ANOVA & Experimental Design Lab

Compare means across multiple groups using one-way Analysis of Variance. Enter data for 2 to 6 groups, compute the F-statistic and p-value, measure effect size with eta-squared, and visualize distributions with side-by-side box plots.

Guided Experiment: Comparing Group Means

Hypothesis

Setup

Run Experiment

Analyze

Conclude

If three fertilizers have different effects on plant height, the between-group variation should be large relative to within-group variation. What F-statistic and p-value would indicate a real difference?

Write your hypothesis in the Lab Report panel, then click Next.

Controls

Significance Level (α)

Presets

Groups (3)

n = 5

ANOVA Results

Source	SS	df	MS	F	p-value
Between	4.4680	2	2.2340	52.7717	< 0.0001
Within	0.5080	12	0.0423	—	—
Total	4.9760	14	—	—	—

Reject H₀ at α = 0.05

There is statistically significant evidence that at least one group mean differs from the others.

F-statistic

52.7717

p-value

< 0.0001

Grand Mean

4.5600

Effect Size (η²)

0.8979(large)

Key Formulas

F = \frac{MSB}{MSW} = \frac{2.23}{0.04} = 52.77

\eta^2 = \frac{SSB}{SST} = \frac{4.47}{4.98} = 0.8979

Group Summaries

Group	n	Mean	Std Dev
Fertilizer A	5	4.38	0.28
Fertilizer B	5	5.30	0.16
Fertilizer C	5	4.00	0.16

Visualization

Box Plots by Group

Group means connectedGrand meanGroup mean

F-Distribution (df₁ = 2, df₂ = 12)

F-distribution p-value area

Data Table

(0 rows)

#	Trial	Groups	F-statistic	df_B	df_W	p-value	η²	Conclusion

Hypothesis

0 / 500

Observations

0 / 500

Conclusions

0 / 500

Reference Guide

The ANOVA Table

ANOVA partitions the total variation in data into between-group and within-group components. The ANOVA table summarizes these sources of variation.

SST = SSB + SSW

SSB = \sum n_i(\bar{x}_i - \bar{x})^2, \quad SSW = \sum\sum(x_{ij} - \bar{x}_i)^2

Between-group variation (SSB) measures how far group means are from the grand mean. Within-group variation (SSW) measures spread within each group.

F-Distribution and Hypothesis Test

The F-statistic is the ratio of between-group to within-group mean squares. Under the null hypothesis (all group means are equal), F follows an F-distribution.

F = \frac{MSB}{MSW} = \frac{SSB / (k-1)}{SSW / (N-k)}

A large F-statistic (small p-value) provides evidence that at least one group mean differs from the others. The null hypothesis states all group means are equal.

Effect Size (Eta-Squared)

Statistical significance alone does not tell you how important the effect is. Eta-squared measures the proportion of total variation explained by group membership.

\eta^2 = \frac{SSB}{SST}

Guidelines for interpreting eta-squared: small (0.01), medium (0.06), large (0.14 or greater). A value of 0.14 means 14% of the total variation is due to group differences.

Conditions for ANOVA

One-way ANOVA assumes three conditions. Violations can reduce the reliability of the test.

Independence — Observations within and across groups are independent of each other.
Normality — The data in each group are approximately normally distributed (less critical with larger samples due to the Central Limit Theorem).
Equal variances — The population variances are roughly equal across groups (homogeneity of variance). A common rule of thumb is that the largest variance should be no more than 4 times the smallest.