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# What If the Nazis Bombed Britain Earlier? Now There's Math for That

Statistical modeling calculates how the Battle of Britain could have played out if the Luftwaffe changed its tactics.

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• Math could settle, once and for all, the speculative debate over the Battle of Britain.
• The battle was a pivotal point where Britain held off the invading Nazi Luftwaffe.
• Statistical bootstrapping lets mathematicians accurately multiply sample size into something much larger and more precise.

Researchers have built a mathematical model to try to understand how changes would affect the Battle of Britain in World War II. Both armchair and pedigreed historians have argued for decades that small changes in the strategy of Nazi Germany’s air force, the Luftwaffe, could have turned a British victory into a Nazi one. Mathematicians from the University of York worked on the model, which uses a statistical method called bootstrapping.

If the Luftwaffe began earlier and focused its bombing on Britain's dozens of airfields, the researchers say, outcomes from any beginning likelihood would be drastically different.

"[H]ad the likelihood of a British victory in the actual battle been 50 [percent], these two tactical changes would have reduced it to less than 10 [percent]. If the real probability of British victory was 98 [percent], the same changes would have reduced this to just 34 [percent]," the press release reads.

Coauthor Jamie Wood said in a release that the bootstrapping technique allowed his team to effectively move a slider on parameters like the date when the Luftwaffe first attacked, and how long each facet of their attack lasted. The Battle of Britain was a major strategic victory for the Allies, but also a huge morale victory in Britain, which has been defined for thousands of years by its position as an island fortified by defenses like Dover Castle.

Monobody aircraft—instead of the biplanes of the Red Baron—were new in World War II and changed the landscape of war completely. Instead of single or small groups of planes doing targeted missions, dedicated aeronautic branches of the military waged entire air campaigns. To defend itself from the air, Britain had to adapt millennia of ground defense and turn to the sky. With so many new and rapidly changing technologies and dynamics in play, and so much pivoting on it within World War II, this single battle is heavily studied with a speculative eye.

Enter mathematics. “The weighted bootstrap technique allowed us to model alternative campaigns in which the Luftwaffe prolongs or contracts the different phases of the battle and varies its targets,” Wood said in the statement. In bootstrapping, statisticians can pull samples from an overall pool of data and replace them in the data, meaning the same data point can be used again.

That means one large dataset can turn into thousands of bootstrap samples, pulling a different random sampling each time and multiplying one economical sample study into a more robust dataset. It’s a bunch of new sample surveys that use the original sample as the population. And computers have made bootstrapping into a powerful tool. In a way, bootstrapping works the way TV forensics use “zoom and enhance,” except that bootstrapping is real: The more bootstrap sets you continue to add, the larger the size and the more precise the statistical inferences.

The new paper serves to introduce the usefulness of this kind of analysis. The study showed that certain changes shifted the probable outcomes of particular points during the Battle of Britain, but more importantly, “We show here how the statistical technique of weighted bootstrapping can be used to create a new quantitative basis to help address such controversies,” the authors explain.

But if centuries of kibbitzing and beer-fueled debates are any indication, even math can’t stop speculation about what might have been.