Analysis of Variance
What is Analysis of Variance?
A statistical method used to test differences between the means of two or more groups by analyzing the variances within and between those groups.
Key formula / rule: Total Sum of Squares (SST)
Key points
- Understand the purpose and application of ANOVA.
- Learn to partition total variance.
- Calculate and interpret the F-statistic.
- State and test hypotheses using ANOVA.
Common exam trap
Assuming equal variances when they are not (violates assumption).
Definitions
- Term
Analysis of Variance (ANOVA)
- Meaning
A statistical method used to test differences between the means of two or more groups by analyzing the variances within and between those groups.
- Term
F-statistic
- Meaning
The test statistic used in ANOVA, calculated as the ratio of the variance between groups to the variance within groups.
- Term
Null Hypothesis (H0)
- Meaning
The hypothesis that there is no significant difference between the means of the groups being compared.
- Term
Alternative Hypothesis (Ha)
- Meaning
The hypothesis that at least one group mean is significantly different from the others.
- Term
Homogeneity of Variances
- Meaning
The assumption that the variances of the populations from which the samples are drawn are equal.
Learning objectives
Understand the purpose and application of ANOVA.
Learn to partition total variance.
Calculate and interpret the F-statistic.
State and test hypotheses using ANOVA.
Identify the assumptions of ANOVA.
Formulae
- Name
Total Sum of Squares (SST)
- Note
Measures total variation in the data.
- Expression
SST = Σ(xi - x̄)²
- Name
Sum of Squares Between Groups (SSB)
- Note
Measures variation between group means and the overall mean.
- Expression
SSB = Σ ni(x̄i - x̄)²
- Name
Sum of Squares Within Groups (SSW)
- Note
Measures variation within each group (error variation).
- Expression
SSW = Σ Σ (xij - x̄i)²
- Name
Degrees of Freedom Between Groups (dfB)
- Note
k is the number of groups.
- Expression
dfB = k - 1
- Name
Degrees of Freedom Within Groups (dfW)
- Note
N is the total number of observations.
- Expression
dfW = N - k
- Name
Mean Square Between Groups (MSB)
- Note
Estimate of variance between groups.
- Expression
MSB = SSB / dfB
- Name
Mean Square Within Groups (MSW)
- Note
Estimate of variance within groups (error variance).
- Expression
MSW = SSW / dfW
- Name
F-statistic
- Note
Ratio of between-group variance to within-group variance.
- Expression
F = MSB / MSW
Prerequisites
Basic statistics (mean, variance, standard deviation).
Understanding of hypothesis testing (null and alternative hypotheses).
Familiarity with probability distributions (especially F-distribution).
Basic algebra for calculations.
Common mistakes
Assuming equal variances when they are not (violates assumption).
Incorrectly calculating the F-statistic.
Misinterpreting the p-value or F-critical value.
Applying ANOVA when only two groups are present (t-test is more appropriate).
Confusing between-group variance with within-group variance.
Keywords
ANOVA
F-statistic
Variance
Hypothesis Testing
Group Means
One-way ANOVA
Two-way ANOVA
R.A. Fisher
Statistical Significance
Practice preview
In Analysis of Variance (ANOVA), which of the following is the primary purpose of the F-test?…
easy
A researcher conducts a One-Way ANOVA with 4 groups, each containing 5 subjects. If the Sum of Squares Between (SSB) is 60 and the Sum of Squares Within (SSW) is 80, what is the calculated F-value?…
medium
In a Two-Way ANOVA without replication, if there are 'r' rows and 'c' columns, what is the degrees of freedom associated with the Error term?…
hard