What you are about to read is not the blog posting that I set out to write. As I was writing the second installment of my Canons of Proportions article, I began to realize that there was very little hard evidence one way or another to support what I was saying. Although I had looked at the issue enough to feel fairly confident in what I was saying, I realized I couldn’t really prove it, and therefore didn’t really know it. So I set out to find some good data to prove what I was saying, and that led me down a different, but interesting path.

Finding the Data

The need for hard evidence with regards to human proportions led to a very long quest to find good, reliable anthropometric data. Most of what I was able to find on the web was not raw data, but high-level aggregations, and I was highly skeptical of some of the aggregations I found, such as the NIST’s 1977 study of children that (when calculated out) seemed to indicate that by the age of sixteen, the average kid was well over eight heads tall.

Much of the good anthropometric data I found was only available for purchase or to subscribers. Now, don’t get me wrong, I love you all, but I’m not about to spend several thousands of dollars to show you that I know what I’m talking about. Eventually, I managed to find some relatively good, free data courtesy of the United States Army. The U.S. Military has done several very detailed anthropometric studies over the years, to ensure that their machines of war are nice and comfy. The results of one of those studies conducted in 1988 are available online, as were some very detailed notes of their measurement methodology. This gave me a decent starting point for some very non-artsy data analysis.

This dataset, although definitely showing a bias toward people of European descent, does contain a relatively diverse sampling of adults (male and female) thanks to America’s “melting pot” status and the fact that the military had aggressive equal opportunity programs in place well before 1988 when this survey was conducted. By the nature of military service and the physical requirements for entry, this dataset also excludes many outliers that might have skewed the results such as those that would be caused by physical deformities.

One Fudge

Unfortunately, out of several hundred measurements, the military did not choose to include a caliper measurement based on the height of the head. They did, however, include the Menton-Sellion caliper measurement, which is measured from two facial landmarks and is equivalent almost exactly to half of the head height. I hated to use an assumption in place of actual measurement for such an important metric, but absent a better data source, it is a necessary assumption. It may skew the results a tiny bit, but over a population of several thousand people, it shouldn’t skew the aggregations by more than a centimeter or two at most, which is close to a meaningless distance in terms of the overall height of the body measured in heads.

It is, perhaps, not a perfect data set, but it is certainly a serviceable one at the least and certainly provides a better statistical basis than what most statements of human proportions are based on. The dataset does NOT include children, but if anyone can point me to a good source of reliable, free anthropometric data about kids, I’d love to do a similar analysis of proportions for children.

Height in Heads

The most basic measurements of every canon of proportion since the Greek Canon has been the overall height stated as a number of units based on the height of the head. Polykleitos said that a person was seven heads tall; Vitruvius said eight. Most books today either give seven and one-half or eight heads as the measure of the typical person.

So, what did the data show?

Well, interestingly enough, it seems to show that the Polykleitos got things pretty close to right, but that Vitruvius wasn’t completely outside the realm of the possible.

Men:

Mean: 7.22
Median: 7.2
Minimum: 6.11
Maximum: 8.98
Standard Deviation: 0.4
Mean +2 Std Deviation 8.02
Mean -2 Std. Deviation 6.42


Women:

Mean: 7.2
Median: 7.18
Minimum: 5.93
Maximum: 8.77
Standard Deviation: 0.39
Mean +2 Std Deviation 6.41
Mean -2 Std. Deviation 7.98


For those of you who didn’t take statistics, don’t remember it, or simply don’t want to think about it, I’ll translate this into non-technical terms. Basically, the average person, both male and female, is right about 7.2 heads tall. The standard deviation, which is an indication of how spread out the data is, is much higher than I expected. A normal distribution (such as this appears to be) has about 95% of the population falling within two standard deviations of the mean. So, this seems to indicate that “typical” people fall roughly between 6.4 to 8.0 heads tall, but we saw at least one woman who was 5.9 heads tall, and at least one man who who was 8.9 heads tall, so our population is quite diverse in terms of height in heads.

Polykleitos was pretty darn close with his "7 heads for men, seven heads or a little less for women". Vitruvius and Da Vinci’s eight head “ideal”, however, does fall within the confines of our actual population, albeit at the upper end of the range, and the very high end of the of “typical” population. That is not entirely unexpected: We call it the “Ideal” because it was intended not to describe the everyday person but a person of better-than-average (or “ideal”) stature and appearance.

So, basically, it’s not impossible to reconcile either Vitruvius’ or Polykleitos’ canons, at least in terms of head height, if we assume that Polykleitos was trying to capture an “average” person, and Vitruvius an “ideal” one.

Midpoint of the Body

At the end of the last article, I left you with a little clue to what I consider the most glaring error of Vitruvius' De Architectura. I think I made it obvious that the midpoint of the body as measured from foot to head, is not at the navel, but rather several inches lower.

Now, some people would argue that Vitruvius was using the center of the circle that circumscribes the figure with arms up and extended, and that may have been his thinking, however it simply means that his error was not in calculation but in defining the problem to be solved. Others have argued that he was defining the center of mass, not the vertical center of the body, but I’m even more skeptical of that claim, because the math needed to calculate the center of mass of the human body didn’t exist until Newton and Leibniz came up with Calculus in the eighteenth century.

Either way, let’s look at the claim that the navel is the middle of the body. According to our data set, the location as the navel as a percentage of overall height is:

Men

Mean: 60.29
Median: 60.29
Minimum: 56.31
Maximum: 64.86
Standard Deviation: 1.22
Mean +2 Std Deviation 62.73
Mean -2 Std. Deviation 57.84


Women

Mean: 60.26
Median: 60.28
Minimum: 55.43
Maximum: 64.35
Standard Deviation: 1.25
Mean +2 Std Deviation 57.75
Mean -2 Std. Deviation 62.77


I think we can all agree that any midpoint, expressed as a percent, is 50%. The mean and median naval position, expressed as a percent, both fall at over 60% in both men and women. Our minimum in the sample falls at 56%. So, it seems pretty fair to say the midpoint is not at the navel.

Arm Span

Another thing Vitruvius mentions but which, to my knowledge, is not addressed in Polykleito’s Canon is the proportion of the arm spans measured from middle finger to middle finger with the arms outspread. Virtuvius states that this will be equal to the figure’s height.

Well, let’s see what the data has to say. Here is the arm span as a percentage of height.

Men:

Mean: 103.84
Median: 103.64
Minimum: 96.26
Maximum: 113.29
Standard Deviation: 2.71
Mean +2 Std Deviation 109.25
Mean -2 Std. Deviation 98.42


Women:

Mean: 102.61
Median: 102.45
Minimum: 93.43
Maximum: 113.68
Standard Deviation: 3.08
Mean +2 Std Deviation 96.46
Mean -2 Std. Deviation 108.77


This one honestly surprised me. I had expected the average arm span to be slightly less than the height, but it turns out that it’s a little bit more for both men and women and “typical” arm span runs from just almost 99% to a little over 109% for men, and from a little over 96% to a little under 109% for women. Well, it looks like Vitruvius got pretty close on that one.

Let's see... what else can we look at? Vitruvius says that the maximum width of the shoulders is a quarter of a man's height. Well, it just so happens that we have a measurement of the width of the shoulders in our data, so let’s see how the data compares to Vitruvius

Men:

Average: 28.03
Median 28
Min 24.19
Max 32.73
Std. Deviation 1.45
Mean + 2 Std Dev 30.92
Mean - 2 Std. Dev 25.14


Women:

Mean: 26.57
Median: 26.51
Minimum: 22.20
Maximum: 31.74
Standard Deviation: 1.39
Mean +2 Std Deviation 23.80
Mean -2 Std. Deviation 29.34


Once again, Vitruvius isn’t too far off, all things considered. He was close, but actually a little under, even if we look at the mean shoulder width of the female population. I would tend to think that an “ideal” male figure would be slightly broader across the shoulders than the average so was a little surprised to see that Vitruvius fell short of reality on this particular measurement of the “ideal”.

Now shoulder width is often expressed in head heights in modern canons. Typically, the "realistic" or "academic" male figure is typically stated to have a shoulder width of two head-lengths, with the "ideal" figure being two and a third.

Above, for example, you can see the normal and idealistic proportions as presented in Andrew Loomis’ seminal book on figure drawing Figure Drawing for All It’s Worth. Does the data support his assertion?

Men:

Average: 2.02
Median 2
Min 0.13
Max 2.59
Std. Deviation 0.14
Mean + 2Std Dev 2.3
Mean - 2 Std. Dev 1.74


Hot diggity! Couldn’t have been much closer on this one. Both the mean and median, when rounded to one decimal place, come out to exactly two head-lengths wide, and the mean plus two standard deviations almost exactly matches Loomis’ “ideal” figure.

Andrew Loomis’ proportions also show the nipple line as a landmark falling at the five and a half head-units from the ground for the standard figure, and six head units from the ground for the ideal figure. And the data says:

Men:

Average: 5.24
Median 5.23
Min 4.37
Max 6.48
Std. Deviation 0.31
Mean + 2Std Dev 5.86
Mean - 2 Std. Dev 4.63


Well, it’s not an exact hit. Loomis’ “academic” figure’s nipple line falls well within the typical range of the data, however the “ideal” figure’s nipple line falls a bit above, although still within the range of possible value.

What’s It All Mean?

I could go on for several more pages, giving you statistics about where parts of the body fall in our population, but I’d bore the few of you still reading. At this point, I’d like to take a step back from the numbers

None of the canons of proportion fall consistently at the same place according to our sample population. Vitruvius, for example, falls at the upper range for head height, but well below the average for arm span and shoulder breadth. Of course, this doesn’t really impact the utility of the canons. If you create a figure accurately to any of the canons, your figure will be proportioned in such a way as to look like a real person and to fall well within the realm of how real people are proportioned.

The more important thing to notice, in my mind, is that nasty Standard Deviation number most of you have been pretending I never mentioned. In every instance, it was fairly large: larger than I expected it to be. What that means, basically, is that "normal" people fall into a broader range of proportions than the idea of a “canon” conveys. We saw people in our sample set — a sample set that, by its nature, excludes many outliers caused by such things as macrocephaly, microcephaly, and dwarfism — adults that range from under six to nearly nine heads tall, and the two standard deviation range around the mean that encompasses 95% of the sample set goes from less than six and a half head to more than eight, which is really quite a range.

The thing that you should take away from this is that Canons of Proportion are a good tool, but don't be beholden to them. Real people come in a wide variety of shapes - probably a wider variety than most of us even realize, so don’t be scared to deviate from “standard” proportions as set out in your various books on drawing and sculpting the figure.

A Statistical Canon of Proportion

Since I had all this aggregated measurement data, I thought it would be fun to try and find parallels in the data for the various Canons, and actually it wasn’t that hard. The “academic canon” lines up pretty well with the mean plus one standard deviation. The “ideal (Vitruvian) canon” matches up fairly close to the mean plus two standard deviations and the “heroic” nine heads canon matches up with the proportions of the population maximum. Realizing this, I decided to create a spreadsheet that would allow you to type in the height of your sculpture, and have it give you the various measurements based on these different “statistical canons”. You can download it:

Apple iWork Numbers Spreadsheet
Microsoft Excel Spreadsheet

There are two different sheets, one for calculating male proportions and another for calculating female proportions. I’ve also included the raw data in two separate sheets so anyone who wants to double-check my calculations can do so.

At some point, I’d like to turn this into a web-based calculator with some additional features, but hopefully some of you will find this spreadsheet to be a useful tool in the meantime. Feedback is always welcome.

Whoah!... after spending a little more time thinking about this, I think there are some serious flaws with all but the "realistic" proportions in the spreadsheet, most especially with the "max" or "heroic" canon. The problem is that it takes the "Max" proportions for each measurements, not the measurements off the person with the max head-heights, so the they will give an oddly proportioned figure, probably a fat one. Feel free to download the spreadsheet, but don't use any of the "statistical canons" yet except for the one based on the average values... I come up with a better way to derive proportions for the other canons. Sorry for the inconvenience.