# Taylor's law

Template:Distinguish Taylor's law is an empirical law in ecology that relates the between-sample variance in density to the overall mean density of a sample of organisms in a study area. Taylor described this relationship in 1961 and it has been found to be true for many species since. It has also been found to be true in other areas including transmission of infectious diseases, human sexual behavior, childhood leukemia, cancer metastases, blood flow heterogeneity, genomic distributions of single nucleotide polymorphisms and gene structures. This law is also known in the literature as the power law (in the biological literature) or the ﬂuctuation scaling law (in the physics literature).

It is possible to derive this law if it is assumed that the organisms of interest form clusters that obey a Poisson distribution. Alternative suggestions for its origin have also been proposed.

## History

Barlett proposed a relationship between the sample mean and variance

in 1936.

Smith while studying crop yields proposed a relationship in 1938 similar to Taylor's. Smith proposed the relationship

$\log V_{x}=\log V_{1}+b\log x\,$ where Vx is the variance of yield for plots of x units, V1 is the variance of yield per unit area and x is the size of plots. The slope (b) is the index of heterogeneity. The value of b in this relationship lies between 0 and 1. Where the yield are highly correlated b tends to 0; when they are uncorrelated b tends to 1.

Bliss while studying Japanese beetles found the relationship

This relationship was used by Yates and Finney in 1942. The Bliss and Yates and Finney studies were later cited by Taylor as examples of this relationship.

Fracker and Brischle in 1944 and Hayman and Lowe in 1961 independently described relationships between the mean and variance that are now known as Taylor's law.

The law itself is named after the ecologist L. R. Taylor (1924–2007). The name 'Taylor's law' was coined by Southwood in 1966. Taylor's original name for this relationship was the law of the mean.

It appears that Taylor's law is an example of Stigler's law of eponymy.

## Mathematical formulation

In symbols

$s_{i}^{2}=am_{i}^{b}$ where si2 is the variance of the density of the ith sample, mi is the mean density of the ith sample and a and b are constants.

In logarithmic form

$\log s_{i}^{2}=\log a+b\log m_{i}$ ### Extensions and refinements

A refinement in the estimation of the slope b has been proposed by Rayner.

$b={\frac {f-\varphi +{\sqrt {(f-\varphi )^{2}-4r^{2}f\varphi }}}{2r{\sqrt {f}}}}$ where r is the Pearson moment correlation coefficient between log(s2) and log m, f is the ratio of sample variances in log(s2) and log m and φ is the ratio of the errors in log(s2) and log m.

Ordinary least squares regression assumes that φ = ∞. This tends to underestimate the value of b because the estimates of both log(s2) and log m are subject to error.

A extension of Taylor's law has been proposed by Ferris et al when multiple samples are taken

$s^{2}=(cn^{d})(m^{b})$ where s2 and m are the variance and mean respectively, b, c and d are constants and n is the number of samples taken. To date this proposed extension has not been verified to be as applicable as the original version of Taylor's law.

### Interpretation

Slope values (b) significantly > 1 indicate clumping of the organisms.

In Poisson distributed data b = 1. If the population follows a lognormal or gamma distribution then b = 2.

Populations that are experiencing constant per capita environmental variability the regression of log( variance ) versus log( mean abundance ) should have a line with b = 2.

Most populations that have been studied have b < 2 (usually 1.5–1.6) but values of 2 have been reported. Occasionally cases with b > 2 have been reported. b values below 1 are uncommon but have also been reported ( b = 0.93 ).