# (b) Mixtures of unknown composition

If the composition *p* or *P* of the mixture is not known the variance *a _{R}^{2}* 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 <7

_{R}

^{2}:

The time dependence of s*^{2} is then determined. If after a certain time f_{M} 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 > t _{M}* the measurement error s

_{M}

^{2}is very much less than

*s*

***and that there is no systematic demixing. In particular cases this must be investigated by testing a mixture of known composition.

^{2}# (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 *n _{x}* 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)/o _{x}* and

*о*=

^{2}*p{*1 -

*p)/n*can be used if both

*n*and

_{x}*n*as well as

_{y},*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 *±Ax _{g}.* The corresponding relative deviation about the expected value

*p*is

*f*= A

_{x}*xjp.*The following expressions, in which

*Ax*has been replaced by p, give the required sizes of the samples:

_{g}/o_{x }

The probability *P(p) - P(-p)* that *Ax < Ax _{g}* or

*Ax/p*x can be found from tables of the binomial, Poisson and normal distributions.