Calibration

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Citation: Neuhauser, C. Calibration
Created: October 18, 2009 Revisions:
Copyright: © 2009 Neuhauser. This is an open-access article distributed under the terms of the Creative Commons Attribution
Non-Commercial Share Alike License, which permits unrestricted use, distribution, and reproduction in any medium, and allows
others to translate, make remixes, and produce new stories based on this work, provided the original author and source are
credited and the new work will carry the same license.
Funding: This work was partially supported by a HHMI Professors grant from the Howard Hughes Medical Institute. Page 1

Calibration

Learning Objectives
After completing this module, the student will be able to
- explain the purpose of calibration
- find a calibration curve using the Excel function trendline
- write a macro in Excel
- explain the meaning of R
2

- explain sources of error when estimating the independent
variable value
- find a confidence interval for the independent variable value

Knowledge and Skills
- trendline calculation
- linear regression
- coefficient of determination
- calibration

Prerequisites
- linear equation
- average and standard deviation
- normal distribution


Citation: Neuhauser, C. Calibration
Created: October 18, 2009 Revisions:
Copyright: © 2009 Neuhauser. This is an open-access article distributed under the terms of the Creative Commons Attribution
Non-Commercial Share Alike License, which permits unrestricted use, distribution, and reproduction in any medium, and allows
others to translate, make remixes, and produce new stories based on this work, provided the original author and source are
credited and the new work will carry the same license.
Funding: This work was partially supported by a HHMI Professors grant from the Howard Hughes Medical Institute. Page 2

Pre-assessment
Before completing the module test whether you master the prerequisites. Linear Equation
1. Find the equation of a horizontal line that goes through the point (2,4).
2. Find the equation of a vertical line that goes through the point (-1,3).
3. Determine the equation of the line passing through (-2,1) and (3,-1/2).
4. Determine the equation of the line passing through (1,-2) and (-2,4).
5. Determine the equation of the line with slope 3 and vertical intercept (0,2).
6. Determine the equation of the line passing through (-1,-1) and parallel to the line passing through
(0,1) and (3,0).
7. Graph of the line given by the equation 2 1 = + y x .
8. Graph the line given by the equation 3 4 1 0 ÷ + = x y .
Average and Standard Deviation
9. Find the average and sample standard deviation of the following data set: 2,4,5,6,6,7,8
10. Write down the equation for calculating the average and the sample standard deviation of a data set
of size n:
1 2
, ,...,
n
x x x
Normal Distribution
11. Suppose X is normally distributed with mean 2 and standard deviation 1. Find (a) the 75
th
percentile,
(b) the 95
th
percentile, and (c) the 99
th
percentile.
12. Suppose X is normally distributed with mean 3 and variance 4. Find the probability that X is between
1 and 4, that is, find (1 4) s s P X .
13. Suppose X is normally distributed with mean -1 and standard deviation 4. Find an interval centered
about the mean so that with probability 0.95 X is contained in that interval.
14. Suppose that the number of seeds a plant produces is normally distributed with mean 142 and
standard deviation 31. Find the probability that a randomly sampled plant will produce more than
200 seeds.

Citation: Neuhauser, C. Calibration
Created: October 18, 2009 Revisions:
Copyright: © 2009 Neuhauser. This is an open-access article distributed under the terms of the Creative Commons Attribution
Non-Commercial Share Alike License, which permits unrestricted use, distribution, and reproduction in any medium, and allows
others to translate, make remixes, and produce new stories based on this work, provided the original author and source are
credited and the new work will carry the same license.
Funding: This work was partially supported by a HHMI Professors grant from the Howard Hughes Medical Institute. Page 3

Calibration
According to the NIST handbook
(http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd133.htm), “[t]he goal of calibration is to
quantitatively convert measurements made on one of two measurement scales to the other
measurement scale.” The relationship between two measurements is used to convert one measurement
into the other measurement. You saw one such example in your chemistry lab where you measured
absorbance to find the concentration of an unknown sample. In this case, the relationship between
absorbance and concentration was linear. You derived the relationship by measuring absorbance of
standard samples of known concentration. The resulting line is called calibration curve. The basis for the
calibration curve is Beer’s Law, which states that there is a direct linear relationship between
absorbance (A) and concentration (c): When if we graph absorbance as a function of concentration, a
straight line with positive slope provides a good fit. To illustrate this, we provide in the following table
absorption measurements of standard samples:
Concentration
[μmole L
-1
]
Absorbance
0 0
20 0.2356
40 0.4725
60 0.7127
80 0.9507

If we graph the data points and fit a straight line through the points (Figure 1), we find that the equation
of the straight line is 0.0119 0.0014 A c = ÷ .

Figure 1: Straight line fit
Citation: Neuhauser, C. Calibration
Created: October 18, 2009 Revisions:
Copyright: © 2009 Neuhauser. This is an open-access article distributed under the terms of the Creative Commons Attribution
Non-Commercial Share Alike License, which permits unrestricted use, distribution, and reproduction in any medium, and allows
others to translate, make remixes, and produce new stories based on this work, provided the original author and source are
credited and the new work will carry the same license.
Funding: This work was partially supported by a HHMI Professors grant from the Howard Hughes Medical Institute. Page 4


This curve is called a standard curve and is used to infer the unknown concentration of a solution. For
instance, if we find that the absorbance A of an unknown solution is 0.6386, we find for the
concentration c

0.6386 ( 0.0014)
53.8
0.0119
c
÷ ÷
= =
The data in our example fits Beer’s Law extremely well. The data was generated using a Virtual Lab on
Spectrophotometry (http://www.chm.davidson.edu/vce/Spectrophotometry/UnknownSolution.html).
When data are obtained in actual lab experiments, measurement errors need to be taken into account.

A Model for Linear Calibration
We assume in the following that we measure a signal y that depends linearly on a quantity x. We call the
quantity x the independent variable and the quantity y the dependent variable. We assume that we
measure x without error and that the quantity y is measured with an error ε that is normally distributed
with mean 0 and standard deviation σ. The relationship between the two quantities is then
y a bx c = + +
To get a sense for the measurement uncertainty when inferring the quantity x from the measurement y,
we begin with simulating an experiment in which we have a set of n standard samples and for each
sample we measure the signal m times.

In-class Activity 1
In the spreadsheet CalibrationWorkbook under the tab “Simulation,” you will find the simulation of
standard samples with values 10,20,40,60,80 x = and 90 and where the intercept 0 a= and the slope
1 b= . Each signal is measured 3 times. The simulated data are in the gray-colored box. The input
parameters for the slope, the intercept, and the standard deviation s.d. for the error are in the yellow-
colored box. The trendline is calculated using the Excel function LINEST. (This function is difficult to use
and you will not need to learn how at this point.)
To investigate how the estimated value of the independent variable x depends on the error ε, we
proceed as follows. We assume that the (unknown) value of the independent variable x is equal to 50
Citation: Neuhauser, C. Calibration
Created: October 18, 2009 Revisions:
Copyright: © 2009 Neuhauser. This is an open-access article distributed under the terms of the Creative Commons Attribution
Non-Commercial Share Alike License, which permits unrestricted use, distribution, and reproduction in any medium, and allows
others to translate, make remixes, and produce new stories based on this work, provided the original author and source are
credited and the new work will carry the same license.
Funding: This work was partially supported by a HHMI Professors grant from the Howard Hughes Medical Institute. Page 5

(Cell F11). Using the equation y ax b c = + + with 1 a= and 0 b= with s.d. 1, we can calculate the
measured value of the quantity y (Cell F12). We can then use the estimated trendline to find the
estimate for x (Cell F13) . The graph displays the simulated data from the calibration experiment, the
trendline, and the data point corresponding to the unknown sample.
When you press F9, you will see that Excel runs another simulation. By repeatedly pressing F9, you can
get a sense for the variability of the estimated value of x in our simulation experiment. It is tedious to
record manually the values of repeated simulations. Excel has a feature, called Macro, that records
repeated key strokes. Let’s write a macro to record the outcome of repeated simulations for the
estimate of x.
(a) To write a macro to simulate values of x, proceed as follows:
1. Open the Developer tab and click on Record Macro in the Code group.
2. Give the macro a name and select a key, for instance, Ctrl-a works.
3. Select the Home tab.
4. Copy the value of x from Cell F13.
5. Paste the value of x into Cell Q3 as Paste Value.
6. Click on Insert in the Cells group and click on Shift cells down in the Insert window.
7. Go to the Developer tab and click on Stop Recording in the Code group.
If you press Ctrl-a, the simulated values will be copied into your spreadsheet in Column Q. Repeat the
simulation 100 times. (The numbers in Column P help you keep track of the simulations.)Sort the
simulated values from Smallest to Largest. Find the middle 90%.
(b) Repeat the simulation when 15 x = . Are the inferred values of x more or less spread out compared
to when 50 x = ?
(c) Change the standard deviation to see how an increase/decrease in the measurement error affects
the uncertainty in the calculation of x.
Citation: Neuhauser, C. Calibration
Created: October 18, 2009 Revisions:
Copyright: © 2009 Neuhauser. This is an open-access article distributed under the terms of the Creative Commons Attribution
Non-Commercial Share Alike License, which permits unrestricted use, distribution, and reproduction in any medium, and allows
others to translate, make remixes, and produce new stories based on this work, provided the original author and source are
credited and the new work will carry the same license.
Funding: This work was partially supported by a HHMI Professors grant from the Howard Hughes Medical Institute. Page 6


Figure 2: Screenshot of the simulation. The input parameters are listed in the yellow box; the simulated data are
listed in the gray box; the estimated values of the slope and vertical intercept are listed in the green box together
with the calculation of the unknown quantity x based on the measurement of the unknown sample y. The graph
displays the simulated data (blue symbols), the trendline (black line), and the unknown measurement (red data
point).

Linear Regression
When two quantities are linearly related, such as absorbance and concentration, a straight line provides
a good fit. In Excel, a straight line can be fitted using the Trendline option. The Trendline option is under
the Layout in the Chart Tools. When clicking on the blue triangle under Trendline and choosing More
Trendline Options, a window opens that offers additional options, such as Display Equation on chart
and Display R-squared value on chart. We already know the meaning of the Equation. We will now look
at the meaning of R-squared.
Assume a linear model y a bx c = + + where the error has mean 0 and standard deviation o . We
obtained data points ( , )
j j
x y , 1,2,..., j n = , and used the Trendline option to fit a straight line. This results
in estimates for the slope and the intercept. We denote the estimated value of the intercept by ˆ a and
the estimated value of the slope by
ˆ
b .

Citation: Neuhauser, C. Calibration
Created: October 18, 2009 Revisions:
Copyright: © 2009 Neuhauser. This is an open-access article distributed under the terms of the Creative Commons Attribution
Non-Commercial Share Alike License, which permits unrestricted use, distribution, and reproduction in any medium, and allows
others to translate, make remixes, and produce new stories based on this work, provided the original author and source are
credited and the new work will carry the same license.
Funding: This work was partially supported by a HHMI Professors grant from the Howard Hughes Medical Institute. Page 7

How does Excel estimate the slope and the intercept?
The method that Excel uses to estimate the slope and the intercept is called method of least squares.
The method says: Find ˆ a and
ˆ
b so that the expression

2
1
ˆ
ˆ ( )
n
j j
j
y a bx
=
(
÷ +
¸ ¸
¿

is as small as possible. We say that the sum of the squared deviations is minimized. Expressions for the
estimated intercept and slope can be given. It is not important to memorize the expressions.
The least square line (or linear regression line) is given by

ˆ
ˆ y a bx = +
with

1
2
1
( )( )
ˆ
( )
ˆ
ˆ
n
j j
j
n
j
j
x x y y
b
x x
a y bx
=
=
÷ ÷
=
÷
= ÷
¿
¿


To measure how good the fit is we calculate a quantity called the coefficient of determination, which is
abbreviated as R
2
. For each data point ( , )
j j
x y , we can define
ˆ
ˆ ˆ
j j
y a bx = + . We introduce the deviation of
the measured y-values from their mean,
j
y y ÷ , which we can write as

ˆ ˆ ( ) ( )
j j j j
y y y y y y ÷ = ÷ + ÷
A somewhat lengthy calculation shows that the total sum of squared deviations
2
1
( )
n
j
j
y y
=
÷
¿
can be
written as a part that is explained by the linear model (Explained) and a part that reflects the stochastic
errors (Unexplained)

2 2 2
1 1 1
Total Explained Unexplained
ˆ ˆ ( ) ( ) ( )
n n n
j j j j
j j j
y y y y y y
= = =
÷ = ÷ + ÷
¿ ¿ ¿

The ratio between the explained variation and the total variation is the coefficient of determination
Citation: Neuhauser, C. Calibration
Created: October 18, 2009 Revisions:
Copyright: © 2009 Neuhauser. This is an open-access article distributed under the terms of the Creative Commons Attribution
Non-Commercial Share Alike License, which permits unrestricted use, distribution, and reproduction in any medium, and allows
others to translate, make remixes, and produce new stories based on this work, provided the original author and source are
credited and the new work will carry the same license.
Funding: This work was partially supported by a HHMI Professors grant from the Howard Hughes Medical Institute. Page 8


2
1 2
2
1
ˆ ( )
Explained
Total
( )
n
j
j
n
j
j
y y
R
y y
=
=
÷
= =
÷
¿
¿

The coefficient of determination
2
R is the proportion of variation that is explained by the model.

In-class Activity 2
Return to the spreadsheet CalibrationWorkbook. Under the tab “Simulation,” you have already worked
on the simulation of standard samples with values 10,20,40,60,80 x = and 90 and where the intercept
0 a= and the slope 1 b= . Each signal is measured 3 times. The simulated data are in the gray-colored
box. The graph has a small textbox where the equation of the trendline and the coefficient of
determination is listed. You will see that when you increase the standard deviation, the coefficient of
determination decreases. Give a verbal explanation as to why you would expect this.

The Chemistry Calibration Lab
In your Calibration Lab, you were asked to prepare a calibration curve. The spreadsheet
CalibrationLab.xlsx will help you do the analysis. Open the spreadsheet. The Calibration Lab Analysis
sheet is set up so that you can enter your data into the yellow cells. To calculate the calibration curve,
enter the data from the absorbance measurements of the standard samples into C4:C21 (Step 2). The
spreadsheet will calculate the slope and intercept in the cells I19 and I20, respectively, (see blue cells
and Step 3). In Step 4, the spreadsheet calculates the coefficient of determination. Compare the values
in the cell to the textbox in the figure that has the same information.
(a) To include the uncertainty of the calibration curve in your lab report, record the coefficient of
determination together with the equation of the trendline. Explain in words the meaning of the
coefficient of determination.
(b) In the chemistry lab, you then determined the concentration of an unknown sample based on the
calibration curve. Enter the three measurements into cells B25-B27 (Step 5). The spreadsheet is set up
so that it calculates the estimated concentration. Use paper and pencil to verify the result in Cell B 31
(estimated concentration) the spreadsheet.
(c) While the theory is beyond this course, the spreadsheet is set up to calculate a confidence interval
for the estimated concentration * x . In Cell K25, you can set the confidence level, for instance 95%. The
lower and upper limits of the confidence interval are listed in Cells K27 and K28, respectively. Record the
Citation: Neuhauser, C. Calibration
Created: October 18, 2009 Revisions:
Copyright: © 2009 Neuhauser. This is an open-access article distributed under the terms of the Creative Commons Attribution
Non-Commercial Share Alike License, which permits unrestricted use, distribution, and reproduction in any medium, and allows
others to translate, make remixes, and produce new stories based on this work, provided the original author and source are
credited and the new work will carry the same license.
Funding: This work was partially supported by a HHMI Professors grant from the Howard Hughes Medical Institute. Page 9

confidence interval. The Cell K26 contains the value of half the length of the confidence interval, which
we denote by
x
C . We can thus report the result also as *
x
x C ± .
If you want to read more about Linear Calibration, consult the statistics and data analysis paper by
Burke, S. Regression and Calibration. LC GC Europe Online Supplement.

Homework
1. Find a linear regression line through the given points and compute the coefficient of determination
x -3.0 -2.0 -1.0 0.0 1.0 2.0
y -6.3 -5.6 -3.3 0.1 1.7 2.1

2. To determine whether the frequency of chirping crickets depends on temperature, the following
data were obtained by Pierce, 1949 (The Songs of Insects. Cambridge, Mass. Harvard University
Press):
Temperature (F) 69 70 72 75 81 82 83 84 89 93
Chirps/sec 15 15 16 16 17 17 16 18 20 29

Fit a linear trendline and find the coefficient of determination.
3. To determine the glucose in a wine sample an enzyme spectroscopy method is used. The calibration
curve is obtained from the following data
Added glucose,
[glucose] (mM)
0.000 0.050 0.100 0.200 0.300 0.400
Absorbance 0.231 0.279 0.314 0.423 0.540 0.665

(a) Find the equation of the calibration curve and the coefficient of determination.
(b) Suppose the absorbance of an unknown sample is measured as 0.356. Use the calibration curve
to estimate the glucose level.


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