The American mathematician Jordan Ellenberg of the University of Wisconsin–Madison relates how entire classes of students were given assessment tests to measure teaching quality. Someone had the idea of ranking the results by class size. Classes with fewer students topped the rankings, and some educators argued that teachers in smaller classes could give students more individual attention, leading to better results. The interpretation seemed plausible… until a statistician brought fluctuations into the picture. By chance, some students have already taken the test and earn a high score, while others have a migraine or have misplaced their pen and are thrown off as a result. These random fluctuations have a greater effect when there are fewer individuals: a single good or bad result makes a relatively larger difference to the overall result, which explained why some small classes ranked higher. To convince his audience, the statistician pointed out that classes with fewer students also appeared at the bottom of the rankings!
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**Proportions of heads in a coin-tossing experiment. For each of the ten thousand experiments consisting of two hundred tosses, the proportion of heads is calculated. The probability of tossing two hundred consecutive heads is 1/2200, or approximately 10–60. Extremely rare, but not impossible.
In quantitative terms, statisticians have proved that the standard deviation, which measures the magnitude of fluctuations around the mean, is inversely proportional to the square root of the number of elements in the sample.
