What is ANOVA (Analysis of Variance) in Statistics ? | Explained with Examples (ANOVA F - test)

ANOVA, or Analysis of Variance, is a statistical method used to compare means across multiple groups. It helps determine if there are significant differences between group averages by analyzing variance within and between the groups. Different types of ANOVA include one-way ANOVA for single factors and two-way ANOVA for examining interactions between two variables. Practical examples illustrate how this tool can be applied in real-world scenarios to draw meaningful conclusions from data.

ANOVA, or Analysis of Variance, developed by Ronald Fisher, is a statistical method used to test the significance of differences among more than two sample means. It extends the t-test capabilities beyond just comparing two population means. While both ANOVA and t-tests yield similar results for two samples, relying on multiple t-tests when dealing with three or more samples increases the risk of Type I errors due to compounded error rates. Therefore, ANOVA provides a more reliable approach in such scenarios.

One Way ANOVA involves a single independent variable and compares three or more levels of that factor, while Two Way ANOVA extends this by analyzing the effects of two independent variables simultaneously. The F test is used for statistical significance in both methods, assessing variance between group means against overall variance. Key assumptions include normal distribution within populations, random sampling, common variances across groups, and independence of data points. The null hypothesis states all group means are equal; if any mean significantly differs from others, it leads to rejection of the null hypothesis.

To compare three sample means (X1, X2, and X3) for differences in population origin, one must assess variance within each distribution versus variance between them. Variance within reflects the internal spread of variation among samples while variance between examines how far each mean deviates from the overall mean. The null hypothesis posits that all means are equal (μ1 = μ2 = μ3), whereas the alternative suggests at least one differs. ANOVA calculates a ratio of these variances; if this ratio is greater than or equal to 1, it indicates significant differences warranting rejection of the null hypothesis. A higher F statistic signifies substantial group mean variation compared to individual variations.

Comparing Study Methods Through Mean Scores To determine if three study methods yield different mean exam scores, 30 students are assigned to each method. The group means for methods A, B, and C are calculated as 8.7, 8.6, and 8.5 respectively with an overall mean of 8.6. Between-group variation is computed at 0.2 while within-group variation totals to 28 after calculating individual variances for each method's observations against their respective means.

F-Test Analysis on Hypothesis Testing Using the F-test approach reveals that the ratio of between-group variance (0.2) to within-group variance (28) results in an F-statistic of approximately 0.0071 which is less than the critical value from the F-table (3.35). With a significance level set at alpha = .05 and degrees of freedom determined accordingly—numerator: samples minus one; denominator: total values minus number of groups—the conclusion drawn is that there’s insufficient evidence to reject the null hypothesis regarding equal means across study methods.

Total variation (SST) is divided into between-group variation (SSC) and within-group variation (SSE). The sum of squares reflects this variation, influenced by sample size and degrees of freedom. ANOVA calculates parameters summarized in a one-way ANOVA table, which includes variations between groups, within groups, and total variations. Key terms include mean square for both types of variance calculated from their respective sums of squares adjusted by degrees of freedom. The F ratio compares the mean square values to determine if group means are significantly different; if they are equal, the null hypothesis holds true.

A quick quiz tests understanding of statistical significance and analysis methods. The first question asks how to determine the significance of an over test, with options including T statistics, F statistics, or chi-square statistics. The second question addresses whether analysis of variance compares several population means or other factors. Lastly, it inquires about what the sum of squares measures regarding variability around treatments.