Analysis of variance — Edexcel A-Level Statistics
Test yourself on Analysis of variance with PEARSON EDEXCEL A-Level practice questions.
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Analysis of variance explained
One-way ANOVA compares the means of three or more treatments in a completely randomised design, where each unit is assigned at random to exactly one treatment.
Read the full explanation
The underlying model is Yᵢⱼ = μ + τᵢ + εᵢⱼ: the observation equals an overall mean μ, an additive treatment effect τᵢ, and an experimental error εᵢⱼ. Additive effects mean the treatment shifts the response by a constant amount. Errors are independent and distributed as N(0, σ²). Partition the total variation into between-treatment and within-treatment sums of squares, form mean squares by dividing by their degrees of freedom, and compare the F ratio with the F distribution. For example, testing four fertilisers on 20 plots gives 3 and 16 degrees of freedom.
Your focus
- Set up the one-way ANOVA model for a completely randomised design and state its assumptions.
- Carry out the sum of squares partition and compute the F statistic correctly.
- Interpret the outcome of the F test in the context of the experiment.
Analysis of variance exam tips
Marking Points
- State the model Yᵢⱼ = μ + τᵢ + εᵢⱼ and identify μ as the overall mean, τᵢ as the additive treatment effect and εᵢⱼ as the experimental error.
- State the assumptions: errors are independent and distributed as N(0, σ²), with constant variance across treatments.
- Partition the total sum of squares into between-treatment and within-treatment (residual) sums of squares.
- Compute the treatment and residual mean squares using their degrees of freedom, then form the F ratio of treatment mean square to residual mean square.
- Compare the calculated F statistic with the critical value from the F distribution at the chosen significance level and state the conclusion in context.
- Recognise that a completely randomised design assigns each experimental unit at random to exactly one treatment.
Examiner Tips
- 💡Write out the model equation and the error assumption before doing any arithmetic, so the structure of the test is clear.
- 💡Lay out the ANOVA table with sources, sums of squares, degrees of freedom, mean squares and the F ratio to avoid losing track of the partition.
- 💡Interpret the F test in the context of the experiment, naming the treatments and the response variable rather than only quoting a p-value.
Common Mistakes
- Treating the additive model as including an interaction term; the one-way model has only an overall mean, a treatment effect and an error, so interaction is not part of it.
- Assuming the errors are normally distributed with mean equal to the treatment mean; the errors are distributed as N(0, σ²), centred on zero.
- Dividing each sum of squares by the number of observations rather than by the correct degrees of freedom when forming mean squares.
- Concluding that a non-significant F proves all treatment means are equal; it only means there is insufficient evidence of a difference.