Alex's Adventures in Numberland - Alex Bellos [159]
Galton called this phenomenon ‘regression towards mediocrity in hereditary stature’. The concept is now more generally known as regression to the mean. In a mathematical context, regression to the mean is the statement that an extreme event is likely to be followed by a less extreme event. For example, when I measured a Greggs baguette and got 380g, a very low weight, it was very likely that the next baguette would weigh more than 380g. Likewise, after finding a 420g baguette, it was very likely that the following baguette would weigh less than 420g. The quincunx gives us a visual representation of the mechanics of regression. If a ball is put in at the top and then falls to the furthest position on the left, then the next ball dropped will probably land closer t the middle position – because most of the balls dropped will land in the middle positions.
Variation in human height through generations, however, follows a different pattern from variation in baguette weight through the week or variation in where a quincunx ball will land. We know from experience that families with above-average-sized parents tend to have above-average-sized kids. We also know that the shortest guy in the class probably comes from a family with adults of correspondingly diminutive stature. In other words, the height of a child is not totally random in relation to the height of his parents. On the other hand, the weight of a baguette on a Tuesday probably is random in relation to the weight of a baguette on a Monday. The position of one ball in a quincunx is (for all practical purposes) random in relation to any other ball dropped.
In order to understand the strength of association between parental height and child height, Galton came up with another idea. He plotted a graph with parental height along one axis and child height along the other, and then drew a straight line through the points that best fitted their spread. (Each set of parents was represented by the height midway between mother and father – which he called the ‘mid-parent’). The line had a gradient of . In other words for every inch taller than the average that the mid-parent was, the child would only be of an inch taller than the average. For every inch shorter than the average the mid-parent was, the child would only be of an inch shorter than the average. Galton called the gradient of the line the coefficient of correlation. The coefficient is a number that determines how strongly two sets of variables are related. Correlation was more fully developed by Galton’s protégé Karl Pearson, who in 1911 set up the world’s first university statistics department, at University College London.
Regression and correlation were major breakthroughs in scientific thought. For Isaac Newton and his peers, the universe obeyed deterministic laws of cause and effect. Everything that happened had a reason. Yet not all science is so reductive. In biology, for example, certain outcomes – such as the occurrence of lung cancer – can have multiple causes that mix together in a complicated way. Correlation provided a way to analyse the fuzzy relationships between linked sets of data. For example, not everyone who smokes will develop lung cancer, but by looking at the incidence of smoking and the incidence of lung cancer mathematicians can work out your chances of getting cancer if you do smoke. Likewise, not every child from a big class in school will perform less well than a child from a small class, yet class sizes do have an impact on exam results. Statistical analysis opened up whole new areas of research – in subjects from medicine to sociology, from psychology to economics. It allowed us to make use of information without knowing exact causes. Galton’s original insights helped make statistics a respectable field: ‘Some people hate the very name of statistics, but I find them full of beauty