Two-Way ANOVA: Student Performance by Region and Curriculum

Introduction and Research Objectives

The objective of this statistical analysis is to learn more about the relationships between variables that may influence student performance in the schools under study. The specific variables of interest are the region of the county in which schools are located and the curriculum being used at each school. One key question is whether differences in student performance can be attributed to the regional location of a school or to the curriculum in use. A secondary question explores whether student performance appears to change based on the combination of curriculum and region — that is, whether an interaction effect exists between the two factors.

Data Sample and Hypotheses

A web-based randomization generator was used to select the sample. Twenty individual students were selected, with each student identified only by the number indicating their position in the complete data set, which includes all schools participating in the study.

The hypotheses being tested in this statistical procedure are as follows:

Null Hypothesis (Region): H₀: μ₁ = μ₂ = … μₖ — No differences in student performance can be accounted for by the region in which the student's school is located.

Null Hypothesis (Curriculum): H₀: μ₁ = μ₂ = … μₖ — No differences in student performance can be accounted for by the curriculum being used at the student's school.

Alternative Hypothesis (Curriculum): Hₐ: μᵢ ≠ μⱼ — The curriculum used at a school accounts for some of the differences measured in student performance.

Alternative Hypothesis (Region): Hₐ: μᵢ ≠ μⱼ — The region in which a school is located accounts for some of the differences measured in student performance.

ANOVA Methodology

The Excel ANOVA tool was accessed through the StatPlus add-in for Mac. The results populate a standard Analysis of Variance (ANOVA) table. The test statistic used in an ANOVA is an F-test with k − 1 and N − k degrees of freedom, where N is the total number of subjects and k is the number of groups. A low p-value generally indicates reason to reject the null hypothesis in favor of the alternative — that is, there is evidence that at least one pair of group means is not equal.

A two-way analysis of variance enables the simultaneous investigation of two factors within a single experiment. This approach avoids the need to run two separate one-way ANOVAs and then attempt to combine the results afterward. A two-way ANOVA provides more information and is more economical and efficient, since the effects of both factors — and their potential interaction — are explored at the same time.

Statistical Results

The first calculation in an ANOVA is the grand mean — the mean of all observations regardless of group. The grand mean for this analysis is 1,502. Following this, the sum of squares (SS) is calculated as an estimate of the variation among groups, specifically representing the deviation of group means from the grand mean.

The sum of squares values are as follows: SS = 223,222 for Factor 1 (RE — regional area); SS = 65,040 for Factor 2 (CU — curriculum); SS = 1,830 for the interaction term (RE × CU); and SS = 128,839 for within-groups variation.

The F-value is the key test statistic in an ANOVA and is calculated as the ratio of among-groups variation to within-groups variation. A large F-value indicates that variation among groups substantially exceeds variation within groups, suggesting that a given variable or experimental manipulation — rather than chance — accounts for the observed differences. Conversely, when among-groups and within-groups variation are similar, the F-value is small, indicating that differences among groups are more likely attributable to measurement error or natural chance rather than a specific variable.

The F-values and critical F-values (F crit) are as follows:

Factor 1 (RE): F = 1.2617; F crit = 5.2407
Factor 2 (CU): F = 7.068; F crit = 6.8879
Factor 1 + 2 (RE × CU): F = 0.09945 (≈ 0.10); F crit = 5.2407