File:SampleBiasCoefficient.png

An estimate of expected error in the (sample estimate of the population) mean of variable A sampled at N locations in a parameter space x, can be expressed in terms of sample bias coefficient ρ defined as the average auto-correlation coefficient ρ_AA[Δx] over all sample point pairs. This generalized error in the mean is the square root of: sample variance (treated as a population) times (1+(N-1)ρ)/((N-1)(1-ρ)). The ρ=0 line (of log-log slope -½) is the more familiar standard error in the mean for samples that are uncorrelated.

Example: For a 3D fragment of size D taken from a collection of differing and randomly-distributed but homogeneous grains of size d, sample bias may be approximated by ρ ≈ 1-(D/d)+(2/7)(D/d)² when D≤d and ρ ≈ (2/7)(d/D)⁵-(d/D)⁴+(d/D)³ when D>d. Hence an equidimensional fragment about 4.2 times the grain size will have ρ~0.01 and leave you with uncertainty in the population mean of no less than about 10% of the sample standard deviation, regardless of how many points within that fragment are sampled.