One of the first steps many people take when analyzing a set of data is to calculate the average. You can compare your height to the average height of people where you live or brag about your favorite baseball player’s batting average. But while averaging can help you examine a set of data, it has important limitations.
Using the average while ignoring these limitations has led to serious problems such as discrimination, injury, and even life-threatening accidents.
For example, the US Air Force was designing its planes for the “average person”, but gave up this practice when the pilots could not control their planes. The mean has many uses, but it tells you nothing about the variability in a data set.
I am a discipline-specific educational researcher, meaning I study how people learn with an engineering focus. My research includes studies of how engineers use averages in their work.
Using the mean to summarize data
The mean has been around for a long time, with its use documented as early as the ninth or eighth century BC. First, the Greek poet Homer estimated the number of soldiers on ships by taking an average.
Early astronomers wanted to predict the future positions of stars. But to make these predictions, they first needed accurate measurements of the stars’ current positions. Multiple astronomers made position measurements independently, but they often arrived at different values. Since a star has only one true position, these inconsistencies were problematic.
In 1632, Galileo was the first to strive for a systematic approach to these measurement differences. His analysis was the beginning of error theory. Error theory helps scientists reduce uncertainty in their measurements.
Error theory and mean
According to error theory, researchers interpret a series of measurements as falling around a true value that is distorted by error. In astronomy, a star has an actual position, but early astronomers’ hands may have been sources of error due to fickle, blurry telescope images and bad weather.
To deal with errors, researchers often assume that measurements are unbiased. In statistics, this means that they are evenly distributed around a central value. Unbiased measurements still have error, but they can be combined to better estimate the true value.
Let’s say three scientists each took three measurements. Viewed individually, their measurements may appear random, but when unbiased measurements are put together, they are evenly distributed around a middle value: the mean.
When measurements are unbiased, the mean will tend to lie in the middle of all measurements. In fact, we can show mathematically that the mean is the closest of all possible measurements. Therefore, the average is an excellent tool for dealing with measurement errors.
statistical thinking
Error theory was considered revolutionary in its time. Other scientists admired the precision of astronomy and sought to bring the same approach to their own disciplines. 19th-century scientist Adolphe Quetelet applied ideas from error theory to study people and came up with the idea of averaging people’s height and weight.
The mean helps make comparisons between groups. For example, taking averages from a dataset containing the heights of men and women might show that the men in the data set are, on average, taller than the women. However, the average does not tell us everything. In the same data set we can probably find individual women who are taller than men.
So you can’t just consider the average. By thinking statistically, you should also consider the spread of values. Statistical thinking is defined as thinking carefully about variation or the tendency for measured values to be different.
For example, different astronomers making measurements of the same star and recording different positions is an example of variation. Astronomers had to think carefully about where the variations were coming from. Since a star has only one true position, they can safely assume that its variations are due to error.
Averaging measurements makes sense when variation arises from sources of error. However, when there is real variability, researchers need to be careful when interpreting the mean. For example, in the case of height, individual women may be taller than men, even though men are taller on average. Focusing solely on the average ignores variability, which causes serious problems.
Quetelet not only derived the practice of calculating averages from error theory. He also took the assumption of a single true value. He elevated the ideal of the “average man” and suggested that human variability is fundamentally error, that is, non-ideal. According to Quetelet, if you’re not exactly your average height, there’s something wrong with you.
Researchers who study social norms note that Quetelet’s ideas about the “average person” contributed to the modern meaning of the word “normal” – normal height and normal behavior.
These ideas were used by some, such as early statisticians, to divide populations into two groups: those that were superior in some way and those that were inferior.
For example, the eugenics movement, a despicable effort to prevent “inferior” people from having children, bases its thinking on these ideas about “normal” people.
While Quetelet’s idea of variation as error supports discriminatory practices, Quetelet-like uses of the mean also have direct connections to modern engineering failures.
average failures
In the 1950s, the U.S. Air Force designed its aircraft for the “average man.” It was assumed that an airplane designed for the average of average height, average arm length, and some other basic dimensions would work for most pilots.
This decision caused up to 17 pilots to crash in one day. While the “average person” could operate the plane perfectly, true diversity prevented this. A shorter pilot will have difficulty seeing, while a pilot with longer arms and legs will have to squish themselves to accommodate.
While the Air Force assumed most of its pilots would be near average on all key dimensions, it found that zero of 4,063 pilots were average.
The Air Force solved the problem by designing for diversity; designed adjustable seats to account for real variability between pilots.
While adjustable seats may seem obvious now, this “average person” idea still causes problems today. In the United States, women are nearly 50% more likely to be seriously injured in automobile accidents.
The Government Accountability Office blames this disparity on crash testing practices in which female passengers are crudely represented using a scaled version of a male dummy, similar to the Air Force’s “average male.” The first female crash test dummy was introduced in 2022 and has yet to be adopted in the US.
Average is useful but has its limitations. The mean is powerful for estimating true values or making comparisons between groups. But for individuals who exhibit real variability, the average has little meaning.
This article is republished from The Conversation, an independent, nonprofit news organization providing facts and authoritative analysis to help you understand our complex world. 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.