The Navy body fat formula and its error bars

The US Navy circumference method estimates body-fat percentage from a tape measure. No calipers, no tank, no scanner. Just a few girths and a height, run through a logarithmic formula.

It is a genuinely clever piece of work and it has a specific kind of uncertainty attached to it, which is the part worth understanding before you read the output.

Where the equation comes from

The method was developed for the US Navy in the 1980s at its health research centre, in work associated with Hodgdon and Beckett. The approach was regression: measure a large group of people by a reference method — underwater weighing, which was the practical gold standard at the time — also measure their circumferences with a tape, then fit an equation that predicts the first from the second as closely as the data allows.

That is the origin of every constant in the formula. They are not physical quantities. They are fitted coefficients, and they carry the shape of the particular group of people the fitting was done on.

The arithmetic

The male equation uses waist, neck and height, all in centimetres, and base-ten logarithms:

495 / (1.0324 − 0.19077 * log10(waist − neck) + 0.15456 * log10(height)) − 450

Worked through for a waist of 90, a neck of 38 and a height of 178:

1. 90 − 38 = 52, and log10(52) = 1.7160033
2. log10(178) = 2.2504200
3. 0.19077 * 1.7160033 = 0.3273620
4. 0.15456 * 2.2504200 = 0.3478249
5. 1.0324 − 0.3273620 + 0.3478249 = 1.0528630
6. 495 / 1.0528630 = 470.146657, then − 450 gives 20.146657

The female equation is a different fit with different constants and it adds hip circumference, so the logarithm is taken of waist + hip − neck. With a waist of 90, hip of 95, neck of 38 and height of 178 it returns 28.498935. The two equations are separate fits against separate data, which is why the numbers are not comparable across them.

Six decimal places, three significant limitations

The tool prints 20.146657%. That figure is arithmetically exact for the inputs given. It is not accurate to a millionth of a percentage point, and the gap between those two statements is the whole point of this section.

• Regression error. The equation was fitted to a population, and a fit has residuals. An individual result can sit several percentage points from what a reference method would report on the same person. The tool says as much in its own notes: within a few percentage points when measured carefully.
• Measurement error. A centimetre of difference in where you place the tape, or how tightly you pull it, moves the answer. Tape position is not standardised by the tool because the tool cannot see you.
• Population fit. The coefficients came from the group they were fitted on. How well they transfer to a body shaped differently from that group is exactly the open question you would expect it to be.

None of that makes the method useless. It makes it a different kind of instrument than a scan. A scan measures. This estimates, from proxies, using a relationship inferred from other people.

The one thing it is genuinely good at

Because the errors are largely systematic — the same fit, the same tape, the same landmarks — repeated measurements taken the same way tend to move together. That makes the method more informative about change over time than about the absolute level at any single moment. Our tool describes itself the same way: a trend indicator rather than a lab number, with consistency of measurement mattering most.

Running it

The Navy body fat estimator takes sex, waist, neck, height and hip in centimetres, picks the matching equation and computes it on your own device. It prints one percentage and nothing else — no band, no label, no judgement.

It is informational and it is not medical advice. What a body-composition figure means for one person, and whether a tape estimate is the right instrument for the question at all, is something to ask a clinician rather than a calculator.