Visualising Twofold Dependencies by Fan Charts

To measure the dependency of observed values on one or more variables, regression calculations are applied. If there is only one independent variable, a sequence of box plots can be used for visual control. If there are two independent variables, a glyph with constant size is more practical and allows a denser presentation.

The following example1 shall demonstrate if the mathematics achievment scores of the sample are only dependent on the socio-economic status of the students or if they are influenced by the average socio-economic status of all students at the same school.

[Tab. 1] To represent the distributions, «fan charts»2 are used. They allow to report in a dense way the same distribution numbers as box plots, thus especially median and quartiles. The circular symbol allows direct comparisons within not only one but two directions: horizontally and vertically.

As scale, a circular line is used. It begins at the left with the starting value, e. g. with zero. The following values are applicated clockwise. The white tail of the diameter indicates the median. The dark fan indicates the distribution of the middle half of the observed values; thus it encompasses the values from the first to the third quartile. The white feathers indicate the distribution of the middle 90 % of the observed values. The length of the white part of the diameter corresponds with the number of observations.

The following example presents data from the data set MathAchieve which is part of the R package nlme of Jose Pinheiro et al.3 It contains mathematics achievment scores of 7185 students. The students are categorised according to sex and membership of a minority ethnic group.4

[Tab. 2] The graphics show the mathematics achievement scores in dependency on the socio-economic status of the students (x axis) and on the average socio-economic status of all students at the same school (y axis). The four graphic panels differentiate the students according to sex and membership of a minority ethnic group.

The largely opened fans attract our first attention: The achievement scores have large distributions within nearly all subgroups (with the cells). A comparison of the four panels indicates that the achievement scores tend to be better for students which are not member of a minority and that men have slighty higher scores when compared to women.

Within the panels, it can be seen that both the socio-economic status of the students and – not so strongly – the average the socio-economic status of all students at the same school correlate positively with the achievement scores.

The fan charts shown reveal how the median value is partially following a big main tendency while the values within the single subgroups scatter largely what could lead to doubts about a possible correlation. (This can seen especially well on the two lower panels.)