Partitions of group totals

In many data sets, the values within each group are partitions of a total. If the values are counts (forming a frequency table), the two-way table of categories by groups is called a contingency table, but there are other tables in which the values can be treated as partitions of a group total.

Measurement    Group       Categories    Total in group
Number of households Region Household size Households in the region
Rainfall Country Months Annual rainfall
GDP Year Sector of economy National GDP

Only first example above would form a contingency table but the others also describe partitions of a meaninful total.

Proportions within groups

If the totals vary widely in the different groups, bar charts and tables of the raw values often give a visual impression that is dominated by differences between the group totals.

Rather than tabulating and graphing the raw values for each group, it is more informative to tabulate the proportions within the groups. Each value in the table is therefore divided by the total for that group.

Even when the group totals are similar, it is often better to examine proportions rather than the raw values.


Severity of injuries on New Zealand roads

In New Zealand, there is a legal obligation to report traffic accidents in which someone has been injured. The table below describes the types of road user injured and the severity of these injuries in 2006.

There were far fewer reported injuries of cyclists and pedestrians than of drivers and passengers, making it difficult to compare injury severity for the four groups of road user. The proportions within each group of road users are easier to compare than the raw frequencies — click on any row to see how the proportions are obtained.

Select the Propn within Road user group from the pop-up menu to display the resulting proportions. This scales each row, making all row totals the same, 1.0. Select Percent within Road user group to show these proportions as percentages — clearer since leading zeros are omitted.

Observe that a much higher proportion of pedestrians are killed or have serious injuries than other road users.

Bar chart of proportions

The diagram below shows bar charts for the four types of road users who were injured.

The large number of drivers and passengers who were injured or killed is evident in the bar chart, but it does not effectively allow comparison of the injury types of the four groups of road users.

Select Propn within Road user group or Percent within Road user group from the pop-up menu. The diagram now shows more clearly the larger proportion of pedestrians who were killed or had serious injuries, and the low proportion of cyclists who were killed.

Interpretation

Attributing a reason to the greater severity of pedestrian injuries is difficult. Perhaps pedestrians are more likely to suffer serious inuries and death, but there could also be under-reporting of minor injuries by pedestrians.

Warning: Always think carefully about possible reasons for differences that you observe.