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 1

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 y

2 x

1 .
8.
Graph the line given by the equation 3 x

4 y 1 0 .
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
, ,..., 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) .
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 2

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 ]
0
20
40
60
80
Absorbance
0
0.2356
0.4725
0.7127
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 A

0.0119
c

0.0014
.
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 3

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 c
 
53.8
0.0119
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.
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
 
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 x

10,20,40,60,80 and 90 and where the intercept a

0 and the slope b

1 . 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 yellowcolored 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 4

(Cell F11). Using the equation y ax b

with a

1 and b

0 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 x

15 . Are the inferred values of x more or less spread out compared to when x

50 ?
(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 5

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
 
where the error has mean 0 and standard deviation

. We obtained data points ( , ) j

1,2,..., n , and used the Trendline option to fit a straight line. This results the estimated value of the slope by ˆb .
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

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. n  j

1

 y j
(
ˆ j
)


2 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 with
ˆ

 n j  1

( x j

)( j
 y n j

1
( x j
 x )
2
)
To measure how good the fit is we calculate a quantity called the coefficient of determination, which is abbreviated as R 2 ˆ j
ˆ ˆ j
. We introduce the deviation of the measured y-values from their mean, y j
 y , which we can write as y j y ( y j
 y j
 y j
 y )
A somewhat lengthy calculation shows that the total sum of squared deviations
 j n

1
( y j
 y )
2
can be written as a part that is explained by the linear model (Explained) and a part that reflects the stochastic errors (Unexplained) j n 
 1
( y j
 y )
2  n  j  1
( y ˆ j
 y )
2  n  j  1
( y j
 ˆ j
)
2
Total Explained Unexplained
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 7

R
2 
Explained

Total
 n j

1
 n j

1
( ˆ j
 y )
2
( y j
 y )
2
The coefficient of determination
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 x  10,20,40,60,80 and 90 and where the intercept a

0 and the slope b

1 . 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 * . 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 8

confidence interval. The Cell K26 contains the value of half the length of the confidence interval, which we denote by C . We can thus report the result also as *

C x
.
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 y -6.3
-2.0
-5.6
-1.0
-3.3
0.0
0.1
1.0
1.7
2.0
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)
Absorbance
0.000 0.050 0.100 0.200 0.300 0.400
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.
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