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(b) Mixtures of unknown composition

If the composition p or P of the mixture is not known the variance aR2 of the composition of the samples from the uniform random mixture cannot be calculated. Despite this, in many cases we may want to know whether we actually have a uniform random mixture. The empirical variance based on the average value of the particular measured concentrations can then be used as an estimator of <7R2:

The time dependence of s*2 is then determined. If after a certain time fM a statistically constant value of s*2 is obtained, i.e. if the values of s*2 lie with a frequency of 95% within a range given by the %2 distribution, then it is assumed that a uniform random mixture has been achieved. Here it must be assumed that at t > tM the measurement error sM2 is very much less than s*2 and that there is no systematic demixing. In particular cases this must be investigated by testing a mixture of known composition.

(c) The distribution of the composition of the samples from a uniform random mixture

If samples containing n particles are taken from a uniform random mixture of composition p then the distribution of their composition is the binomial distribution

where nx is the number of particles of component (x) in the sample. If the number of particles n in the sample is large then two approximations to the binomial distribution can be used. The Poisson distribution

is valid provided not only that n » 1 but also that p « 1 so that np is always a small number; its variance is о 2 = p(l - p)/n = p/n. The standardized normal distribution

with t= (x - p)/ox and о 2 = p{ 1 - p)/n can be used if both nx and ny, as well as n, are very much greater than unity.

(d) Sample size

The sample size n selected must be large enough to ensure that the deviation Дл: = x - p will fall with a given probability within predeter?mined limits ±Axg. The corresponding relative deviation about the expected value p is fx = Axjp. The following expressions, in which Axg/ox has been replaced by p, give the required sizes of the samples:

The probability P(p) - P(-p) that Ax < Axg or Ax/px can be found from tables of the binomial, Poisson and normal distributions.

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