Final answer:
The 'within' sum of squares measures the variability around group means due to chance factors, not the treatment itself. SS within is crucial in ANOVA for estimating the unexplained variation within treatment groups and differentiating it from variation caused by treatments (SS between).
Step-by-step explanation:
The within sum of squares measures the variability of the observed values around their respective treatment means. The SS within, also known as the sum of squares within groups, represents the variation in outcome that cannot be attributed to the different treatments but due to individual or chance factors among the samples in each group. This is in contrast to SS between, which measures the variability among different samples, attributed to the treatments given.
To calculate SS within, you would start by finding the deviation of each observation from its respective group mean, square these deviations, and then sum them up for each group. This provides a measure of the unexplained variation or the variance due to error within each treatment group. Conversely, the SS between represents the explained variation or variance due to treatment among the groups. When calculating the F-ratio in an analysis of variance (ANOVA), these two types of sum of squares are crucial as they provide estimates of variance within and between samples, helping to determine if there are significant treatment effects.