# Basic Statistical Tools

Error is the collective noun for any departure of the result from the “true” value*. Analytical errors can be: 1. Random or unpredictable deviations between replicates, quantified with the “standard deviation”. 2. Systematic or predictable regular deviation from the “true” value, quantified as “mean difference” (i. e. the difference between the true value and the mean of replicate determinations). 3. Constant, unrelated to the concentration of the substance analyzed (the analyte). 4. Proportional, i. e. related to the concentration of the analyte.

* The “true” value of an attribute is by nature indeterminate and often has only a very relative meaning. Particularly in soil science for several attributes there is no such thing as the true value as any value obtained is method-dependent (e. g. cation exchange capacity). Obviously, this does not mean that no adequate analysis serving a purpose is possible. It does, however, emphasize the need for the establishment of standard reference methods and the importance of external QC (see Chapter 9). 6. 2. 2 Accuracy The “trueness” or the closeness of the analytical result to the “true” value.

It is constituted by a combination of random and systematic errors (precision and bias) and cannot be quantified directly. The test result may be a mean of several values. An accurate determination produces a “true” quantitative value, i. e. it is precise and free of bias. 6. 2. 3 Precision The closeness with which results of replicate analyses of a sample agree. It is a measure of dispersion or scattering around the mean value and usually expressed in terms of standard deviation, standard error or a range (difference between the highest and the lowest result).

6. 2. 4 Bias The consistent deviation of analytical results from the “true” value caused by systematic errors in a procedure. Bias is the opposite but most used measure for “trueness” which is the agreement of the mean of analytical results with the true value, i. e. excluding the contribution of randomness represented in precision. There are several components contributing to bias: 1. Method bias The difference between the (mean) test result obtained from a number of laboratories using the same method and an accepted reference value.

The method bias may depend on the analyte level. 2. Laboratory bias The difference between the (mean) test result from a particular laboratory and the accepted reference value. 3. Sample bias The difference between the mean of replicate test results of a sample and the (“true”) value of the target population from which the sample was taken. In practice, for a laboratory this refers mainly to sample preparation, subsampling and weighing techniques.

Whether a sample is representative for the population in the field is an extremely important aspect but usually falls outside the responsibility of the laboratory (in some cases laboratories have their own field sampling personnel). The relationship between these concepts can be expressed in the following equation: Figure The types of errors are illustrated in Fig. 6-1. Fig. 6-1. Accuracy and precision in laboratory measurements. (Note that the qualifications apply to the mean of results: in c the mean is accurate but some individual results are inaccurate)