groups	$SS_{within}$	$N - k$	$\frac{SS_{within}}{N-k}$	$\frac{MS_{between}}{MS_{within}}$
Total	$SS_{total}$	$N - 1$	—	—

Note: $SS_{total} = SS_{between} + SS_{within}$

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: Complete ANOVA Table

Source	SS	df	MS	F	p-value
Between Groups	16.17	2	8.08	5.71	0.025
Within Groups	12.75	9	1.42	5.71	0.025
Total	28.92	11	—	—	—

Note: df between = k - 1 = 3 - 1 = 2; df within = N - k = 12 - 3 = 9

Step 8: 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)

data <- tibble(group, values)
anova_result <- aov(values ~ group, data = data)
summary(anova_result)

Python

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

Verification

One-Way ANOVA

Created:September 8, 2024

Last Updated:September 10, 2025

What You'll Get:

Complete ANOVA Table: F-statistics, p-values, and effect sizes
Assumption Testing: Normality and homogeneity of variance checks
Visual Analysis: Group comparison charts and distribution plots
Effect Size Interpretation: Know if differences are practically meaningful
Next Steps Guidance: Recommendations for post-hoc tests when needed
APA-Ready Report: Publication-quality tables and results you can copy directly

💡 Pro Tip: If you only have two groups to compare, use ourTwo-Sample T-Test Calculatorinstead for more appropriate analysis.

Ready to analyze your groups? to see how it works, or upload your own data to discover if your groups truly differ.

Calculator

1. Load Your Data

2. Select Columns & Options

Select group column:

Select value column:

Significance Level:

Exclude Outliers

Related Calculators

Independent T-Test Calculator

Tukey's HSD Calculator

Two-Way ANOVA Calculator

Kruskal-Wallis Test Calculator

Learn More

One-way ANOVA

Definition

Why Do We Need ANOVA?

P(\text{at least one Type I error}) = 1 - (1 - \alpha)^k

where k is the number of tests and α is the significance level. Probability of Type I Error with Multiple Comparisons

For example:

With 3 groups (3 pairwise tests): 14.3% chance
With 4 groups (6 pairwise tests): 26.5% chance
With 5 groups (10 pairwise tests): 40.1% chance

These probabilities assume a significance level (α) of 0.05 for each test.

How Does ANOVA Work?

To compare means, ANOVA cleverly compares variances. If group means are truly different, then the variation between groups should be much larger than the variation within groups.

Between-group variance: How much do group means differ from the overall mean?

Within-group variance: How much do individual observations vary within each group?

F = \frac{\text{Between-group variance}}{\text{Within-group variance}}

Interactive ANOVA Explorer

Configure Groups

Group 1

Mean8

Standard Deviation0.5

Group 2

Mean11

Standard Deviation0.5

Group 1
Group 2

Understanding ANOVA Scenarios

Large mean difference (8 vs 11) + Small spread (SD=0.5) = Strong evidence of group differences

When group means are far apart and individual measurements have minimal variation, differences become clearly distinguishable.

Small mean difference (8 vs 9) + Small spread (SD=0.5) = Moderate evidence of group differences

When group means are relatively close but individual measurements show little variation, meaningful differences may still be detectable.

Large mean difference (8 vs 11) + Large spread (SD=2) = Weak evidence of group differences

When individual measurements vary widely within groups, even substantial differences between group means can be difficult to detect reliably.

Note: This is a visual demonstration only.

Current scenario: Strong evidence of group differences

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

ANOVA Table Structure

Source of Variation	Sum of Squares (SS)	Degrees of Freedom (df)	Mean Square (MS)	F-statistic
Between Groups	$SS_{between}$	$k - 1$	$\frac{SS_{between}}{k-1}$	$\frac{MS_{between}}{MS_{within}}$
Within Groups	$SS_{within}$	$N - k$	$\frac{SS_{within}}{N-k}$	$\frac{MS_{between}}{MS_{within}}$
Total	$SS_{total}$	$N - 1$	—	—

Note: $SS_{total} = SS_{between} + SS_{within}$

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: Complete ANOVA Table

Source	SS	df	MS	F	p-value
Between Groups	16.17	2	8.08	5.71	0.025
Within Groups	12.75	9	1.42	5.71	0.025
Total	28.92	11	—	—	—

Note: df between = k - 1 = 3 - 1 = 2; df within = N - k = 12 - 3 = 9

Step 8: Draw Conclusion

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

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)

data <- tibble(group, values)
anova_result <- aov(values ~ group, data = data)
summary(anova_result)

Python

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

Verification

Source of Variation

Sum of Squares (SS)

Degrees of Freedom (df)

Mean Square (MS)

F-statistic

Between Groups

SS_{between}

k - 1

\frac{SS_{between}}{k-1}

\frac{MS_{between}}{MS_{within}}

Within Groups

SS_{within}

N - k

\frac{SS_{within}}{N-k}

Total

SS_{total}

N - 1

—

Group A

Group B

Group C

Source

p-value

Between Groups

16.17

8.08

5.71

0.025

Within Groups

12.75

1.42

Total

28.92

—

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}')

Source of Variation

Sum of Squares (SS)

Degrees of Freedom (df)

Mean Square (MS)

F-statistic

Between Groups

SS_{between}

k - 1

\frac{SS_{between}}{k-1}

\frac{MS_{between}}{MS_{within}}

Within Groups

SS_{within}

N - k

\frac{SS_{within}}{N-k}

Total

SS_{total}

N - 1

—

Group A

Group B

Group C

Source

p-value

Between Groups

16.17

8.08

5.71

0.025

Within Groups

12.75

1.42

Total

28.92

—

One-Way ANOVA

What You'll Get:

Calculator

1. Load Your Data

2. Select Columns & Options

Related Calculators

Independent T-Test Calculator

Tukey's HSD Calculator

Two-Way ANOVA Calculator

Kruskal-Wallis Test Calculator

Learn More

One-way ANOVA

Definition

Why Do We Need ANOVA?

How Does ANOVA Work?

Interactive ANOVA Explorer

Configure Groups

Group 1

Group 2

Understanding ANOVA Scenarios

Formula

ANOVA Table Structure

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: Complete ANOVA Table

Step 8: Draw Conclusion

Effect Size

Code Examples

Alternative Tests

Verification

View Verification Details

One-Way ANOVA

What You'll Get:

Calculator

1. Load Your Data

2. Select Columns & Options

Related Calculators

Independent T-Test Calculator

Tukey's HSD Calculator

Two-Way ANOVA Calculator

Kruskal-Wallis Test Calculator

Learn More

One-way ANOVA

Definition

Why Do We Need ANOVA?

How Does ANOVA Work?

Interactive ANOVA Explorer

Configure Groups

Group 1

Group 2

Understanding ANOVA Scenarios

Formula

ANOVA Table Structure

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: Complete ANOVA Table

Step 8: Draw Conclusion

Effect Size

Code Examples

Alternative Tests

Verification

View Verification Details

One-Way ANOVA

What You'll Get:

Calculator

1. Load Your Data

2. Select Columns & Options

Related Calculators