Editing Harmonic mean (section)

===Size biased sampling===

Assume a random variate has a distribution ''f''( ''x'' ). Assume also that the likelihood of a variate being chosen is proportional to its value. This is known as length based or size biased sampling.

Let ''μ'' be the mean of the population. Then the [[probability density function]] ''f''*( ''x'' ) of the size biased population is
: <math>f^*(x) = \frac{x f(x)}{\mu}</math>

The expectation of this length biased distribution E<sup>*</sup>( ''x'' ) is<ref name="Zelen1972"/>
: <math>\operatorname{E}^*(x) = \mu \left[ 1 + \frac{\sigma^2}{\mu^2} \right]</math>

where ''σ''<sup>2</sup> is the variance.

The expectation of the harmonic mean is the same as the non-length biased version E( ''x'' )
:  <math> E^*( x^{ -1 } ) = E( x )^{ -1 } </math>

The problem of length biased sampling arises in a number of areas including textile manufacture<ref name=Cox1969>Cox DR (1969) Some sampling problems in technology. In: New developments in survey sampling. U.L. Johnson, H Smith eds. New York: Wiley Interscience</ref> pedigree analysis<ref name=Davidov2001>Davidov O, Zelen M (2001) Referent sampling, family history and relative risk: the role of length-biased sampling. Biostat 2(2): 173-181 {{doi|10.1093/biostatistics/2.2.173}}</ref> and survival analysis<ref name=Zelen1969>Zelen M, Feinleib M (1969) On the theory of screening for chronic diseases. Biometrika 56: 601-614</ref>

Akman ''et al.'' have developed a test for the detection of length based bias in samples.<ref name=Akman2007>Akman O, Gamage J, Jannot J, Juliano S, Thurman A, Whitman D (2007) A simple test for detection of length-biased sampling. J Biostats 1 (2) 189-195</ref>