This chapter develops weighted averages as a way of combining values when different parts of the data do not have equal importance or equal numbers of observations. The weighted mean of values x1, x2, ... with corresponding weights w1, w2, ... is obtained by multiplying each value by its weight, adding those products, and dividing by the sum of the weights. This is different from simply averaging the values without considering their weights.
The weighted mean is useful when combining averages from groups of different sizes, calculating concentrations in mixtures, combining marks from assessments, and designing rating schemes. The key idea is that a larger weight gives a value a larger influence on the final result. The chapter also explains that an average of averages can be found correctly using a weighted mean when the groups have different numbers of observations.
The chapter then turns to bar-chart representations. Stacked bar charts allow totals and the composition of those totals to be seen together. A 100% stacked bar chart instead represents each total as 100% and shows the proportion contributed by each category. This makes it useful for comparing fractions or percentages within each group.
An important limitation is that a 100% stacked bar chart does not reveal the absolute total of each group. Two groups can have the same 100% bar while having very different actual totals. Consequently, a percentage comparison across groups does not automatically tell us which group has the larger number of items in a category. The chapter uses such examples to show that mathematical graphs must be interpreted according to what they actually encode, rather than what a viewer might assume.
When data are grouped, the number of observations in a group acts as an important weight. Combining two group averages by simply averaging the two averages can give a misleading result if one group contains many more observations than the other. The weighted mean corrects this by giving each group influence proportional to its size or to the chosen importance of the group. The same reasoning supports rating systems in which different aspects are deliberately assigned different weights.
The chapter also stresses the difference between what a chart shows and what it does not show. A stacked bar can preserve information about totals as well as the parts making up those totals. A 100% stacked bar normalises every total to the same full height, so it is especially suited to comparing composition. Because the total size is hidden, conclusions about absolute counts require additional information.