## Replicates

If a certain analysis is frequently carried out, it may be advantageous to keep in stock some well-mixed bulk samples (low, medium and high range values), sub-samples of which should be included with every fresh batch of analyses, and again well-mixed before weighing. After a statistically significant number of analyses, an accepted average value for the bulk sample is obtained, and the amount of scatter of results computed as a standard deviation, s, where n = total number of results x = observed value x = mean value of observed concentrations (x - x) = deviation from mean

If subsequent analyses of the bulk sample deviate by more than a predetermined amount, the whole batch of results is rejected. Results are thus only accepted if they fall between specified values of s above and below the mean, where 1s includes 68%, 2s includes 95% (the normally accepted value), and 3s includes 99.7% of results. The scatter of results usually assumes a symmetrical normal or Gaussian distribution about the mean, as shown in Figs 12.1 and 12.2.

Bulk samples are repeated, and if still outside the acceptable limits of precision, the methodology must be examined for sources of error; this was considered fully in an early paper by Büttner (1968).

Scatter of results from replicate analyses with random errors

Scatter of results from replicate analyses with random errors x t

### Concentration

Fig. 12.1. Typical scatter of results from nine replicate analyses. a = absolute error of the determination, d = systematic error, n = total number of results (=9 in Fig. 12.1; see under 'Replicates' in the text), t = true value, x = mean observed concentration value, x = observed value (see under 'Replicates' in text).

Concentration

Fig. 12.1. Typical scatter of results from nine replicate analyses. a = absolute error of the determination, d = systematic error, n = total number of results (=9 in Fig. 12.1; see under 'Replicates' in the text), t = true value, x = mean observed concentration value, x = observed value (see under 'Replicates' in text). 