Florence Nightingale Essay

The Polar subject Diagrams of Florence nightingaleIf you read the article on Florence nightingale in The Childrens Book of Famous Lives1 you will not learn that she had to action with her parents to be allowed to study mathematics. If you read the Ladybird book Florence Nightingale2 you will not wear that she was the starting line fair sex to be elected as a Fellow of the Royal Statistical Society. In looking virtually for an part of research I was intrigued to disc over that Florence Nightingale, who I always thought of as the lady with the lamp, was a competent Mathematician who created her own type of statistical plot which she used to save thousands of soldiers from needless wipe start. Florence Nightingale headed a group of 38 nurses who went to clean up the hospitals for the British soldiers in the Crimea in 1854. She fix that most of the stopping points were due to diseases which could be prevented by basic hygiene, such as typhus and cholera. Her improvements wer e simple that they had an enormous effect She and her nurses washed and bathed the soldiers, laundered their linens, gave them clean beds to lie in, and fed them3.When she returned to Britain she made a detailed declare to the Government specialiseting out what conditions were like and what needed to be make to reduce deaths in the hospitals. zip was done, so she tried again, making an different(a) statistical report and included in it three stark naked statistical plots to make entropy collated by William Farr more than accessible to people who could not get their minds around tables of figures. These were her icy surface field of view draws or rose diagrams, neartimes also known as coxcombs. The first showed how legion(predicate) men had died over the two years 1854-5, the plump for showed what proportions of men had died from wounds in battle, from disease and from other causes, the tierce showed how the degene outrank of deaths had decreased once sanitary im provements4 had been introduced. I trenchant I would try to act as the flake of these diagrams which is the most complicated and the most shocking. It is called Diagram of the causes of mortality rate in the array in the vitamin E. A copy of it is belowMathematics SL and HL t individuallyer condesc prohibit real1 showcase 6 Student motionFigure 1The basic ideas are very simple. The good-for-nothing area represents deaths due to disease, the red area represents death due to wounds in battle and the sullen area represents death due to other causes. I tried to find a copy of the data which this diagram represented, still I had no luck, so I decided to make sure I understood exactly how the diagram was made and to make my own recitation of some data which I did study to hand. erstwhile I tried to understand the diagram in detail I found there were some problems. The First ProblemI wasnt sure whether the black area in a variety such as this was supposed to bethis areaor this areaMathematics SL and HL t severallyer hold back textile2Example 6 Student drillIn other words, were the cogitation separate, or overlapping? The articles I read didnt make it clear. OConnor says that The area of separately color w pass on, measured from the centre as a greenness point, is in proportion to the statistic it represents5, which makes it debatem that all color are wedged figure of speechd, or sectors, so the burnishs overlap. However, Lienhard commented that in the November 1854 section battle deaths take up a very small portion of each cut of meat6, which makes it sound as though the slice has three separate portions, and Brasseur says that she also divided the areas within each of the wedges to show which portion of the mortality data for that month could be allotted to each cause of death4.I decided to construct polar area diagrams for a set of data with the colours separate and with the colours overlapping to match if putting theory into practic e would make it clearer to me. The data I used was taken from the IB grade distribution statistics for the historic 15 years at my own institution. I used the song fetching higher(prenominal) Mathematics, cadence Mathematics and mathematical Studies to be represented by my three colours. I took the old Mathematical Methods note to be the uniform as type Mathematics. To fit 15 sectors into the circle I needed each arc to subtend an angle of 2 radians at 151 2 the centre. The area of each sector would then be A = r 2 = 2 where r is the r 2 15 15 radius of the sector. Since the area needs to be proportional to the statistic, I needed to 15A and right used a photographic plate which would allow me to draw find the radius, so I used r =a reasonable sized diagram. To create a polar area diagram with overlapping sectors I retributive used this formula on each of the snatchs of students pickings the various options. dos winning Mathematics year on year Numbers (A) Higher St udies Standard 1995 1 24 0 1996 4 15 0 1997 8 10 0 1998 6 31 0 1999 9 17 0 2000 10 20 0 2001 4 31 1 2002 5 21 2 2003 4 15 4 2004 5 29 5 2005 1 28 0 2006 3 16 2 2007 8 13 0 2008 11 29 14 2009 10 23 15 universal gas constant ( r ) Higher Studies Standard 2.2 10.7 0.0 4.4 8.5 0.0 6.2 6.9 0.0 5.4 12.2 0.0 6.6 9.0 0.0 6.9 9.8 0.0 4.4 12.2 2.2 4.9 10.0 3.1 4.4 8.5 4.4 4.9 11.8 4.9 2.2 11.6 0.0 3.8 8.7 3.1 6.2 7.9 0.0 7.2 11.8 8.2 6.9 10.5 8.5Mathematics SL and HL teacher plump for material3Example 6 Student workI then used a geometric program (GeoGebra) to draw the sectors all with a common 2 centre, each with an angle of radians, and with the radii as given in the table. I drew 15 the Higher sectors first with the Studies on top of these, and the Standard on top of these. This was the result Figure 2 Polar area diagram to show students victorious Mathematics at one school (colours overlapping)Blue represents the go of students taking Higher Maths. Brown represents the number of stude nts taking Mathematical Studies. Green represents the number of students taking Standard Maths.The colours are not solid, so where colours overlap there is a different colour. The blue overlapping the brown makes a knock here, and the green overlapping the blue makes a darker green. In 2003 and in 2004 there were an equal number of students taking Higher and Standard so three separate colours cannot be seen on the diagram. Next I worked out the radii needed if the colours were not to overlap. For this I used cumulative areas to work out the radii. R1 = R3 = 15 ( A1 + A2 + A3) 15 A1 15 ( A1 + A2 ), R2 =and.Radii R2 10.9 9.5 9.3 13.3 11.1 12.0 12.9 11.1 9.5 12.7 11.8 9.5 10.0 13.8 12.6Numbers taking Mathematics year on year Numbers (A) Higher (A1) Studies (A2) Standard (A3) 1995 1 24 0 1996 4 15 0 1997 8 10 0 1998 6 31 0 1999 9 17 0 2000 10 20 0 2001 4 31 1 2002 5 21 2 2003 4 15 4 2004 5 29 5 2005 1 28 02006 3 16 2 2007 8 13 0 2008 11 29 14 2009 10 23 15R12.2 4.4 6.2 5.4 6.6 6.9 4.4 4.9 4.4 4.9 2.2 3.8 6.2 7.2 6.9R3 10.9 9.5 9.3 13.3 11.1 12.0 13.1 11.6 10.5 13.6 11.8 10.0 10.0 16.1 15.1Mathematics SL and HL teacher buy at material4Example 6 Student workThis gave a diagram with Higher numbers at the centre and Standard numbers at the edge, like thisFigure 3 Polar area diagram to show students taking Mathematics at one school (colours separate)Blue represents the number of students taking Higher Maths. Brown represents the number of students taking Mathematical Studies. Green represents the number of students taking Standard Maths.This diagram is incomplete in that it has not got the dates on it, but I was interested in the basic shape it would make rather than seeing it as a finished article to represent the data. I decided to do the same thing but with Studies in the middle and Higher at the edge to see how different it would look.Figure 4 Polar area diagram to show students taking Mathematics at one school (colours separate)Blue represents the number of stud ents taking Higher Maths. Brown represents the number of students taking Mathematical Studies. Green represents the number of students taking Standard Maths.This feels very different. The blue section actually looks less significant, to my eye, macrocosm put at the edges. This made me think of something else I had read in Brasseurs article, Nightingale pose these colored areas so that the main cause of death (and the largest sections)deaths by diseasewould be at the end of the wedges and would be more easily noticed.4 I am sure that Brasseur thought that the colours were separate, and not overlapped. However, comparing my diagrams to Nightingales original in Figure 1, IMathematics SL and HL teacher support material5Example 6 Student workbecame sure that she did mean them to be overlapped. I noticed that in the lefthand rose in figure 1 (representing the second year) there is a wedge with blue at the edge followed by a wedge with blue at the edge Figure 5 A zoom in of part of figu re 1This can croak in a diagram like my figure 2 of overlapping colours, but would be out of the question if the colours are separate as in figures 3 and 4. From this I deduced that the colours on the diagram must be overlapping. The Second Problem My diagrams were unlike Nightingales ones in that the total area of the sectors in figure 2 represented the total number of students taking the IB at this school over the 15 years. Nightingales statistics were order of mortality. Basically they can be thought of as percentages of soldiers who died, but, as before, when I read through the articles again, I was unsure what they were percentages of. Gill and Gill have table (Table 2) in their article with headings No. of soldiers admitted to the hospital and No. (%) of soldiers who died3.This might suggest that Nightingale was working with percentages of soldiers who were admitted into hospital. Lewi is more defined and refers to the actual statistic of one wedge of the third of Nighting ales polar area diagrams as follows The mortality during the first period was 192 per 1,000 hospitalized soldiers (on a yearly basis)9. However, Brasseur refers to the statistic in a wedge of Nightingales first diagram as being the ratio of mortality for every 1,000 soldiers per annum in the field4, in other words a percentage of the force actually on duty. I decided to create a polar area diagram to act as an analogy to the possible situations as follows Nightingales data My data Number of soldiers in the array in a month Number of students taking the IB in a year Number of soldiers taken to hospital Number of students taking Maths Studies Number of soldiers dying of wounds Number of students gaining grade 7 Number of soldiers dying of disease Number of students gaining grade 6 Number of soldiers dying for other reasons Number of students gaining grade 5My analogy of drawing a diagram showing the numbers of soldiers dying as a percentage of those admitted to hospital would then b e the number of students gaining a grade above 4 as a percentage of those taking Mathematical Studies. I decided to do this one by hand, partly to prove I could, and partly to see if it would spend any extra light on the construction of the diagrams.Mathematics SL and HL teacher support material6Example 6 Student workI gathered the data, found the percentages and used the percentages as A in the usual 15A to find the radii needed to construct the diagram. The data is here formula r =Numbers gaining top three grades in Mathematical Studies As percentage of those taking Studies radius needed for each Taking Total Grade 7 Grade 6 Grade 5 Studies in year % grade 7 % grade 6 % grade 5 R7 R6 R5 1995 7 10 4 24 25 29.16667 41.66667 16.66667 11.80 14.10 8.92 1996 2 9 3 15 19 13.3333360.00000 20.00000 7.98 16.93 9.77 1997 1 4 2 10 18 10.00000 40.00000 20.00000 6.91 13.82 9.77 1998 5 12 11 31 37 16.12903 38.70968 35.48387 8.78 13.60 13.02 1999 2 6 7 17 26 11.76471 35.29412 41.17647 7.49 12. 98 14.02 2000 3 4 7 20 30 15.00000 20.00000 35.00000 8.46 9.77 12.93 2001 3 8 8 31 36 9.67742 25.80645 25.80645 6.80 11.10 11.10 2002 1 8 4 21 28 4.76190 38.09524 19.04762 4.77 13.49 9.54 2003 0 1 8 15 23 0.00000 6.66667 53.33333 0.00 5.64 15.96 2004 3 9 7 29 34 10.34483 31.03448 24.13793 7.03 12.17 10.74 2005 1 11 9 28 29 3.57143 39.28571 32.14286 4.13 13.70 12.39 2006 2 4 5 16 21 12.50000 25.00000 31.25000 7.73 10.93 12.22 2007 1 8 3 13 22 7.69231 61.53846 23.07692 6.06 17.14 10.50 2008 0 3 17 29 54 0.00000 10.34483 58.62069 0.00 7.03 16.73 2009 0 5 5 23 48 0.00000 21.73913 21.73913 0.00 10.19 10.19And the diagram came out like thisFigure 6 Polar area diagram to show percentages of students taking Mathematical Studies who gained grades above 4Red represents the number of students gaining grade 7. Blue represents the number of students gaining grade 6. Green represents the number of students gaining grade 5. The purple areas represent coinciding numbers of students gaining grade 5 and 6.Mathematics SL and HL teacher support material7Example 6 Student workOne thing which I learnt from this exercise is that you have to be very careful about your scale and think through every move before you start if you dont want to fall off the edge of the paper It is a far more tense experience drawing a diagram by hand because you know that one slip will make the whole diagram flawed. A computer slip can be corrected before you print out the result. My hold for Florence Nightingales draftsmanship was heightened by doing this. The other thing which drawing by hand brought out was that, if you draw the arcs in in the appropriate colours, the colouring of the sectors sorts itself out. You colour from the arc inwards until you come to some other arc or the centre. The only problem came when two arcs of different colours came in exactly the same place. I got around this problem by colouring these areas in a totally different colour and saying so at the side. At this point in my research someone suggested some more possible websites to me, and following these up I found a copy of Nightingales second diagram which was clear enough for me to read her notes, and a copy of the original data she used.The first of these was in a letter by Henry Woodbury suggesting that Nightingale got her calculations wrong and the radii represented the statistics rather than the area.7 The letter had a comment posted by Ian petty which led me to an article by him8 giving the data for the second diagram and explaining how it was created. The very clear reproduction of Nightingales second diagram in Woodburys letter7 shows that Miss Nightingale wrote beside it The areas of the blue, red and black wedges are each measured from the centre as the common vertex.This makes it quite clear that the colours are overlapped and so solves my first problem. She also wrote In October 1854 & April 1855 the black area coincides with the red. She coloured the first of these in red and the secon d in black, but just commented on it beside the diagram to make it clear. The article by Short8 was a joy to read, although I could only work out the mathematical equations, which were written out in a way which is strange to me ( for sample $$ extArea of sector B = fracpi r_B23=3$$8 ) because I al plant knew what they were (The example had a sector B in a diagram which I could see had 1 2 2 2 = = areaB rB rB ). The two things I found exciting from this article were the 2 3 3 table of data which Nightingale used to create the second diagram, and an explanation of what rates of mortality she used.She described these as follows The ratios of deaths and admissions to Force per jet per annum are calculated from the monthly ratios given in Dr. Smiths Table B4 and I had not been able to understand the consequence of this from the other articles. (Brasseur adds that Dr. Smith was the late director-general of the army.4). Using Shorts article I was able to work out what it meant. I will use an example of data taken from the table in Shorts article, which is in turn taken from A contribution to the sanitary history of the British army during the late war with Russia by Florence Nightingale of 18598. In February 1855 the average size of the army was 30919. Of these 2120 died of zymolytic diseases, 42 died of wounds & injuries and 361 died of all other causes. This gives a total of 2120 + 42 + 361 = deaths. 2523 2523Mathematics SL and HL teacher support material8Example 6 Student work2523 81.6003 men died per 1000 men in the army in 1000 = 30919 that month. If the size of the army had stayed at 30919, with no more men being shipped in or out, and the death rate had continued at 81.6 deaths per 1000 men per month over 12 months, the number of deaths per annum would have been 81.6003 12 = 979.2 per 1000 men in the army. In other words 979.2 deaths per 1000 per annum. out of 30919 means thatThis understanding of the units used allowed me to final examinationly underst and why OConnor says of the death rate in January 1855, if this rate had continued, and troops had not been replaced frequently, then disease alone would have killed the entire British army in the Crimea.5 The number of deaths due to disease in January 1855 was 2761 and the 2761 average size of the army was 32393. This gives a rate of 1022.8 1000 12 = 32393 deaths from disease per 1000 per annum.Another way of looking at it is that if 2761 had dies each month from disease, 276112 = 33132 would have died in 12 months, but there were only 32393 in the army As an aside, I noticed that OConnor quoted the mortality rate for January 1855 as 1,023 per 10,000 being from zymotic diseases5. Another example that we should not trust everything we see in print. Having sorted this out I was ready to attempt my recreation of figure 1. I decided to do the right hand rose only, screening April 1854 to sue 1855. The following table shows the data taken from Shorts article in blue and my calculation s in blackAverage Wounds size of Zymotic & Z/S*1000*12 Radius W/S*1000*12 Radius O/S*1000*12 Radius (Az) (Aw) (Ao) for army diseases injuries Other for for Month (S) (Z) (W) (O) (1 d.p.) Zymtotic (1 d.p.) Wounds (1 d.p.) Other Apr-54 8571 1 0 5 1.4 2.3 0.0 0.0 7.0 5.2 May-54 23333 12 0 9 6.2 4.9 0.0 0.0 4.6 4.2 Jun-54 28333 11 0 6 4.7 4.2 0.0 0.0 2.5 3.1 Jul-54 28722 359 0 23 150.0 23.9 0.0 0.0 9.6 6.1 Aug-54 30246 828 1 30 328.5 35.4 0.4 1.2 11.9 6.7 Sep-54 30290 788 81 70 312.2 34.5 32.1 11.1 27.7 10.3 Oct-54 30643 503 132 128 197.0 27.4 51.7 14.1 50.1 13.8 Nov-54 29736 844 287 106 340.6 36.1 115.8 21.0 42.8 12.8 Dec-54 32779 1725 114 131 631.5 49.1 41.7 12.6 48.0 13.5 Jan-55 32393 2761 83 324 1022.8 62.5 30.7 10.8 120.0 21.4 Feb-55 30919 2120 42 361 822.8 56.1 16.3 7.9 140.1 23.1 Mar-55 30107 1205 32 172 480.3 42.8 12.8 7.0 68.6 16.2Az is the death rate per 1000 per annum from disease, Aw is the death rate per 1000 per annum from wounds and Ao is the death rate per 1000 per annum from other causes. For 2 this diagram there are 12 divisions so each sector has an angle of = and an area of 12 6 12A 1 2 2 . r = r . So for each radius r = 26 12Mathematics SL and HL teacher support material9Example 6 Student workI will show my final polar area diagram side by side with Nightingales original versionFigure 7. Nightingales original Diagram of the causes of mortality in the army in the tocopherol and my recreation. I have to admit that I felt rather proud once I had done this However, looking at the September 1854 wedge I realised that the two diagrams didnt correspond. In Nightingales original diagram I can see that there are more deaths from other causes than from wounds. In my version there are fewer deaths from other causes than fromwounds. All other versions of the original in other articles I looked at ( Gill and Gill3, Brasseur4, OConnor5, Woodbury7, Riddle10, Small11, Lienhard6) are as the original, but the table in Short definitely shows fewer deaths fro m other causes than from wounds8. Conclusion I started out to try to lean how to urge the polar area diagram which Florence Nightingale made to communicate to other people just how mediocre the situation was in army hospitals. This diagram shouts a need for reform. Look at it.The blue represents deaths which could be avoided with a bit of organisation and care. The red represents deaths due to the actual battles. Florence Nightingale had copies of her report containing her diagrams published at her own expense and sent them to doctors, army officers, members of parliament and the Queen. Following her persistent lobbying a commission was set up to improve military barracks and hospitals, sanitary codes were established and procedures were put in place for more set up collection of medical statistics4. It is a very shocking picture with a huge snowball of social change behind it. It has been an exciting adventure to drill down to a real understanding of its construction.Mathematic s SL and HL teacher support material10Example 6 Student workHowever, the biggest lesson I have learnt from this research is that you cant trust what you read. As I have argued in the main text, I am moderately sure that Brasseur thought the colours of the second diagram did not overlap4, I think OConnor got his death rates wrong for January 18555, and I think Short may have transcribed the data incorrectly for September 18548. According to Brasseur, Florence Nightingale cross checked her data and was regular about addressing objections to her analysis4. Everyone can make mistakes, and errors can propagate if we just quote what someone else says without looking for corroboration. I have been left with a desire to find out more about this tenacious woman who wouldnt let society mould her into a genteel wife. Also, if I ever get the chance, I would like to get a look at one of the 2000 copies of Notes on Matters Affecting the Health, Effiency and Hospital disposal of the British Army . Founded Chiefly on the Experience of the Late War which Florence Nightingale had published in 1858, to see the actual table of data and check the numbers for September 1854.Mathematics SL and HL teacher support material11Example 6 Student workReferences/Bibliography1.Duthie, Eric ed. The Childrens Book of Famous Lives.Odhams evoke Ltd, London 1957 2. Du Garde Peach, L. Florence Nightingale. Wills & Hepworth Ltd, Loughborough, 1959 3. Gill, Christopher J. and Gill, Gillian C. Nightingale in Scutari Her Legacy Reexamined Center for Internatinal Health, Boston University School of Public Health, Boston, Massachusetts, viewed 26th July 2009 4. Brasseur, Lee, Florence Nightingales Visual Rhetoric in the Rose Diagrams. Technical Communication Quarterly, 14(2), 161-182, Lawrence Erlbaum Associates, Inc, 2005, viewed 26th July 2009 5. OConnor, J.J. and Robertson, E.F., Florence Nightingale. viewed 26 July 2009 6. Lienhard, John H., Nightingales Graph, The Engines of Our Ingenuity. 2002 v iewed 26th July 2009 7. Woodbury, Henry, Nightingales Rose. American Physical SocietyLaunches Dynamic Diagrams plan of Physical Review Letters, January 9, 2008, 405 pm, filed under Information Design, Visual Explanation Viewed 30 July 2009 8. Short, Ian, Mathematics of the Coxcombs. November 5th, 2008 viewed thirtieth July 2009 9. Lewi, Paul J. Florence Nightingale and Polar Area Diagrams, Speaking of Graphics. 2006 www.datascope.be/sog/SOG-Chapter5.pdf viewed 26th July 2009 10. Riddle, Larry, Polar-Area Diagram. 2006 , viewed 26th July 2009 11. Small, Hugh, Florence Nightingales statistical diagrams. Presentation to Research Conference organized by the Florence Nightingale Museum St. Thomass Hospital, 18th March 1998 viewed 26th July 2009Mathematics SL and HL teacher support material