12.3: Chapter 12 - Key Terms and Symbols

Key Terms

Alternative Hypothesis: The statement that at least one group mean is different from the others. It contrasts with the null hypothesis in ANOVA.

Between-Group Variance: A measure of variability between the means of different groups. It reflects how much the group means differs from the grand mean.

Degrees of Freedom: The number of independent values that can vary in an analysis. ANOVA has degrees of freedom for between-group, within-group, and total variation.

Dependent Variable: The variable being measured or tested in the experiment. It is affected by changes in the independent variable.

F-Distribution: A probability distribution used in ANOVA to compare variances. It is skewed right and used to determine critical values in F-tests.

F-Ratio: The test statistic in ANOVA, calculated by dividing the between-group variance by the within-group variance.

F-Test: A statistical test that compares the variances of two or more groups to determine if at least one group mean is different.

Grand Mean: The overall mean of all the data points from all groups combined.

Independent Variable: The variable that is manipulated or categorized to observe its effect on the dependent variable. It defines the groups being compared.

Level of Significance: The probability threshold (commonly denoted as alpha, such as 0.05) used to decide whether to reject the null hypothesis.

Mean Square: The variance estimate in ANOVA, calculated by dividing the sum of squares by the corresponding degrees of freedom. Includes between-group and within-group mean squares.

Null Hypothesis: The statement that all group means are equal. It is tested against the alternative hypothesis in ANOVA.

One-Way ANOVA: An analysis of variance method used to compare means across one independent variable with two or more levels or groups.

Within-Group Variance: A measure of variability within each group. It reflects differences among individual scores in the same group.

Key Symbols

\( \bar{x} \) – sample mean of a group

\( \bar{x}_{GM} \) – grand mean (mean of all data across all groups)

\( k \) – number of groups or levels of the independent variable

\( n \) – number of observations per group

\( N \) – total number of observations across all groups

\( S^2_B \) – Between-group variance.

\( S^2_W \) – within group variance

\( dfn \) – degrees of freedom of the numerator (\( k - 1 \))

\( dfd \) – degrees of freedom of the denominator (\( N - k \))

\( F \) – F-ratio or test statistic (\( S^2_B / S^2_W \))

\( \alpha \) – significance level (probability of Type I error)

\( p \) – p-value.