Has any variation been explained by the groups?
We have distinguished between two concepts (without formally defining them).
In practice, the group means are unlikely to be exactly equal, so there will always be some variation between the group means even if there is the groups are basically identical.
The amount of between-group variation is an important indicator of whether the groups really differ, but the within-group variation must also be taken into account to interpret it correctly, as shown by the diagrams below.
The relative size of between-group and within-group variation is what holds information about whether the groups differ. |
Variation between groups
The jittered dot plots below show 10 numerical measurements from each of 4 groups.
Use the slider to alter the difference between the group means. As the differences between groups increase,
Variation within groups
The diagram below is similar, but the slider adjusts the spread of values within each group, leaving the group mean unaltered.
As the within-group variation decreases,
Are the groups identical?
The evidence for a difference between the groups depends on both the variation between and within groups. It is strongest when:
Signal and noise
In the field of communications, the signal in a recorded or transmitted message (e.g. music) is defined to be the information about which we are interested. Other variability in the received message about which there is no interest but which can potentially obscure or corrupt the signal is called noise.
Applying this terminology to the comparison of several groups,
The greater the noise, the harder it is to detect or estimate the signal. We will next present numerical summaries of the signal and noise in multi-group data.