Statistics of Random Mixing

If random samples of N particles are taken from the known mixture of average composition q, where the fraction of the first component is P and second component is 1 — P, the composition of the samples will be normally distributed with 

As can be seen, as the sample size decreases, the variance or standard deviation increases. In a mixture of total mass W, if m random samples are analyzed for concentration of component 1 (concentration Ci, C2, C3, C4,..Cm), the sample mean can be calculated as 

It is assumed here that the sample concentrations are normally distributed about the mean. 

It is known that multiple samples taken from a bulk (population) will give a distribution of measurements. For completely random mixture, such as a mixture of two free-flowing granular materials of equal size and density, the variance can be calculated theoretically as 

For nonrandom mixtures, Williams (1986) found that the relationship above no longer holds. The relationship between sample size and variance is 

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