The Fugue Counterpoint by Hans Fugal


Calipers and Science

Just for kicks I dug up the original Jackson/Pollock paper for skinfold measurements for determining body fat percentage. Turns out there's also a 7-point equation that also takes circumference of waist and forearm into account.

Here's a snapshot of the equations for men from the paper ("Generalized equations for predicting body density of men" by A.S. Jackson and M.L. Pollock, 1978. I couldn't find the PDF for the women paper online).
Generalized body density equations

Important notes: skinfolds are in millimeters, circumferences are in meters, and log is the natural log (ln in most computer languages). I plugged my values from two weeks back into a spreadsheet and got the following results:

JP Equation Density %BF
Sum of seven skinfolds
S, S^2, age 1.0518 20.62%
S, S^2, age,C 1.0476 22.51%
log S, age 1.0506 21.15%
log S, age, C 1.0482 22.25%
Sum of three skinfolds
S, S^2, age (5) 1.0607 16.69%
S, S^2, age,C (6) 1.0549 19.24%
log S, age (7) 1.0578 17.95%
log S, age, C (8) 1.0574 18.14%

The most interesting thing here is that there's a large difference between 7 and 3 site measurements, and the 3 site range is significantly larger. Also very interesting to note is that the one-site (suprailiac) AccuMeasure chart is, for me, in line with the 7-site measurement (22.1%). Given other measurements I've taken and just general guesswork based on what I see in the mirror, I think that is a decent estimate.

It's also curious that there are two sets of equations given, one using logs and one using squares.

Moral of the story: more data is better, sometimes not-enough more data is worse than a simpler estimate, and interesting things can be learned when you go to the original source. (This is just a quick note, but the paper is very interesting and reading it will be an interesting exercise that sets proper expectations for, and understanding of, the JP7 skinfold method).


Measure body fat with only a gallon jug (and a couple thousand tons of water)

I finally got around to uploading the PDF version of my body fat measurement quest, and also a simplified one-page PDF for those of you that just want to try it out and don't want to wade through all the physics and my ramblings. While I was at it I threw together an spreadsheet to do all the heavy math for you too. Now the only difficulty is finding a couple thousand tons of water laying around. I put together a simple page with links to all that stuff I just mentioned (except the water).

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Body Density Measurement Uncertainty

A couple days back I posted my idea for measuring body density and estimating
Dad, who has a set of skinfold calipers gave it a try and gave me comparative
results, and asked the question on everbody's mind: just how accurate is it,
especially with that pretty blatant guess at residual lung volume?

So I took some time to learn how to account for uncertainty and take a stab at
pinning a confidence interval on the technique. First of all, I didn't realize
how complicated uncertainty propogation is. Partial derivatives, squares and
square roots, etc. Luckily, I came across some lecture or presentation notes
detailing a sequential perturbation method (instead of an analytical method). I
could have talked Jacob into walking me through the partial derivatives, but
this method is easy to code and a find in and of itself. Read about it in this

I coded up the formula and ran some test data through it. Here's the equation
again for review: ρ = m / ((m + mc)/ρw - (va +
vc + vr)) Here's the values and uncertainty I attribute
to each variable:

  • m = 121.29 ±0.02 kg
  • ρw = 0.997 ±0.001 kg/l
  • va = 1.13 ±ٍ0.01 l
  • vr = 1.87 ±0.5 l
  • mc = 0 ±0.02 kg
  • vc = 0 ±0.01 l

I didn't actually use a counterbalance, but I included the uncertainty in
measuring its mass and volume as if I had, just for completeness. As suspected,
vr has the largest uncertainty. I calculated the uncertainty if
vr were magically accurate, and found that the uncertainty was
0.0014 kg/l. This translates to about 0.65% body fat with Siri's equation
(ignoring the uncertainty inherent in that equation, which is a constant bias
accross measurements for one person on any given day).

Note that I give ρw this time, instead of whisking it away with a
magical 1 kg/l. I picked an average value between 72°F and 84°F (most pools are
in this range), with an uncertainty (due to water temperature) of about 0.001
kg/l. If you use 1 kg/l instead you are introducing a bias of about 0.9% body
fat. So I was wrong about that being insignificant.

Now, I found a better estimate
(why better? because it seems to come from a more reputable source than
Wikipedia) for residual lung volume: vr = RV = 0.24 VC. So I may
have overestimated my RV last time by ½ liter.

(Update: I think that must be a typo on that page, they probably mean 24% or 28% of total capacity instead. This fits in much better with the rest of the literature that I have found, e.g. Quanjer and Paoletti.) That seems like a generous
uncertainty measure for RV, too. With that uncertainty factored in, we get an
uncertainty of about 2.1% body fat, or about 5% is you are on the slight side of average (the less you weigh, the more difference that 1/2 liter makes).

So, Dad, let's bump your score up by about 1% for the density of water and then
tack an uncertainty of 2% onto it, you have a body fat of 26.3% ±2%. I'm no
expert on using calipers, but one paper's
indicates that the
skinfold method uncertainty is about 3%. I've seen 10% tossed around casually
too, but have no reliable source to back that up. That puts the two methods
within the appropriate reach of eachother, which is heartening. It's
interesting to note that BMI is overestimating Dad's fat, because he's more
lean than the average couch potato. Imagine the difference if the subject were
someone completely nuts, like a young triathlete, who has body fat of about
15%. Even better, if you are such a nut you could do the experiment and post
your results (and BMI) here as a comment for us to see.


Measure Your Body Density

When you try to lose weight, what you are really trying to do is lose fat.
Weighing yourself is a first approximation of your progress, but a better
indicator is your body fat percentage (%BF). Unfortunately, measuring %BF can
be expensive and/or difficult. It doesn't need to be so. All you need is a body
of water (e.g. a swimming pool), a gallon jug, and your bathroom scale.

One of the most accurate ways to measure body fat is hydrostatic weighing. You are weighed underwater and on land, and your body's density is determined. Then body fat is estimated from the measured density. This is the same basic technique that we will use, but we don't require an underwater scale or special tank.

First the how, then I'll give you the physics. Get in the pool and exhale all
the air you can, and allow yourself to sink. You will sink unless you're
particularly obese. Take note of the sensation of sinking. Then do the same
thing but with your lungs full. Take note of the sensation of floating. Now, we
want to reach the point of neutral buoyancy when your lungs are empty, where
you are neither sinking nor floating. You will be weightless under the water.
Take the gallon jug and hold it under the water, then exhale completely. If the
jug is full of air, you will probably float (unless you are quite lean, in
which case you'll need two jugs). Keep adding water and repeating until you
reach neutral buoyancy. If you sink, add air (pour out some water). If you
float, add water. Once you've found the magic amount of water, use this
equation to calculate your body density (ρ):

rho=m/(m liters/kg - v_\text{air}))

where m is your mass (what the scale tells
you), v is the volume of you and your buoy combined, and vair is the
volume of air in your buoy. If you have ¾ gallon of water in your gallon jug,
then vair is ¼ gallon. ρw=1 kg/liter for all the
precision we need.

Once you have density, you may like to estimate your body fat. The equation for that is Siri's equation, which says

This equation assumes your lungs are completely empty, which they can't be, so we need to introduce a term for the residual volume of your lungs. This is about ¼ of your total lung capacity, or ⅓ of your vital lung capacity. You can measure your vital lung capacity with a balloon or a by blowing air through a straw into an inverted container filled with water. The average residual lung capacity for an american male is 1.2 liters; mine is about 1.9 liters. So we can adjust the formula for density as follows:

rho=m/(m/rho w - v a - v_r)

If you do this experiment you will probably find that your estimated %BF is not too far from your BMI, which is a statistical tool for estimating %BF. It can be wildly inaccurate for statistical outliers (e.g. people who are actually in shape), but it's easy to calculate and a decent sanity check in this case.

Here's what's going on. We're using Archimedes' principle: the buoyant force on a submerged object is equal to the weight of the fluid displaced. When the buoyant force balances the force of gravity, we have neutral buoyancy. The buoyant force is expressed as F_b = -rho_w v g. Substitute weight for the buoyant force and solve for the volume of the body (v = vbody + vair), then substitute that into the definition of density (m/v), and you get the formula I gave you above (if you consider the mass of air and the gallon jug as negligible). I glossed over that—if you'd like me to go into more detail say so in the comments.

If you're particularly obese and don't sink when you exhale completely, then all is not lost. You just need some counterbalance. The modified equation is:

rho = m<em>b / ((m</em>b + m<em>c)/rho</em>w - v<em>a - v</em>c).

You can find the volume of your counterbalance by taking a cue from Archimedes and measuring displacement.

About accuracy: the biggest variable in this process is how much air is left in your lungs. You will find with practice that you are able to exhale more air, which will lower your %BF estimation, as if by magic. However it always overestimates and once you figure out how to completely exhale will be very consistent. Siri's equation is the next place to look—it basically takes the density of fat and the density of muscle and ignores bone mass and density, what you ate for lunch, etc. It will also almost certainly overestimate %BF. The astute reader will wonder about air compression in the milk jug. I measured this and found that when the jug is held within a foot or so from the surface, it does compress. However, the amount it compresses conveniently offsets the extra capacity of the jug (they don't pack milk spilling over the brim of the jug, after all). All in all I think it's accurate within a few percentage points for %BF, gives you an upper bound (i.e. you are free to brag about the number you get, even if it may be slightly high), and is more accurate than BMI.