# Between Group Variation

Between group variation is a term used in ANOVA (analysis of variance) for variation that is measured between separate groups of interest. Unlike within group variation, where the focus is on the differences between a population and its mean, between group variation is concerned with finding how the means of groups differ from each other.

## Understanding Between Group Variation

The formula for between-group variation is given by

and is called the sum of squares between groups, or SS(B). This measures the interaction between the groups or samples. If the group means don’t differ greatly from each other and the grand mean, the SS(B) will be small.

Note that for k groups, there will be k-1 degrees of freedom. The between groups variance is the variation, or SS(B), divided by its degree of freedom. We sometimes refer to the between groups variance as sb 2.

## Between Group Variation, Within Group Variation, and the F-ratio

Between group variation is important in ANOVA because it is compared to within group variation to determine treatment effect.

We can calculate what is called the F-ratio as (between group variation)/(within group variation). This is equivalent to (treatment effect+error)/(error). If the treatment effect goes to zero, the F ratio will be (error)/(error) and go to 1. If the treatment effect increases toward infinity, the F ratio will also go towards infinity.

## References

Garbin, Cal. Between Groups and Within Groups ANOVA. Lecture Notes. Retrieved from http://psych.unl.edu/psycrs/941/q3/bg-wg.pdf on August 17, 2018.

Deater-Deckard et al. (2017). Within- and between-person and group variance in behavior and beliefs in cross-cultural longitudinal data. Journal of Adolescence. 62. 10.1016/j.adolescence.2017.06.002. Retrieved from https://www.researchgate.net/publication/317940722_Within-_and_between-person_and_group_variance_in_behavior_and_beliefs_in_cross-cultural_longitudinal_data on August 17, 2018.

Webb, Barbara. ANOVA – Analysis of Variance