Measurement Technologies: by Hassanain Ghani Hameed

Measurement Technologies
By
Hassanain Ghani Hameed
Lecture Five
Calibration
5.1- Introduction
Every measuring instrument is subject to ageing as a
result of mechanical, chemical or thermal stress and
thus delivers measured values that change over time.
This cannot be prevented, but it can be detected in
good time by calibration.
Calibration is the process of making an adjustment or
marking a scale so that the readings of an instrument
agree with the accepted and the certified standard. The
standard of device with which comparison is made is
called standard instrument. The instrument which is
unknown and is to be calibrated is called test
instrument. Thus in calibration, test instrument is
compared with the standard instrument.
There are two fundamental methodologies of
calibration. They are:
 Direct comparison
 Indirect comparison
5.2 Direct Comparison:
In a direct comparison, a source or generator applies

a known input to the meter under test. The ratio of
what meter is indicating and the known generator
values gives the meter error. In such a case meter is test
instrument, while generator is the standard instrument,
this is shown in figure (5-1). The deviation of meter
from the standard value is compared with the allowable
performance limit.
If meter deviation exceeds the allowance, then the
meter is considered to be out of tolerance.
With the help of direct comparison a generator or
source also can be calibrated. In such calibration, the
meter act as a standard instrument while the generator
act as a test instrument, see figure below.
Transducer converts the signal from one form to another.
Hence if transducer to be calibrated using direct
comparison then both generator as well as meters are
the standard instruments while the transducer act as a
test instrument.
The transducer characteristics are then expressed as a
ratio between the device output to its input, in the
appropriate input and output measurement units as
shown in figure below.
5.3 Indirect Comparison:
In the indirect comparison, then test instrument is
compared with the response of standard instrument of
same type. If test instrument is meter, standard
instrument is also meter, if test instrument is generator;
standard instrument is also generator and so on.
If the test instrument is a meter then the same input is
applied to the same meter as well as a standard meter.
Thus the indication of test meter is compared with the
indication of the standard meter for the same input.
The magnitude of input is not important. This is shown
in figure (5-4).
In case of generator calibration, the output of the
generators, test as well as standard, are set to same
nominal levels. Then a transfer meter is used which
measures the outputs of both standard and test
generators. From the linearity and the resolution of the
transfer meter, the generator is calibrated.
The set up for the generator calibration as shown in
figure (5-5)
The transducer calibration using indirect method is
similar to the generator calibration. The same input is
given, from the source, to the test transducer and the
standard transducer. These are measured using the
standard transfer meter. The sensitivity of the test
transducer is obtained by multiplying the determined
ratio of the two outputs by the known sensitivity of the
standard. The set up as shown in the following figure
5.4 Preparation of Calibration Curves:
Instrument calibration is an essential stage in most
measurement procedures. It is a set of operations that
establish the relationship between the output of the
measurement system (e.g., the response of an
instrument) and the accepted values of the calibration
standards (e.g., the amount of analytic present).
A large number of analytical methods require the
calibration of an instrument. This typically involves the
preparation of a set of standards containing a known
amount of the analytic of interest, measuring the
instrument response for each standard and establishing
the relationship between the instrument response and
analyte concentration.
This relationship is then used to transform
measurements made on test samples into estimates of
the amount of analyte present, as shown in Figure
below
5.4.1 The Calibration Process:
There are a number of stages in the process of
calibrating an analytical instrument. These are
summarized below:
 Plan the experiments;
 Make measurements;
 Plot the results
 Carry out statistical (regression) analysis on the data

to obtain the calibration function;
 Evaluate the results of the regression analysis;
 Use the calibration function to estimate values for
test samples
 Estimate the uncertainty associated with the values
obtained for test samples.
5.4.1.1 Planning the experiments
The issues an analyst needs to consider when planning
a calibration study are as follows:
 The number of calibration standards;
 The concentration of each of the calibration
standards;
 The number of replicates at each concentration;
 Preparation of the calibration standards;
5.4.1.2 Making the Measurements
It is good practice to analyse the standards in a
random order, rather than a set sequence of, for
example, the lowest to the highest concentration.
5.4.1.3 Plotting the results
It is always good practice to plot data before carrying
out any statistical analysis. In the case of regression this
is essential, as some of the statistics generated can be
misleading if considered in isolation, see figure (5-8).
The horizontal axis is defined as the x-axis and the
vertical axis as the y-axis.
When plotting data from a calibration experiment, the
convention is to plot the instrument response data on
the y-axis and the values for the standards on the x-
axis. This is because the statistics used in the regression
analysis assume that the errors in the values on the x-
axis are insignificant compared with those on the y-axis.
5.4.1.3.1 Evaluating the scatter plot
The plot of the data should be inspected for possible
outliers and points of influence. In general, 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
calibrations points.
A point of influence is a calibration point which has a
disproportionate effect on the position of the
regression line. A point of influence may be an outlier,
but may also be caused by poor experimental design.
Points of influence can have one of two effects on a
calibration line – leverage or bias. See the figures (5-9)
and (5-10)
5.4.1.4 Carrying out regression analysis
In relation to instrument calibration, the aim of linear
regression is to establish the equation that best
describes the linear relationship between instrument
response (y) and analyte level (x). The relationship is
described by the equation of the line, i.e., y = mx + c,
where m is the gradient of the line and c is its intercept
with the y-axis.
Linear regression establishes the values of m and c
which best describe the relationship between the data
sets. The equations for calculating m and c are given in
the table (5-1), as below, but their values are most
readily obtained using a suitable software package.
5.4.1.5 Evaluating the results of the regression analysis
Using software to carry out the linear regression will
result in a number of different statistical parameters
and possibly (depending on the software used) a table
and/or plot of the residuals, as follows:

5.4.1.6 Using the calibration function to estimate
values for test samples
If, after plotting the data and examining the regression
statistics, the calibration data are judged to be
satisfactory the calibration equation (i.e., the gradient
and the intercept) can be used to estimate the
concentration of the analyte in test samples.

5.4.1.7 Estimating the uncertainty in predicted
concentrations
Figure (5-11) below illustrates the confidence interval for
the regression line. The interval is represented by the
curved lines on either side of the regression line and gives
an indication of the range within which the ‘true’ line might
lie. Note that the confidence interval is narrowest near the
center (the point x, y ) and less certain near the extremes.
In addition, it is possible to calculate a confidence
interval for values predicted using the calibration
function. This is sometimes referred to as the ‘standard
error of prediction’ and is illustrated in Figure (5-12).
The prediction interval gives an estimate of the
uncertainty associated with predicted values of x.
5.4.1.8 Standard error of prediction worked example
The table and Figure below show a set of calibration
data which will be used to illustrate the calculation of a
prediction interval.
The data required to calculate a prediction interval are
shown as follows:
The residual standard deviation is calculated as:
By applying the following eq., the prediction interval for

a sample which gives an instrument response of 0.871
(((0.32+0.591+0.92+1.135+1.396)/5)=0.871), is:
Note that a single measurement is made on the sample
so N = 1.
Expressed as a % of xpred, Sxo = ((0.0141/7.76)*100)
= 0.53%.
The uncertainty in predicted values can be reduced by
increasing the number of replicate measurements (N)
made on the test sample. Table below shows how Sxo
changes as N is increased.
References
• IAEA - CANDU I & C SNERDI, Shanghai,
INSTRUMENTATION EQUIPMENT
• Guide for the Use of the International System
of Units (SI). NIST Special Publication 811 2008
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Measurement Technologies

In a direct comparison, a source or generator applies

 Carry out statistical (regression) analysis on the data

Using software to carry out the linear regression will

result in a number of different statistical parameters

and possibly (depending on the software used) a table

and/or plot of the residuals, as follows:

values for test samples

If, after plotting the data and examining the regression

statistics, the calibration data are judged to be

satisfactory the calibration equation (i.e., the gradient

and the intercept) can be used to estimate the

concentration of the analyte in test samples.

By applying the following eq., the prediction interval for

Expressed as a % of xpred, Sxo = ((0.0141/7.76)*100)

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