One-Way ANOVA

Compare means across three or more independent groups simultaneously.

Test Overview

What it tests:

One-way ANOVA (Analysis of Variance) is a fundamental statistical procedure that simultaneously compares the means of three or more independent groups. Rather than performing multiple t-tests (which increases Type I error), ANOVA provides a single, comprehensive test to determine if any group means differ significantly from the others while controlling overall error rate.

Statistical Hypothesis:

Hypotheses

Applications:

• • Comparing 3+ treatment groups
• • Multi-group experimental designs
• • Quality control across shifts/batches
• • Treatment effectiveness studies
• • Performance analysis across departments
• • Manufacturing process comparisons

Critical Assumptions:

• • Normality within each group
• • Equal variances (homoscedasticity)
• • Independent observations
• • Random sampling from populations
• • Adequate sample sizes per group

Statistical Foundation

ANOVA F-Test Statistic:

ANOVA partitions total variability into between-group and within-group components, comparing their ratio to determine if groups differ significantly.

ANOVA F-Statistic

MSB

= mean square between groups

MSW

= mean square within groups

Variance Decomposition:

xij

= j-th observation in group i

x..

= grand mean

Degrees of Freedom:

= number of groups

= total sample size

ANOVA Table Structure:

Between Groups

SSB

k-1

SSB/(k-1)

MSB/MSW

Within Groups

SSW

N-k

SSW/(N-k)

Total

SST

N-1

Reading the One-Way ANOVA Group Plot

Post-Hoc Analysis & Effect Size

When ANOVA is Significant (p < 0.05):

A significant F-test indicates that at least one group mean differs from the others, but doesn't specify which groups differ. Post-hoc tests identify specific group differences:

Conservative Tests:

• • Tukey HSD: Controls family-wise error rate
• • Scheffe: Most conservative, any contrasts
• • Bonferroni: Conservative multiple comparisons

Specialized Tests:

• • Dunnett: Compare all to control group
• • Fisher LSD: Less conservative, more power
• • Games-Howell: Unequal variances

Effect Size Interpretation:

Eta-Squared (η²):

• Small: η² = 0.01 (1% of variance)

• Medium: η² = 0.06 (6% of variance)

• Large: η² = 0.14 (14% of variance)

• Formula: η² = SSB / SST

Omega-Squared (ω²):

• Less biased estimate than η²

• Preferred for population inference

• Formula: ω² = (SSB - (k-1)MSW) / (SST + MSW)

• Always smaller than η²

Decision Guidelines:

• Significant F-test: At least one group differs - proceed to post-hoc tests

• Non-significant F-test: No evidence of group differences - avoid post-hoc testing

• Effect Size: Consider practical significance even with statistical significance

• Assumption Violations: Use Welch ANOVA or Kruskal-Wallis for robust alternatives

Implementation Guide

Example Data Layout for One-Way ANOVA

Separate columns for each group. Use as many columns as you have groups to compare (3 or more). Sample Size Recommendation: Minimum 5-10 observations per group for reliable ANOVA results, with equal group sizes preferred for optimal F-test performance.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.