by When analyzing a data set, one of the first steps many people take is to calculate a mean. You could compare your height to the average height of the people you live with, or brag about the batting average of your favorite baseball player. But while the average can help you study a data set, it has important limitations.
Due to the uses of media that ignore these boundaries, there are serious issues such as discrimination, injury and even life threatening accidents.
For example, the US Air Force used to design its planes for “the common man,” but abandoned the practice when pilots were unable to control their aircraft. The mean has many uses, but it tells you nothing about the variance in a data set.
I am a disciplinary educational researcher, meaning I study how people learn, with a focus on engineering. My research includes studying how engineers use media in their work.
Using the mean to summarize data
The medium has been around for a long time, and its use has been documented as early as the ninth or eighth century BCE. Early on, the Greek poet Homer estimated the number of soldiers on ships by taking an average.
The early astronomers wanted to predict the future locations of the stars. But to make these predictions, they first needed accurate measurements of the stars’ current positions. Multiple astronomers would independently take position measurements, but often arrived at different values. Since a star has only one true position, these discrepancies were a problem.
Galileo in 1632 was the first to try a systematic approach to address these measurement differences. His analysis was the beginning of error theory. Error theory helps scientists reduce uncertainty in their measurements.
Error theory and the mean
Under error theory, researchers interpret that a series of measurements leads to a true value that is contaminated by error. In astronomy, a star has a true position, but early astronomers may have had unsteady hands, blurry telescope images and bad weather – all sources of error.
To deal with error, researchers often assume that the measurements are unbiased. In statistics, this means that they are evenly distributed around a central value. Unbiased measurements still contain error, but can be combined to better estimate the true value.
Say three scientists have made three measurements. When viewed separately, their measurements may appear random, but when unbiased measurements are added together, they are evenly distributed around a mean value: the mean.
When the measurements are unbiased, the mean will usually sit in the middle of each measurement. In fact, we can show mathematically that it is the closest average of all possible measurements. For this reason, the average is an excellent tool for dealing with measurement errors.
Statistical thinking
The theory of error, in its time, was considered revolutionary. Other scientists appreciated the accuracy of astronomy and tried to bring the same approach to their disciplines. The 19th-century scientist Adolphe Quetelet applied ideas from error theory to the study of people and introduced the idea of taking the average person’s height and weight.
The mean helps to make comparisons between groups. For example, if a data set is averaged for the heights of men and women it can be shown that the men in the data set are taller – on average – than the women. However, the average does not tell us everything. In the same data set, we would probably find individual females that are taller than individual males.
Therefore, you can only estimate the average. You should also consider the spread of values by thinking statistically. Statistical thinking is defined as thinking carefully about variability – or the tendency of measured values to be different.
For example, one example of variation is different astronomers measuring the same star and recording different positions. Astronomers had to think carefully about where their diversity came from. Since a star has only one true position, they could safely assume that their divergence was an error.
It makes sense to take the average of the measurements when variation comes from sources of error. But researchers must be careful when interpreting the mean when there is real variation. For example, in the height sample, individual females can be taller than individual males, even if males are taller on average. Focusing on the average alone neglects diversity, which has caused serious issues.
Quetelet not only derived the practice of computing media from error theory. He also accepted the assumption of one true value. He raised the ideal of “the average man” and suggested that human variability was a fundamental error – that is, not ideal. Quetelet, there is something wrong with you if you are not exactly the average height.
Researchers who study social norms note that Quetelet’s ideas about “the average man” contributed to the modern meaning of the word “normal” – normal height, as well as normal behavior.
Some people used these ideas, such as the early statisticians, to divide populations into two halves: people who are better in some way and people who are less.
For example, the eugenics movement – a despicable attempt to prevent “inferior” people from having children – traces its thinking to these ideas about “normal” people.
While Quetelet’s idea of diversity as error supports discriminatory practices, Quetelet-like uses of the medium have direct links to modern engineering failures.
Failures of the medium
In the 1950s, the US Air Force designed its aircraft for “the average man.” He assumed that an airplane designed for average height, average arm length and average along several other key dimensions would work for most pilots.
This decision contributed to as many as 17 pilots crashing in a single day. Although the aircraft could be operated perfectly by “the average man”, there was considerable variation along the way. A shorter pilot would have difficulty seeing, and a pilot with longer arms and legs would have to strain themselves to fit.
Although the Air Force assumed that most of its pilots would be close to the average along all key dimensions, it found that out of 4,063 pilots the average was zero.
The Air Force solved the problem by designing for diversity – it designed adjustable seats to accommodate the true diversity among pilots.
Although adjustable seats seem obvious now, this “middle man” idea still causes problems today. In the United States, women experience about a 50% higher odds of serious injury in car accidents.
The Government Accountability Office blames this disparity on crash test practices, where female passengers are crudely represented using a scaled-down version of a male dummy, similar to the Air Force’s “average man.” The first female crash test dummy was introduced in 2022 and is yet to be adopted in the US
The medium is useful, but it has limitations. For estimating true values or making comparisons across groups, the mean is powerful. However, for individuals who show true diversity, the average is not that much.
This article is republished from The Conversation, a non-profit, independent news organization that brings you reliable facts and analysis to help you make sense of our complex world. It was written by: Zachary del Rosario, Olin College of Engineering
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Zachary del Rosario receives funding from the National Science Foundation, and has worked with Citrine Informatics and Toyota Research Institute.
