Author Archives: u0614258

Who knew that an unlikely friendship and a few games of cricket with one of the greatest mathematicians in the early 20th Century could lead to a breakthrough in population genetics?

Today, it is almost commonplace for us in the scientific community to accept the influence natural selection and Mendelian genetics have on one another, however for the majority of human history this was not the case. Up until the early 1900s, many scientists believed that these concepts were nothing more than two opposing and unassociated positions on heredity. Scientists were torn between a theory of inheritance (a.k.a. Mendelian genetics) and a theory of evolution through natural selection. Although natural selection could account for variation, which inheritance could not, it offered no real explanation on how traits were passed on to the next generation. For the most part, scientists could not see how well Mendel’s theory of inheritance worked with Darwin’s theory of evolution because they did not have a way to quantify the relationship. It was not until the introduction of the theorem of genetic equilibrium that biologists acquired the necessary mathematical rigor to show how inheritance and natural selection interacted. One of the men who helped provide this framework was G.H. Hardy.

G. H. Hardy. Image: public domain, via Wikimedia Commons.

G. H. Hardy. Image: public domain, via Wikimedia Commons.

Godfrey Harold (G.H.) Hardy was a renowned English mathematician who lived between 1877-1947 and is best known for his accomplishments in number theory and for his work with the another great mathematician, Srinivasa Ramanujan. For a man who was such an outspoken supporter of pure mathematics and abhorred any practical application of his work[5], it is ironic that he should have such a powerful influence on a field of applied mathematics and help shape our very understanding of population genetics.

How did a pure mathematician come to work on population genetics? Well it all started with a few games of cricket. Whilst teaching at the University of Cambridge, Hardy would often interact with professors in other departments through friendly games of cricket and evening common meals [1]. It was through these interactions that Hardy came to know Reginald Punnett, cofounder of the genetics department at Cambridge and developer of Punnett Squares, which are named for him, and developed a close friendship with him[13].

Punnett, being one of the foremost experts in population genetics, was in the thick of the debate over inheritance vs. evolution. His interactions with contemporaries like G. Udny Yule, made him wonder why a population’s genotype, or the genes found in each person, did not eventually contain only variations, known as alleles, of a particular gene that are dominant. This was the question he posed to Hardy in 1908, and Hardy’s response was nigh on brilliant. The answer was so simple that it almost seemed obvious. Hardy even expressed that “I should have expected the very simple point which I wish to make to have been familiar to biologists’’ [4]. His solution was so simple in fact that unbeknownst to him, another scientist had reached the same conclusion around the same time in Germany [17]. In time, this concept would be known as Hardy-Weinberg Equilibrium (HWE).

In short, HWE asserts that when a population is not experiencing any genetic changes that would cause it to evolve, such as genetic drift, gene flow, selective mating, etc., then the allele (af) and genotypic frequencies (gf) will remain constant within a given population (P’). To calculate the gf for humans, a diploid species that receives two complete sets of chromosomes from their parents, we simply look at the proportion of genotypes in P’.

0 < gf < 1

To calculate the af, we look at the case where either the gene variation is homozygous and contains two copies of the alleles (dominant—AA || recessive—aa) or heterozygous and only has one copy of each allele (Aa). P’ achieves “equilibrium” when these frequencies do not change.

Hardy’s proof of these constant frequencies for humans, a diploid species that receives two complete sets of chromosomes from its parents, is as follows[1][4]:

If parental genotypic proportions are p AA: 2q Aa: r aa, then the offspring’s would be (p + q)2: 2(p + q)(q + r): (q + r)2. With four equations (the three genotype frequencies and p + 2q + r = 1) and three unknowns, there must be a relation among them. ‘‘It is easy to see that . . . this is q2 = pr” 

Which is then broken down as:

q =(p + q)(q + r) = q(p + r) + pr + q2

Then to:

q2 = q(1- p – r) – pr = 2q2 – pr     ——->   q2 = pr

In order to fully account for the population, the gf and af must sum to 1. And, since each subsequent generation will have the same number of genes, the frequencies remain constant and follows either a binomial or multinomial distribution.

One important thing to keep in mind, however, is that almost every population is experiencing some form of evolutionary change. So, while HWE shows that the frequencies don’t change or disappear, it is best used as a baseline model to test for changes or equilibrium.

When using the Hardy-Weinberg theorem to test for equilibrium, researchers divide the genotypic expressions into two homozygous events: HHο and hhο. The union of each event’s frequency ( f ), is then calculated to give the estimated number of alleles (Nf). In this case, the expression for HWE could read something like this:

Nf = f(HHο)  f(hhο)

However, another way to view this expression is to represent the frequency of each homozygous event as single variable, i.e. p and q. Using p to represent the frequency of one dominant homozygous event (H) and q to represent the frequency of one recessive homozygous event (h), gives the following: p = f(H) and q = f(h). It then follows that p² = f(HHο) and q² = f(hhο). By using the Rule of Addition and Associative Property to calculate the union of the two event’s frequencies, we are left with F = (p+q)². Given that the genotype frequencies must sum to one, the prevailing expression for HWE emerges when F is expanded:

Fp² +2pq + q² = 1

Using this formula, researchers can create a baseline model of P’ and then identify evolutionary pressures by comparing any subsequent frequencies of alleles and genotypes (F) to F. The data can then be visually represented as a change of allele frequency with respect to time.

HWE represents the curious situation that populations experience when their allele frequencies change. This situation is realized by first assuming complete dominance, then calculating the frequency of alleles, and then using the resultant number as a baseline with which to compare any subsequent values. Although there are some limitations on how we can use HWE—namely, identifying complete dominance, the model is very useful in identifying any evolutionary pressures a population may be experiencing and is one of the most important principles in population genetics. Developed, in part, by G.H. Hardy, it connected two key theories: the theory of inheritance and the theory of evolution. Although, mathematically speaking, his observation/discovery was almost trivial, Hardy provided the mathematical rigor the field sorely needed in order to see that the genotypes didn’t completely disappear and, in turn, forever changed the way we view the fields of biology and genetics.


  1. Edwards, A. W. F. “GH Hardy (1908) and Hardy–Weinberg Equilibrium.”Genetics3 (2008): 1143-1150.
  2. Edwards, Anthony WF. Foundations of mathematical genetics. Cambridge University Press, 2000.
  3. Guo, Sun Wei, and Elizabeth A. Thompson. “Performing the exact test of Hardy-Weinberg proportion for multiple alleles.” Biometrics(1992): 361-372.
  4. Hardy, Godfrey H. “Mendelian proportions in a mixed population.” Science706 (1908): 49-50.
  5. Hardy, G. H., & Snow, C. P. (1967). A mathematician’s apology. Reprinted, with a foreword by CP Snow. Cambridge University Press.
  6. Pearson, Karl. “Mathematical contributions to the theory of evolution. XI. On the influence of natural selection on the variability and correlation of organs.”Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character(1903): 1-66.
  7. Pearson, K., 1904. Mathematical contributions to the theory of evolution. XII. On a generalised theory of alternative inheritance, with special reference to Mendel’s laws. Philos. Trans. R. Soc. A 203 53–86.
  8. Punnett, R. C., 1908. Mendelism in relation to disease. Proc. R. Soc. Med. 1 135–168.[PMC free article] [PubMed]
  9. Punnett, R. C., 1911. Mendelism. Macmillan, London.
  10. Punnett, R. C., 1915. Mimicry in Butterflies. Cambridge University Press, Cambridge/London/New York.
  11. Punnett, R. C., 1917. Eliminating feeblemindedness. J. Hered. 8 464–465.
  12. Punnett, R. C., 1950. Early days of genetics. Heredity 4 1–10.
  13. Snow, C. P., 1967. G. H. Hardy. Macmillan, London.
  14. Stern, C., 1943. The Hardy–Weinberg law. Science 97 137–138. [PubMed]
  15. Sturtevant, A. H., 1965. A History of Genetics. Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY.
  16. Weinberg, Wilhelm. “Über vererbungsgesetze beim menschen.” Molecular and General Genetics MGG1 (1908): 440-460.
  17. Weinberg, W. “On the demonstration of heredity in man.” Boyer SH, trans (1963) Papers on human genetics. Prentice Hall, Englewood Cliffs, NJ(1908).

Figure: Wikimedia Commons


The Passionate Statistician

Fig. 1 Portrait of Florence Nightingale

Almost everyone has heard of Florence Nightingale: the Nurse; very few people have heard about Florence Nightingale: the Statistician. Ironically, however, she is one of the most important statisticians to have ever graced the field. Yes, her improvements to sanitation were revolutionary, and surely saved countless lives, but how she was able to bring about those improvements was equally innovative.

Picture this: It’s 1855 during the Crimean War. The air is rank and humid, filled with the smell of blood and sulfur—the fresh, salty aroma of the adjacent Black Sea long forgotten. Although you are required to help care for the many wounded soldiers, you are also charged with collecting, accurately keeping, and even analyzing army mortality records.

After months of attending to these records and conducting your analyses, you are faced with an appalling, yet undeniable, truth: more soldiers are dying from poor sanitation than from combat. In fact, the mortality rates from disease are so great that “during the first seven months of the Crimean campaign, a mortality rate of 60 per cent. per annum from disease alone occurred, a rate of mortality which exceeded even the Great Plague in London…”[1]

Florence Nightingale (FN) believed that allowing 16,000 men[2] to die from causes that were easily prevented with improved sanitation was almost akin to murder, and it would be equally criminal to do nothing to prevent these needless deaths from happening again[3]. She also felt it was downright disgraceful, if not scandalous, for a nation that considered itself the epitome of civilization to be this neglectful of their sanitation policy[4]. Sadly, these poor sanitary conditions were not just associated with the battlefield. As the war ended and FN returned home, she found that army barracks and even the hospitals experienced equally appalling mortality rates from disease[5].

She knew sanitation policies needed to be improved. She knew her statistical analysis was the best tool she could use to convince others of this need. Yet, she also knew that she would have to develop a better way to convey her data to the general public. She realized that although the empirical evidence would easily convince those who knew how to read the data, publishing it in the traditional way would severely limit the amount of people who could actually utilize the information. A smaller group of supporters meant that it would take more time to bring about the necessary improvements, and more time meant that more people would die. No, if her campaign was to succeed, she needed to have something that enabled almost everyone to easily draw the same conclusion she had for herself: their nation’s sanitation methods were claiming more lives than their enemy’s artillery.

She decided to model her data as a graphical illustration, specifically as a Rose chart (also known as a Polar Area or Coxcomb chart).

FN 1858 Rose Chart Fig. 2 Nightingale’s 1858 Rose Chart that graphically illustrates mortality rates Text: The Area of the blue, red, & black wedges are each measured from the centre as the common vertex. The blue wedges measured from the centre of the circle represent area for area the deaths from Preventible or Mitigable Zymotic diseases; the red wedges measured from the centre the deaths from wounds; & the black wedges measured from the centre the deaths from all the other causes. The black line across the red triangle in Nov. 1854 marks the boundary of the deaths from all other causes during the month. In October 1854, & April 1855; the black area coincides with the red; in January & February 1856, the blue coincides with the black. The entire areas may be compared by following the blue, the red & the black lines enclosing them.

At the time, these illustrations were quite an extraordinary feat, and very few statisticians had previously attempted to use these representations. This is because if the statistician was not careful with their calculations, then the representations could very easily mislead the observer. In fact, even FN was mislead at one point. Her initial analysis of the army’s mortality records led her to conclude that it was malnutrition, and not sanitation, that was the major cause of death during the Crimean campaign[6].

As can be seen in this link here [7], a Rose chart consists of multiple wedges. These wedges are called sectors and each one represents a different category in one’s data. It is important to note that the individual value of each category is represented by the sector’s area and not the radius. This was actually what caused FN to misinterpreted her data at first, as she used the radius, instead of the area, to represent the value of each sector. Despite the fact that the calculations can make it tricky to accurately represent numerical data, the chart can, on-the-other-hand, simplify comparisons and enable observers to easily identify causation[8].

Florence Nightingale was a true devotee to both statistical analysis and improved health care. She was an innovative woman, who pioneered the means with which to effectively communicate statistics’ findings that describe social phenomena. She was both revered and admired by her compatriots in the field of mathematics, and her great admiration for statistics earned her the nickname, “The Passionate Statistician.”


[1] Pearson, Egon Sharpe, Maurice George Kendall, and Robert Lewis Plackett, eds. Studies in the History of Statistics and Probability. Vol. 2. London: Griffin, 1970.

[2] Nightingale, F. (1858) Notes on Matters Affecting the Health, Efficiency, and Hospital Administration of the British Army Harrison and Sons, 1858

[3] Nightingale, Florence. Letter to Sir John McNeill. 1 March 1857. Letter H1/ST/NC3/SU74, copy ADD MSS 45768 f29 of Florence Nightingale: The Crimean War: Collected Works of Florence Nightingale, Vol.14. Ed. Lynn McDonald. Ontario, Canada: Wilfrid Laurier Univ. Press, 2010. 498-500. Print.

[4] Nightingale, F. (1859) A Contribution to the Sanitary History of the British Army during the Late War with Russia London: John W. Parker and Son.

[5] Nightingale, Florence. Notes on hospitals. Longman, Green, Longman, Roberts, and Green, 1863.

[6] Cohen, I.B. The Triumph of Numbers: How Counting Shaped Modern Life, W. H. Norton, 2006.

[7] ims25. “Mathematics of the Coxcombs” Understanding Uncertainty,11 May 2008. Web. 18 Dec. 2014.

[8] Small, Hugh (1998). Florence Nightingale: Avenging Angel. Constable, London.

[9] Heyde, Christopher C., and Eugene Seneta, eds. Statisticians of the Centuries. Springer, 2001.

[10] Farr, William. Letter to Florence Nightingale. (1857-1912). MS.8033, Nos. 1-20. Florence Nightingale and William Farr Collection. The Wellcome Library, Archives. London. Print.

[Fig. 1] Wikicommons: Duyckinick, Evert A. Portrait Gallery of Eminent Men and Women in Europe and America. New York: Johnson, Wilson & Company, 1873.

[Fig. 2] Wikicommons: Public Domain