Why calibration is such a common and important
step in analytical methods? How is defined?
Calibration is the process of assigning a value, usually in concentration
units, to an instrument response.
For example, you might calibrate the response of an analytical
device – such as a spectrophotometer  by analyzing different “known” concentrations of an analyte
(i.e. solution of a metal ion such as Zn^{+2}) and establish the
instrument’s response per unit concentration of the metal (Fig. 1). Then, an unknown
concentration of a test sample of the same analyte can be estimated from the
graph by extrapolating the observed absorbance (yaxis) over to the analyte
concentration (xaxis) (yellow line in Fig. 1).
In the graph shown in Fig.
1, the calibration function is established based on points in Table I.1:
Table I.1: Instrument
Response for different analyte A
concentrations
Analyte Concentration (μg/l)

Instrument
Response (Absorbance)

0

0

2

0.120

4

0.240

6

0.360

8

0.480

10

0.600

The line in Fig. 1 is called calibration
line or in general calibration curve.
Fig.1: A calibration line of an instrument's absorbance A vs. Analyte's Concentration 
Calibration curve is the graphical relationship between the known values, such as
concentrations, of a series of calibration standards and their instrument
response.
What is the definition of calibration?
A definition of calibration
according to IUPAC^{1} is as follows:
Calibration in Analytical Chemistry is the operation that determines
the functional relationship between measured values (signal intensities S at
certain signal positions z_{i}) and analytical quantities
characterizing types of analytes q_{i} and their amount (content,
concentration) n. Calibration includes the selection of the model (its
functional form), the estimation of the model parameters as well as the errors,
and their validation.
How an instrument –
analytical device is calibrated?
As mentioned above, normally
a calibration graph of a standard A response versus different concentrations of
the standard A is constructed.
The analyte –standard A  response
may be: i) absorbance in U.V spectrometers
and atomic absortption spectrophotometers ii) peak area (or height) or peak area
(height) ratios to an internal standard in chromatographic assays.
The relationship may be
linear (a straight line) or curved.
In the past – before ready
access to personal computers and graph constructing softwares – calibration
curves were drawn on graph paper and the concentrations interpolated manually.
At that time having a straight line made the task considerably easier and
scientists tried to work within a “linear” region of the curve even for
techniques that are not linear (fluorescence, ECD).
Even today the ideal case
is a linear response that passes directly through the origin comparing to the
nonlinear calibrations since the equation of a line is exactly known and
therefore errors are minimized .
In case that the
relationship is linear will be of the form:
(S_{meas})_{A}
= k * n_{A} + S_{reag}
(1)
or
(S_{meas})_{A}
= k * C_{A}+ S_{reag}
(2)
Notice that the above
equations are of the form: y = m * x + b
(equation of a straight line). That means that k is the slope m of the
straight line and S_{reag} is the intercept b of the line – the point
where the line crosses the y axis.
Where (S_{meas})_{A}
is the measured signal of substance A, k is a proportionality constant, n_{A}
or C_{A} are the moles or concentration respectively of analyte A and S_{reag}
is the signal due to the reagents (constant error)^{2}.
In the special case that S_{reag}
= 0 then the straight line passes through zero as in Figure 1.
The equation of the
calibration line –straight line – in Fig. 1 is given by:
(S_{meas})_{A}
= k * C_{A}+ S_{reag}
or
y = m * x + b (3)
The slope of the line = m =
k = Rise/Run = 0.6/10 = 0.06 (determined directly from the graph in Fig.1). The
intercept b = 0
Therefore (3) becomes y = 0.06 * x
If you plug in values of x
(analyte’s concentration) from Table I.1 in equation (3) you will get the
corresponding values of y (instrument response) shown in the Table.
Normally though, the points
from which the calibration function is determined do not fit so well as
presented above.
Let us consider that the
instrument’s response to analyte’s A concentrations are those given in Table I.2.
Notice that for the standard solution with no analyte present the instrument
response was 0.0123 – this standard is called a blank.
A blank is a specimen that is intended to contain none of the
analytes of interest and which is subjected to the usual analytical or
measurement process to establish a zero baseline or background value.
Table I.2: Instrument
Response for different analyte A
concentrations
Analyte Concentration (μg/l)

Instrument
Response (Absorbance)

0

.0123

2

.1244

4

.2507

6

.3820

8

.4886

10

.5995

In a case like this it is
always good practice to plot data before carrying out any statistical analysis
(regression). Plot just a scatter plot (Figure 2 – a scatter plot shows only
the points and not the best fitted line) using Excel or any statistical
software (SPSS, Sigmaplot…).
Are there any outliers in
the calibration function?
You will note from the graph in Fig. 2 that there is some scattering
around the bestfitted line and the line crosses the y axis not at zero as
before. This is the normal situation for experimental data.
Since the bestfitted line is affected considerably by outliers –
introduce significant errors  we must check the graph for the presence of
possible outliers.
Fig. 2: Absorbance of
analyte A vs. Analyte A Concentration (linear best fitted line)

An outlier is a result
which is significantly different from the rest of the data set. In the case of
calibration, an outlier would appear as a point which is well removed from the
other calibration points.
There is no outlier in Fig 2, just some scattering that is normal
for experimental data.
The effects outliers can have on the best fitted calibration line
are leverage and bias.
For example an outlier at the extremes of the calibration range can
change the position of the calibration line by tilting it upwards or downwards
(Table I.2a, Fig 2a, ). The instrument response at analyte concentration 10 μg/l is such an outlier.
Table I.2a: Instrument Response for different analyte A concentrations
Analyte Concentration (μg/l)

Instrument
Response (Absorbance)

0

.0123

2

.1244

4

.2507

6

.3820

8

.4886

10

.9995

This outlier is said to have a high degree of leverage. Leverage
affects both the slope and the intercept of the calibration line and introduces
errors.
An outlier in the middle of the calibration range will shift the
best fitted line up or down. The outlier in this case introduces a bias into
the position of the line – an additive error. The slope of the line will not be
affected though but the intercept will be wrong.
Fig. 2a: Absorbance of
analyte A vs. Analyte A Concentration (linear best fitted line). The outlier
point (white point at (10, 0.9995)) that corresponds to analyte concentration =
10 μg/l was excluded when the bestfitted line was
drawn (black line). When this outlier point is included in the calculations for
the best fitted line the line changes position and the slope and intercept
change considerably (red line).
How do you correct for outliers in the calibration
graph?
Use one of the following methods, preferably the last three:
1)
Remove with caution points that
are significantly different from the rest of the data. For example the outlier
point in Fig. 2a can be excluded and the best fitted line should be drawn
through the rest of the points.
2) Repeat the calibration
procedure and get the average of 5 or more replicate
measurements at each concentration level (concentration levels ≥ 3). Then
plot the averages of these replicate
measurements at each concentration level.
3)
Use the method of standard additions.
4)
Use the bracketing method.
Then draw the best fitted calibration line checking first if the calibration is linear.
References
1. K. Danzer, L.A. Currie, Pure
& Appl. Chem., Vol. 70, 4, 9931014 (1998)
2. D. Harvey, “Modern Analytical Chemistry”, McGrawHill Companies
Inc., 2000
4.
A. Field, “Discovering
Statistics using SPSS” , Sage Publications Ltd., 2005
Can you please tell me How do you draw the yellow lines in fig 1 using excel ?
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