Sum of squares

Definition

The sum of squares (SS) is a measure of the total variability within a data set. It quantifies how much individual data points deviate from the mean.

5 Must Know Facts For Your Next Test

1. In ANOVA, the Total Sum of Squares (SST) is partitioned into the Between-Group Sum of Squares (SSB) and Within-Group Sum of Squares (SSW).
2. The formula for calculating SST is $SST = \sum_{i=1}^{n} (X_i - \bar{X})^2$, where $X_i$ are individual observations and $\bar{X}$ is the overall mean.
3. SSB measures the variation due to differences between group means and is calculated as $SSB = \sum_{j=1}^{k} n_j(\bar{X}_j - \bar{X})^2$, where $n_j$ are sample sizes and $\bar{X}_j$ are group means.
4. SSW, or residual sum of squares, measures the variation within each group and is calculated as $SSW = \sum_{j=1}^{k} \sum_{i=1}^{n_j} (X_{ij} - \bar{X}_j)^2$.
5. In the context of F-ratio, SS values are used to compute Mean Squares by dividing by their respective degrees of freedom before forming the F-statistic.

Review Questions

• What does the sum of squares measure in a data set?
• How is the Total Sum of Squares partitioned in ANOVA?
• What are the components needed to calculate Between-Group Sum of Squares?

Related terms

F-Ratio: A ratio used in ANOVA to determine if there are significant differences between group variances. Calculated as the ratio of Mean Square Between Groups to Mean Square Within Groups.

Mean Square: An average squared deviation used in variance analysis. Obtained by dividing a sum of squares by its associated degrees of freedom.

Degrees of Freedom: The number of values free to vary when computing a statistic, often denoted as 'df'. In ANOVA, it's crucial for determining mean squares and critical values.