# Mathematics of the Coxcombs

## What are coxcombs?

Coxcombs, just like bar charts or pie charts, are graphical systems of representing data. They are sometimes referred to as polar area charts or roses. (Despite many claims to the contrary, Nightingale did not originally refer to the graphs as coxcombs; see Florence Nightingale's Statistical Diagrams for an explanation of this error.)

Consider the following bar chart.

A simple bar chart

Category A has value 1, category B has value 3, and category C has value 2. In the next figure, this same set of data is represented in a coxcomb.

A simple coxcomb

Each category is represented by a sector of a disc. The sectors each have the same angle. In this case, as there are three sectors, each sector has angle, in degrees,

$$\frac{360}{3} = 120.$$

The area of a sector represents the value for the corresponding category. So, for example, if $r_A$ is the radius of sector A then

$$\text{Area of sector A} = \frac{\pi r_A^2}{3}=1,$$

because 1 is the value of category A. Likewise

$$\text{Area of sector B} = \frac{\pi r_B^2}{3}=3$$

and

$$\text{Area of sector C} = \frac{\pi r_C^2}{3} = 2.$$

We can solve these equations to determine $r_A$, $r_B$, and $r_C$, enabling us to plot the coxcomb.

It is important to notice that it is the area, rather than the radius, of a sector that represents the value of that sector. The values of the sectors are proportional to the squared radii of the sectors.

## Nightingale's coxcombs

In 'Nightingale's Coxcombs' we displayed a pair of coxcombs drawn by Nightingale, and an animated reproduction of this pair of coxcombs. The categories of these coxcombs were the months of the year. There are twelve months in the year, so the angle of each sector is, in degrees,

$$\frac{360}{12} = 30.$$

The data represented by the coxcombs are deaths each month for British soldiers in the Crimean War. Let us consider exactly the mathematics that Nightingale used in order to calculate the data for deaths by diseases (the blue portion of the graph). Suppose that on average during a particular month there were S soldiers, and D of these soldiers died of diseases during that month. Nightingale considered the proportion of soldiers who died of diseases; that is, she considered the quantity D/S. Next, she wished to express this quantity as a figure out of 1000, so she multiplied it by 1000, to obtain N=1000D/S. Thus, in that month, N out of every 1000 soldiers died of diseases. Finally, Nightingale multiplied N by 12 to indicate how many soldiers would have died from diseases throughout the whole year had the circumstances of that month been replicated throughout the year. She described the final quantity that she obtained as the 'annual rate of mortality per 1000 in each month'. As we have seen, it is given by the formula

$$\frac{12000D}{S}.$$

This is the value of the area of the blue sector for the particular month that we are considering. Of course, area depends on the unit of measurement, so Nightingale will have scaled all her values appropriately. Since coxcombs do not have scales, Nightingale need not have multiplied D/S by 12000 in order to plot the coxcombs. The bar chart in our reproduction of Nightingale's work records 'monthly mortality per 1000'; that is, the quantity N=1000D/S.

The figures used to produce the 'Diagram of the Causes of Mortality in the Army in the East' are shown in the following table. The table was taken from 'A contribution to the sanitary history of the British army during the late war with Russia', by Florence Nightingale, 1859, and it was provided to us courtesy of Hugh Small. Note that the second column represents S, the third column represent D, and the sixth column represents 12000D/S.

This table contains the data that Nightingale used to produce 'Diagram of the causes of mortality in the army in the east

Table showing the Estimated Average Monthly Strength of the Army; and the deaths and Annual Rate of Mortality per 1000 in each month, from April 1854, to March 1856 (inclusive), in the Hospitals of the Army in the East
Deaths Annual rate of mortality per 1000
Month Average size of army Zymotic diseases Wounds & injuries All other causes Zymotic diseases Wounds & injuries All other causes
Apr 1854 8571 1 0 5 1.4 0 7.0
May 1854 23333 12 0 9 6.2 0 4.6
Jun 1854 28333 11 0 6 4.7 0 2.5
Jul 1854 28722 359 0 23 150.0 0 9.6
Aug 1854 30246 828 1 30 328.5 0.4 11.9
Sep 1854 30290 788 81 70 312.2 32.1 27.7
Oct 1854 30643 503 132 128 197.0 51.7 50.1
Nov 1854 29736 844 287 106 340.6 115.8 42.8
Dec 1854 32779 1725 114 131 631.5 41.7 48.0
Jan 1855 32393 2761 83 324 1022.8 30.7 120.0
Feb 1855 30919 2120 42 361 822.8 16.3 140.1
Mar 1855 30107 1205 32 172 480.3 12.8 68.6
Apr 1855 32252 477 48 57 177.5 17.9 21.2
May 1855 35473 508 49 37 171.8 16.6 12.5
Jun 1855 38863 802 209 31 247.6 64.5 9.6
Jul 1855 42647 382 134 33 107.5 37.7 9.3
Aug_1855 44614 483 164 25 129.9 44.1 6.7
Sep 1855 47751 189 276 20 47.5 69.4 5.0
Oct 1855 46852 128 53 18 32.8 13.6 4.6
Nov 1855 37853 178 33 32 56.4 10.5 10.1
Dec 1855 43217 91 18 28 25.3 5.0 7.8
Jan 1856 44212 42 2 48 11.4 0.5 13.0
Feb 1856 43485 24 0 19 6.6 0 5.2
Mar 1856 46140 15 0 35 3.9 0 9.1
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