Deriving the variance of TDS samples from monitoring

Variability of TDS food sample concentrations can be derived using concentration distributions for the sub-foods of the TDS food samples (defined by the TDS compositions). For each sub-food, e.g. apple (sub-food of TDS food FruitMix), a coefficient of variation (CV) is specified that is derived using the available monitoring samples. Note that monitoring samples may be composite samples. For apple, composite food samples are measured and each sample contains, for instance, 12 apples with unit weight 200 g. So monitoring concentrations, $c_{mi}$, are based on composite samples with a total weight $w_{mi}$ = 2400 g each.

A TDS food sample is composed of $w_i$ g of food $i$ with $i=1...k,w_i$ represents the PooledAmount in TDS food sample compositions table. Then, the concentration of a TDS food sample may be represented as:

$$c_{\mathit{TDS}}=\sum _{i=1}^k(w_i\cdot c_i)/\sum _{i=1}^k{w}_i$$

with variance:

$$var(c_{\mathit{TDS}})=\sum _{i=1}^k(w_i\cdot var({\mathit{c}}_i))/\sum _{i=1}^k{w}_i$$

and $var({\mathit{c}}_i)$ is the variance of concentrations $c_i$ of food $i$ with portion sample size $w_i$.

It is expected that increasing the number of units in a composite sample will have a reverse effect on the variation between concentrations. Suppose TDS food FruitMix is composed of 2 x 200 = 400 g apple. The expected variation between portion sizes of 400 g will be larger than between portion sizes of 2400 g:

$$var({\mathit{c}}_i)=var({\mathit{c}}_{mi})\cdot w_{mi}/w_i$$

The variance of the monitoring samples are corrected as follows, calculate:

1. $var({\mathit{c}}_{mi})=\log (C{V}_{mi}^2+1)$

2. $var({\mathit{c}}_i)=var({\mathit{c}}_{mi})\cdot w_{mi}/w_i$

3. $C{V}_i=\sqrt{\exp (\mathit{var}(c+i)-1)}$