The number of degrees of freedom mainly depends on the number of available information (A).

(A)    f = n-v-m

f:    degrees of freedom
n:    number of independent observations (measured values)
v:    number of independent variables (for calibration 1)
m:    number of values which affect calculation of the parameters (e.g. for mean 1)

The degree of freedom is the number of parameters of the system, which can be changed without changing the parameter. For example, a mean of 3 measured values has 2 degrees of freedom because 2 observations could change without changing the mean. The parameter is the concentration of the component or analyte, which has to be determined.

The limit of detection (LOD) is the critical concentration of the analyte, which can distinguish between blank values and measured values. The LOD depends on the quality of the blank values or the calibration line (Fig. 6.2-1 and 6.2-2). It is the decision limit for the existence of the analyte.

► An analyte is detected qualitatively or exists, if it is higher than the critical value of the measured value (higher than the sum of the blank values and its uncertainty (with the degree of scatter)).
► The LOD assesses the analysis.

The limit of quantification (LOQ) is that concentration of the analyte, at which the relative uncertainty of the result takes predefined values. The uncertainty of the result results from the (two-sided) prediction range, the level of confidence and the associated analyte concentration. It describes the efficiency of the quantitative procedure.

► An analyte can be quantified, if its concentration can be determined with a relative uncertainty of result, which means that the measured value is higher than the LOQ. The LOQ provides information about feasibility of a quantitative analysis.

The LOD and LOQ do not hold true alone, but always for a certain sample material (water, soils), a certain method of analysis and a fixed level of confidence.
Further information about LOD and LOQ can be found in DIN 32465 (2008).

The blank sample is a sample, which does not contain the analyte, but matches the samples apart from it. Since such sample is rarely existing or cannot be produced, a sample with the lowest possible concentration of the analyte and a matrix similar to sample is used (salinity, pH and suchlike).
The blank value is the arithmetic mean of the blank samples. The blank value method is the direct determination of the LOD and LOQ. The number of measurements (blank sample) determines den safety factor Φ and should be ≥ 10.
The blank value is determined by the concentration of the analyte (in µmol l-1) by a calibration function, without being part of the LOD und LOQ. The LOD is a multiple of the  standard deviation of the procedure (C, blank value method), whereby

The LOQ can be calculated approximately from the LOD (D)

An additional safety factor considers the increase in standard uncertainty by spreading of the calibration lines (E). 

The measured values of the blank samples have to be distributed normally and the variances have to be distributed homogenously. The blank value method should be preferred, if the above-mentioned prerequisites are fulfilled (blank sample same matrix, free from analyte).

The LOD and LOQ can also be directly calculated from the calibration line, if no suited blank samples are available.
The number of measured values of the calibration line should also be ≥ 10. That could be 10 different concentrations (of an equidistant dilution series), but also replicates of less dilution levels.

In suitable programs (not Excel, but Origin) the 95 % confidence band can be plotted around the calibration line (here concentration = x and extinction = y) (Fig. 6.2-1). Draw the point of intersection of the upper  95 % confidence band with the y axis parallel to the x axis to the calibration line. Let fall a perpendicular onto the x axis. The point of intersection with the x axis is the LOD (Fig. 6.2-2).

► It is calibrated with an analyte, which has typically the same or similar occurrence to them in the sample. Its solubility (in the appropriate matrix and the given conditions) has to be taken into consideration.
► A very good straight line can be created, if from a stock solution with the highest measurable concentration (upper limit of the measurement range) the lower concentrations are made in a geometrical dilution series. A geometrical dilution series can be created by repeated mixing of the same volumes (1+1, from the diluted solution 1+1 again, Fig. 6.2-3).

Equidistant dilution series are common in chemical analytics (Fig. 6.2-4). The upper limit of the measurement range should be higher than the probably highest sample concentration to ensure that all measured values can be calculated by interpolation (all measured values are in the calibrated range). The lower limit of the measurement range should be near the blank values.

Pragmatically, this can be done in the following way:

► Determine the volume of the calibrant like it is for the samples. That means 25 ml for phosphate and nitrite. The 17.5 ml for ammonium are not suitable. Therefore, produce 20 or 25 ml for each calibrant. Either increase the amounts of reagents proportionally or discard the surplus volume of each calibrant
► The volume of one part is 1/10 of it, e.g. here 2.5 ml.
► Add 25 ml ultra-pure water (10 parts, 0 µmol l^-1) and subsequently 25 ml analyte (10 parts, 10 µmol l^-1), measured by measuring cylinders.
► Afterwards, pipette the necessary parts in the other 8 vessels. No shortcuts! Really pipette 9 x 2.5 ml.
► Add in reverse order the stock solution, shake well!

Since most methods have a blank value correction (reagent blank value), the blank values have to be subtracted from measured values of the calibration line before graphic representation and calculation of the function. With 10 calibrants the coefficient of determination should be R² > 0,995.
For practical reasons (calculated function can directly be used for calculation of concentration), the dependent variable (extinction) is on the x axis and the independent variable (concentration) on the y axis (Fig. 6.2-5 and 6.2-6). Subsequently, the increase is a conversion factor F from extinction to concentration.
The absolute term of the straight-line equation is commonly not used for calculation of concentration. This term should be nearly zero and is, therefore, negligible. If this term is systematically > 0, there is no analyte-free blank value (in the narrow sense ultra-pure water). This absolute term can be omitted also in this case. If the point of intersection with the ordinate is regularly < 0, there is a problem with too high uncertainty of the position of the calibration lines.

This calibration has to be created and documented for each instrument, for all sample matrices (e.g. liquid solutions of different salinity) and each reagent charge.

Besides, 10 calibration values and 10 blank values for calibration and LOD, several blank values have to be measured for the monitoring of reagents, the ultra-pure waters and cleanliness of vessels at each measurement day. Per measurement series (simultaneously with samples with reagents) 1 blank value is sufficient. Alternatively, (low sample number) at least 3 should be set per measurement day. If the values are low, they are inserted in the blank value – target map (Fig. 6.2-7).

If blank values increase significantly with time, a new charge of reagents ("Shelf Life" out-of-date) has to be prepared. If necessary, the purity of ultra-pure waters has to be controlled as well. Reagents of the ammonium and phosphate determination are very sensitive, for example if ultra-pure waters is stored for a longer time, it can absorb ammonium from the atmosphere and retain it.

Per measurement series at least 1 suited standard value for monitoring of function of reagents and reaction conditions (e.g. temperature, incubation period), but also for troubleshooting should be inserted (e.g. cuvette length, wave length at measurement, zero position of the photometer). For displaying the standards, a set point – target map is a suited instrument (Fig. 6.2-8). This standard is an analyte of medium concentration. Best suited are standards with a certificate (independent verification of analyte, very expensive) or at least out of the same stock solution as the calibrant. If these additional conditions are met, the standard can check the accuracy of measurements.
Suited standards contain at least the analyte. If in an analytical method the analyte still has to be released from different compounds by the digestion or extraction, at least one further compound in the standard should be a compound from which the analyte is still released during the analytical procedure. This compound should not directly react with the reagents.

The reproducibility of samples (= precision) can be monitored in so-called range-target maps (Fig. 6.2-9). A range is the absolute value of the difference of two measured values divided by their mean (equation G).

For 3 and more replicates, the standard deviation can be normalised to the mean. The range can be given in percent. The target map is especially important, if not each sample can be measured in replicates.

The precision of repeated measurement of standards and/or samples is called standard uncertainty. The precision of standards is calculated by the standard deviation, which is normalised to the mean, of all in the sample period measured standards (equation H). The precision of measurement of samples is calculated by the range as mean (equation I). In the material and method part the combined standard uncertainty is shown (as far as measured) (equation J).

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