Special days, anniversaries, and birthdays provide a great opportunity to commemorate people who deserve more thought than they usually receive. Thanks to these occasions we can discover how their solutions have influenced our modern everydays. What’s way more important, we can meet attitudes and philosophies worth adopting as a researcher, software developer, and even as a businessman.

Therefore, using the occasion of his birthday as a reminder, we devote today’s entry to Abraham Wald. The great mathematician advanced the field of decision theory, geometry, and econometrics, and laid the foundation of statistical sequential analysis.

He was born on 31 October 1902 in Cluj in a Hungarian Jewish family. As antisemitism was gaining more and more ground, he had to emigrate to the United States after finishing his university studies in Vienna. Here, he was studying and teaching at Columbia University. During World War II, his statistical knowledge and skills were counted on in solving military problems.

Abraham Wald and the Missing Bullet Holes

This is the time when the most famous anecdote about Wald takes place, the one with the missing bullet holes on US military aircrafts. As the story goes, one day the Statistical Research Group with Wald as a valuable member received the task of determining which part of military aircrafts needs reinforcement based on the planes returning from mission. To preserve a reasonable weight, extra shield could be applied only on the weakest spots of aircrafts. Hits were distributed on the returning planes unevenly. Engines, for instance had fewer holes, while fuselage had more.

The Air Force was adamant to reinforce those parts of the aircrafts which displayed more hits. However, Wald immediately pointed out that they were about to make a huge mistake. He presumed that the planes were hit evenly in combat. Therefore, fewer bullet holes on the engine of returned aircrafts actually meant that a shot engine was more likely to be fatal than an injured fuselage, as these aircrafts couldn’t return from combat. Consequently, parts displaying fewer holes needed more protection.

Missing Data Instead of Missing Bullet Holes

No doubt, the anecdote is based on a true story; reinforcing the engines proved to be a good decision. However, the story has gained some air of Hollywood over the years, which paints a wrong image of Abraham Wald. The sign of Wald’s genius isn’t the unhesitant, immediate good answer. No matter how dramatic gesture it may be, it lacks scientific creditability.

It is possible that the great Hungarian mathematician suspected engines as weak points as soon as he took his first glimpse at the available data. However, he stayed away from any unfounded assumption. A hundred-page-long document showcases the thought process of Wald. He used carefully selected theories and equations to answer the question of aircrafts. Complicated calculations – conducted without modern computational technology – helped him find the most probable assumption.

That’s why the real character of Abraham Wald isn’t reflected by the above anecdote, but more by the words of his fellow statistician, Jacob Wolfowitz:


Wald brought to statistics a very high degree of mathematical ability and knowledge. Along with this, and in spite of his abstract and theoretical bent and predilections, he never, in any statistical investigation, lost sight of the fact that there was a question to be answered and a decision to be made. […] This, in my opinion, was the key to his success – a high level of mathematical talent of the most abstract sort, and a true feeling for, and insight into, practical problems.

Abraham Wald wasn’t the hectic genius fit for big screen, but a humble, brilliant mathematician, who applied his knowledge, skills, and talent to solve real problems. His abstract thinking brought practical solutions to life.

Abraham Wald’s Influence

As far as practical solutions are concerned, Abraham Wald’s Sequential Probability Ratio Test (SPRT) – a statistical decision-making algorithm – is still applied today in several fields.

Quality Control

SPRT is a significant element of quality control. Especially in calculating the sufficient size of the production sample, upon which safe assumptions can be made about the entire batch. The method is implemented both in manufacturing and agriculture. For example, in several South-Eastern Asian regions SPRT provides the frame for measuring and estimating the Cocoa Pod Borer infection on cocoa fields.


Software solutions based on SPRT also play a major role in personalized education, as Computerized Adaptive Testing (CAT) is built on this algorithm. In such testing the difficulty level of the exam changes in accordance with the right or wrong answers given to easier and more complicated questions. The adjusted set of questions result in an exam experience customized to the present knowledge and skillset of the student.

One of the many benefits of CAT method is shorter testing time with more accurate results. Additionally, CAT exams motivate students more than regular testing methods, as examinees don’t need to struggle with devastatingly difficult tasks or get bored with unchallenging questions. They are granted with a sense of achievement.

On Our Company

Abraham Wald’s Sequential Probability Ratio Test is of course used way beyond the above examples, and there certainly are many possible applications yet to be discovered. We, for instance, considered incorporating it to our connectivity manager software solution designed for cell tower base stations to calculate the number and ideal position of serving wire centers needed to maintain the given network. For smaller networks the method worked seamlessly since it swiftly highlighted the most cost-effective deployment method. Finally, we decided on a different approach due to considerations for larger networks.

Even though Abraham Wald’s algorithm hasn’t made it into the final version of our connectivity manager software, we seek to adopt Wald’s take on science. Learning from the great statistician, we always keep our focus on the given task, the question to be answered. We carefully select and employ the available technology and abstract knowledge to develop the most practical solution.

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