In 1832, a Prussian military analyst named Carl von Clausewitz explained that “three quarters of the factors on which action in war is based are wrapped in a fog of . . . uncertainty.” The best military commanders seemed to see through this “fog of war,” predicting how their opponents would behave on the basis of limited information. Sometimes, though, even the wisest generals made mistakes, divining a signal through the fog when no such signal existed. Often, their mistake was endorsing the law of small numbers—too readily concluding that the patterns they saw in a small sample of information would also hold for a much larger sample.
Both the Allies and Axis powers fell prey to the law of small numbers during World War II. In June 1944, Germany flew several raids on London. War experts plotted the position of each bomb as it fell, and noticed one cluster near Regent’s Park, and another along the banks of the Thames. This clustering concerned them, because it implied that the German military had designed a new bomb that was more accurate than any existing bomb. In fact, the Luftwaffe was dropping bombs randomly, aiming generally at the heart of London but not at any particular location over others. What the experts had seen were clusters that occur naturally through random processes—misleading noise masquerading as a useful signal.
That same month, German commanders made a similar mistake. Anticipating the raid later known as D-Day, they assumed the Allies would attack—but they weren’t sure precisely when. Combing old military records, a weather expert named Karl Sonntag noticed that the Allies had never launched a major attack when there was even a small chance of bad weather. Late May and much of June were forecast to be cloudy and rainy, which “acted like a tranquilizer all along the chain of German command,” according to Irish journalist Cornelius Ryan. “The various headquarters were quite confident that there would be no attack in the immediate future. . . . In each case conditions had varied, but meteorologists had noted that the Allies had never attempted a landing unless the prospects of favorable weather were almost certain.” The German command was mistaken, and on Tuesday, June 6, the Allied forces launched a devastating attack amidst strong winds and rain.
The British and German forces erred because they had taken a small sample of data too seriously: The British forces had mistaken the natural clustering that comes from relatively small samples of random data for a useful signal, while the German forces had mistaken an illusory pattern from a limited set of data for evidence of an ongoing, stable military policy. To illustrate their error, imagine a fair coin tossed three times. You’ll have a one-in-four chance of turning up a string of three heads or tails, which, if you make too much of that small sample, might lead you to conclude that the coin is biased to reveal one particular outcome all or almost all of the time. If you continue to toss the fair coin, say, a thousand times, you’re far more likely to turn up a distribution that approaches five hundred heads and five hundred tails. As the sample grows, your chance of turning up an unbroken string shrinks rapidly (to roughly one-in-sixteen after five tosses; one-in-five-hundred after ten tosses; and one-in-five-hundred-thousand after twenty tosses). A string is far better evidence of bias after twenty tosses than it is after three tosses—but if you succumb to the law of small numbers, you might draw sweeping conclusions from even tiny samples of data, just as the British and Germans did about their opponents’ tactics in World War II.
Of course, the law of small numbers applies to more than military tactics. It explains the rise of stereotypes (concluding that all people with a particular trait behave the same way); the dangers of relying on a single interview when deciding among job or college applicants (concluding that interview performance is a reliable guide to job or college performance at large); and the tendency to see short-term patterns in financial stock charts when in fact short-term stock movements almost never follow predictable patterns. The solution is to pay attention not just to the pattern of data, but also to how much data you have. Small samples aren’t just limited in value; they can be counterproductive because the stories they tell are often misleading.