Sampling and Subsampling

Determination of Sampling and Sub-Sampling Errors

Relative error is defined as the standard deviation divided by the mean (sd / X). The relationship between the sampling error and the analytical method (preparation and measurement) is shown in the following expression:

(sd / X)²_total error = (sd / X)²_sampling + (sd / X)²_analytical

Lets take an example where the total relative error (sd / X)² _{total error} has been determined to be 0.23 (23% relative) on a solid catalyst sample submitted in pellet-form for total Ni analysis. The question is, "How much of this error is the analytical error and how much is the sampling error?" An estimation of the analytical error can be made from multiple preparations and measurements, made on either a homogenized sub-sample or a Certified Reference Material of the same composition as the sample. For this example, lets assume that a CRM was not available and multiple measurements of sub-samples (taken from a ground sample to pass a 200 mesh nylon sieve) gave a relative error of 0.07. Using expression 3.1 and solving for the relative sampling error (sd / X)² _sampling we get a value of 0.22. Relative sampling errors that are 3 times that of the analytical error are not uncommon.

Also note that effort to lower the analytical error below 1/3 of the sampling error would only improve the total error to a best possible case of 0.22 (22 % relative) -- i.e - doing so would be a waste of time.

A determination of the relative subsampling error (sd / X _{subsampling error}) would involve the preparation and measurement of four or more sub-samples, where the total relative error is calculated (sd / X _{total error on sub-sample}) and a determination of the average relative measurement precision (sd / X _{measurement error}) is made by calculating the average relative standard deviation of 10 measurements taken on each of the sub-sample measurements. The relationship between the subsampling error and the analytical measurement is shown in the following expression:

(sd / X)²_{total error on sub-sample} = (sd / X )²_{subsampling error} + (sd / X )²_{measurement error}

The above expression is based upon the assumption that neither negative or positive contamination errors are significant during sample preparation or measurement. If contamination is a problem, then it must be lowered to a level of insignificance. Cases where contamination is significant supercede the need for sampling error calculations for obvious reasons.

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