### Replication and successive dilution in constructing calibration curve

**Replication
and successive dilution in constructing calibration curve**

An analytical instrument generally needs to be calibrated before measurements made on prepared sample solutions, through construction of a linear regression between the analytical responses and the concentrations of the standard analyte solutions. A linear regression is favored over quadratic or exponential curve as it incurs minimum error.

**Replication**

Replication
in standard calibration is found to be useful if replicates are *genuinely*
independent. The calibration precision is improved by increasing the number of
replicates, *n*, and
provides additional checks on the calibration solution preparation and on the
precision of different concentrations.

The trend of its precision can be read from the variance of these calibration points. A calibration curve might be found to have roughly constant standard deviations in all these plotted points, whilst others may show a proportional increase in standard deviation in line with the increase of analyte concentration. The former behavior is known as “homoscedasticity” and the latter, “heteroscedasticity”.

It may
be noted that increasing the number of independent concentration points has
actually little benefit after a certain extent. In fact, after having six
calibration points, it can be shown that any further increase in the number of
observations in calibration has relatively modest effect on the standard error
of prediction for a predicted *x* value
unless such number of points increases very substantially, say to 30 which of
course is not practical.

Instead,
independent replication at each calibration point can be recommended as a
method of improving uncertainties. Indeed, independent replication is
accordingly a viable method of increasing *n* when
the best performance is desired.

However, replication suffers from an important drawback. Many analysts incline to simply injecting a calibration standard solution twice, instead of preparing duplicate standard solutions separately for the injection. By injecting the same standard solution twice into the analytical instrument, the plotted residuals will appear in close pairs but are clearly not independent. This is essentially useless for improving precision. Worse, it artificially increases the number of freedom for simple linear regression, giving a misleading small prediction interval.

Therefore ideally replicated observations should be entirely independent, using different stock calibration solutions if at all possible. Otherwise it is best to first examine replicated injections to check for outlying differences and then to calculate the calibration based on the mean value of *y* for each distinct concentration.

There is one side effect of replication that may be useful. If means of replicates are taken, the distribution of errors in the mean tend to be the normal distribution as the number of replicates increases, regardless of parent distribution. The distribution of the mean of as few as 3 replicates is very close to the normal distribution even with fairly extreme departure from normality. Averaging three or more replicates can therefore provide more accurate statistical inference in critical cases where non-normality is suspected.

**Successive dilutions**

A common pattern of calibration that we usually practice is doing a serial dilution, resulting in logarithmically decreasing concentrations (for example, 16, 8, 4. 2 and 1 mg/L). This is simple and has the advantage of providing a high upper calibrated level, which may be useful in analyzing routine samples that occasionally show high values.

However, this layout has several disadvantages. First, errors in dilution are multiplied at each step, increasing the volume uncertainties, and perhaps worse, increasing the risk of any undetected gross dilution error (especially if the analyst commits the cardinal sin of using one of the calibration solutions as a QC sample as well!).

Second,
the highest concentration point has high leverage, affecting both the gradient
and *y*-intercept
of the line plotted; errors at the high concentration will cause potentially
large variation in results.

Thirdly, departure in linearity are easier to detect with fairly even spaced points. In general, therefore, equally spaced calibration points across the range of interest should be much preferred.

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