What it does
Many traits of animals and plants change with body size as a power law, Y = a M^b. Metabolic rate, heart rate, lifespan, home range size, bone length and leaf area all follow this pattern, each with its own exponent b. On a log-log plot a power law is a straight line, so b is the slope and a is the value of Y at a mass of 1. This calculator fits that line to your own data, tells you how precise a and b are, tests whether your exponent differs from a theoretical value such as 3/4 or 2/3, and predicts Y for a new mass. A second panel works out the classic scaling laws for any mass, with coefficients you can edit. Everything runs in your browser.
How to use it
- Paste two columns, mass then Y, or three columns with a name first. A header row is fine and is skipped. You can also open a CSV or Excel file.
- Type the mass unit and the Y name and unit; they label the chart and the report.
- Choose the confidence level, the exponent to test b against, and a mass to predict Y at.
- Download the fit as CSV, or the report as an image, PDF or Word file.
What each result means
- Exponent b with its standard error and confidence limits. b = 1 means Y grows in proportion to mass; b below 1 means bigger animals have less Y per kilogram.
- Coefficient a: Y at M = 1 in your mass unit. Changing the unit (grams instead of kilograms) changes a but not b.
- R² on the log scale, and the scatter about the line as a factor: a scatter of 0.065 log₁₀ units means a typical point is within about ×1.16 of the line.
- Bias correction factor: a prediction made by back-transforming the log line is the median, not the mean. Multiply by this factor, exp((s ln 10)² / 2), if you need the mean.
- b against …: a t test of your slope against the value you typed. If the value sits inside the confidence limits there is no clear difference.
- Prediction limits: the narrower pair is for the line itself; the wider pair is for one new species or individual.
Scaling laws
The lower panel uses well-known exponents: 3/4 for whole-body metabolic rate (Kleiber), −1/4 for metabolic rate per kilogram and for heart rate, 1/4 for lifespan and 2/3 for surface area. The coefficients are illustrative round numbers chosen to show how each law behaves; they are not values for any group of animals. Replace a, and b if you wish, with figures for your group from a source you trust. The table shows Y from 10 g to 1 tonne and how many times the 1 kg value it is.
Worked example
The page opens with 14 made-up species from 48 g to 67 kg. The fit is Y = 3.532 × M^0.7415, with b = 0.7415 ± 0.017 and 95 % limits 0.7036 to 0.7794, a = 3.532 (3.237 to 3.854) and R² = 0.9935. Tested against 0.75, t = −0.489 and p = 0.63, so the exponent is consistent with 3/4; 2/3 lies outside the limits. At 50 kg the predicted metabolic rate is 64.24 W, with limits for the line of 54.40 to 75.85 W and for one new species of 44.57 to 92.58 W. In the scaling-laws panel, the illustrative Kleiber law gives 3.4 × 70^0.75 = 82.28 W for a 70 kg animal.
Frequently asked questions
Why ordinary least squares and not reduced major axis? Least squares is right when you want to predict Y from mass, which is what this tool does. If both variables have large measurement errors and you only want the slope, a reduced major axis line will give a steeper exponent.
Can I mix species of very different groups? You can, but different groups often have different coefficients, and the pooled exponent may mislead. Fit each group separately and compare their limits.
Is my data sent anywhere? No. It stays in your browser, and the page works offline once loaded.
Where the numbers come from
- Allometric equation and log-log fitting: Huxley, J. S. (1932), Problems of Relative Growth; Kleiber, M. (1932), Hilgardia 6: 315–353 (the 3/4 exponent).
- Back-transformation bias correction: Baskerville, G. L. (1972), Canadian Journal of Forest Research 2: 49–53; Sprugel, D. G. (1983), Ecology 64: 209–210.
- Least squares and t-based limits: standard regression formulas. Checked against Python scipy.stats.linregress and scipy.stats.t on the page's example.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.