 Let "μ" be the mean of the population.Then the probability density function "f"*( "x" ) of the size biased population is

The expectation of this length biased distribution E*( "x" ) is

where "σ"2 is the variance.The expectation of the harmonic mean is the same as the non-length biased version E( "x" )

The problem of length biased sampling arises in a number of areas including textile manufacture pedigree analysis and survival analysis

Akman "et al" have developed a test for the detection of length based bias in samples.