Math intuitions
Averages hide the spread
A mean describes a group, not any member of it; without the spread you cannot tell whether anyone is near it.
Gilbert Daniels's 1952 report for the US Air Force, The 'Average Man'?, is the classic demonstration; the median-versus-mean point is older statistics.
Two data sets can share a mean and have little else in common. Household income in the United States in 2023 had a median of 80,610 dollars, meaning half of households earned less, and a mean of 114,500 dollars, 42 percent higher, because the top is long: the average household in the top 5 percent took 526,200 dollars and the average in the bottom fifth 17,650. Quote the mean and the typical household looks far richer than it is. The deeper failure is treating the average as a person. In 1950 the US Air Force measured 4,063 pilots on ten body dimensions to design cockpits for the average man. Gilbert Daniels called a man approximately average on a dimension if he fell within 0.3 standard deviations of the mean, roughly the middle quarter to third of the men, and counted how many were average on all ten. The answer was zero.
The arithmetic explains why. If about 30 percent of men are average on each dimension and the dimensions were independent, the share average on all ten would be 0.3 to the power 10, about six in a million, or 0.02 of a pilot in a group of 4,063. The dimensions are correlated, which helps a little, but Daniels still found 1,055 of average stature, 143 who were also average in chest circumference and sleeve length, and none left by the tenth measurement. His recommendation was to design for the individual rather than the average, which is what adjustable equipment does. The same logic applies to any product, policy or default built for the mean user: the mean locates the group's centre, and only the spread tells you how far from it the actual people stand.
Which summary to report depends on the shape. For symmetric data the mean and median agree and either will do. For skewed data, such as income, response times, file sizes and house prices, the median says what is typical and the mean says what the total is, and the two answer different questions: a payroll needs the mean, a cost-of-living claim needs the median. Ask for a second number beside any average: a standard deviation, an interquartile range, a 95th percentile, or the histogram itself. And be wary of averages over mixed populations. The mean of a group that is half fast and half slow describes a speed nobody runs at, which is the everyday form of the cockpit problem.
Example
In 1950 the Aero Medical Laboratory at Wright-Patterson Air Force Base measured 4,063 Air Force flying personnel, and Lieutenant Gilbert Daniels reported the result in December 1952 under the title The 'Average Man'?. He took ten dimensions relevant to cockpit and clothing design: stature, chest circumference, sleeve length, crotch height, torso, hip, neck, waist and thigh circumferences, and crotch length. Counting a man as approximately average on a dimension if he fell within 0.3 standard deviations of the mean, he found 1,055 of average stature, 143 who were also average in chest circumference and sleeve length, and zero who were average on all ten.
Where it applies
When a dashboard shows a mean latency or a mean score, ask for the percentiles before deciding anything; the 99th percentile is where users suffer and where capacity goes. When evaluating a model, an average accuracy of 90 percent can be 99 percent on common inputs and 40 percent on the rare class you care about, so break the number out by slice.
Think about it
Which average do you rely on at work, and what would you see if you looked at the full distribution behind it?
Applied it to something real? Log it.
Test yourselfOf 4,063 pilots measured on ten dimensions in 1950, how many were in the average range on all ten, and what does that say about designing for the mean?
None. Each dimension's average band held about a third of the men, and the bands intersect in almost nobody, so a design for the average person fits no one.
Related models
- Base ratesHow common something is to begin with matters as much as the evidence in front of you, and people routinely leave it out.
- Fat tailsIn some domains the rare extreme event is so large that it dominates everything else, and the bell curve cannot see it coming.
- The map is not the territoryEvery model, metric or plan is a simplification, and the parts it leaves out can matter more than the parts it shows.
- Power laws and the 80/20 ruleIn many distributions a few items hold most of the total, and doubling the size of an event divides its frequency by a fixed factor.