Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Instructor: Rob O’Brien (Sci 163) 807-9569 email: Rob.OBrien@ubc.ca
Course Description: Overview of instrumental methods of chemical analysis
including spectroscopic methods, mass spectrometry, surface analysis, and
chromatography. OUC equivalent: CHEM 325. [3-3-0]
Required Prerequisites: CHEM 211. STAT 230 or BIOL 304 is recommended.
Lectures: 7:00 pm - 8:30pm; Tuesday, Wednesday. Sci 337
Laboratory: 2:00 pm - 5:00 pm, Thursday: Sci 221
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Required Text / Material:
1. Principles of Instrumental Analysis: Forth Edition, Skoog, D.A., Holler, J.F., &
Nieman, T.A. (QD 79 .I5 S58 1992) Note: Harris is also acceptable.
2. Chemistry 311 Laboratory Manual, Fall 2006 version
Helpful Reference Material:
1. My Chem 311 Web site (http://people.ok.ubc.ca/orcac/chem311.html)
2. Quantitative Chemical Analysis, Sixth Edition, Daniel C. Harris
3. Harris Textbook Web site (http://bcs.whfreeman.com/qca)
4. The ACS Style Guide: A Manual for Authors & Editors, Editor: Janet S. Dodd,
American Chemical Society, 1998 (ISBN: 0-841-23462-0)
5. Statistics and Chemometrics for Analytical Chemistry 4th edition, Miller, J.N. &
Miller, J.C., Prentice Hall, Toronto, 2000 ISBN 0-130-22888-5 (QD 75.4.S8 M54)
Evaluation: Midterms (x2)
Laboratory
Final Exam
30% (October 4, November 6)
30%
40%
Note: The laboratory and the lecture must be passed independently. Students
are expected to use a word processor as well as a spreadsheet program.
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Course Outline:
Topic 1: Basic Statistics and Calibration Techniques: Figures of merit and
calibration techniques.
Skoog Ch. 1
Topic 2: Atomic Spectroscopy: A brief introduction to spectroscopic
instruments followed by a review of atomic spectroscopic techniques; Atomic
Absorption, Graphite Furnace, ICP and XRF.
Skoog Ch. 6-10, 12
Topic 3: Molecular Spectroscopy: A brief survey of major molecular
spectroscopic techniques, FTIR, UV/VIS, and Fluorescence. Skoog Ch. 13-18
Topic 4: Mass Spectrometry: Introduction to Mass Spectrometry.
Quadrupoles, Ion Traps, Magnetic and Electrostatic sectors, Time-of-flight.
Ionization techniques, ESI, EI, CI, FAB, and MALDI.
Skoog Ch. 20, 11
Topic 5: Basic Chromatography: Introductory Theory, Basic Components,
Qualitative and Quantitative applications. HPLC, GC, Ion Chromatography.
Skoog Ch. 26-29
Topic 6: Electrophoresis and Ion Mobility: Brief Introduction to Separations
based on ion flux and applications.
Skoog Ch. 30
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Statistic terms and techniques you need to know;
• The Standard Deviation (s)
• Degrees of Freedom
• Variance - Square of the standard deviation (s2)
• Relative Standard Deviation or the coefficient of variation.
• Gaussian distribution
• Confidence Limits & Confidence Interval
• Student T-test
• F-Test
(see also stats Excel file - web site: http://people.ok.ubc.ca/orcac/stats.xls)
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Summary of Confidence Limit and Data Reporting:
• For data sets where n>30, report confidence limit using standard deviation
and a k factor (typically 2) i.e., 30.1 ± ks where k= 2 for 95% CL
• For data sets where n<30, report confidence limit using standard t value
ts
intervals. i.e., using the formula
µ=X±
√N
• When reporting results indicate confidence
level, confidence
interval, standard deviation and number of measurements, ie.,
o The measured value at the 95% confidence level is;
ƒ 30.1 ± 0.5 where s= 0.41 and n = 5
• The confidence limit value only has one significant figure. The magnitude of
the confidence limit “sets” the number of significant digits in the measured
value. For example, if the standard deviation above was 0.9 rather than
0.41, the reported 95% confidence level would be
ƒ 30 ± 1 where s= 0.9 and n = 5
• It is acceptable to carry an extra, non-significant digit when reporting values.
Ideally that should be subscripted but in practice it is not always, for
example;
ƒ 30.1 ± 0.51 where s= 0.41 and n = 5
ƒ 30.1 ± 1.1 where s= 0.9 and n = 5
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Performance Characteristics of Instruments - Figures of Merit
¾ A figure of merit is a number, derived from measurements, that is used to
evaluate an instrument or an analytical technique.
¾ The number corresponds to a criterion that can be used to evaluate the
performance of the instrumental method.
¾ To decide whether a given instrumental method is suitable for solving an
analytical problem.
Some figures of merit are:
¾ Accuracy
¾ Precision
¾ Bias
¾ Sensitivity
¾ Detection limit
¾ Dynamic range
¾ Selectivity
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Accuracy:
¾ Indicates how close a measured value is to the true value.
¾ Normally expressed as the relative percent error. (can be absolute error)
o 1% error indicates that the measured concentration is within 1% of
the true analyte concentration
Precision
¾ A measure of the reproducibility of the method.
o How close are the results obtained in the same way?
¾ Standard deviation is an effective monitor of random error
¾ Usually expressed as a percent relative standard deviation.
Accurate & Precise
Precise not Accurate
Accurate not Precise
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Error Types:
¾ Two types of error in an analytical measurement
o Random error indicated by Standard Deviation
o Systematic or determinate error indicated by?
Bias
¾ Measure of the systematic (determinate) error of a measurement.
¾ Expressed as an absolute error
o bias = Xave – µ where; Xave = mean measured value, µ = True value
¾ Bias has both a definite magnitude and sign.
¾ Typical sources are Instrumental errors, Personal errors, and Method errors.
Shots very precise but offset by a fixed amount.
Clear Systematic error
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Sensitivity
Indicates the response of the instrument to changes in analyte concentration or a
measure of a method’s ability to distinguish between small differences in
concentration in different samples. Effected by slope of calibration curve &
precision.
¾ Indicated by the slope of the calibration curve or
¾ in other words, a change in analytical signal per unit change in [analyte]
¾ For two methods with equal precision, the one with steeper calibration
curve is more sensitive.
Calibration sensitivity
¾ If two methods have calibration curves with equal slopes, the one with
higher precision is more sensitive. Analytical sensitivity
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Calibration sensitivity (S):
¾ The slope of the calibration curve evaluated in the [analyte] range of interest.
o S = mc + Sbl (m = slope; c = conc; Sbl = Signal of Blank)
¾ Advantage: With linear calibration curve, sensitivity independent of [analyte]
¾ Disadvantage: does not account for precision of individual measurements
Analytical Sensitivity (γ)
¾ Defined by Mandel and Stiehler to include precision in sensitivity definition
o γ = m/Ss (m = slope; Ss is the standard deviation of measurement)
¾ Advantage:
o Insensitive to amplification factors i.e. increasing gain also increases m
but Ss also increases by same factor hence γ stays constant.
o Independent of measurement units for S
¾ Disadvantage: concentration dependent as Ss usually varies with [analyte]
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Detection Limit
¾ The smallest [analyte] that can be determined with statistical confidence.
¾ Analyte must produce an analytical signal that is statistically greater than the
random noise of blank. (i.e. analytical signal = 2 or 3 times std. dev. of blank
measurement (approx. equal to the peak-peak noise level).
¾ Calculation of detection limit
o The minimum detectable analytical signal (Sm) is given by:
Sm = Sbl + k(stdbl); for detection use k =3
¾ To Experimentally Determine
o Perform 20 – 30 blank measurements over an extended period of time.
o Treat the resulting data statistically to obtain Sbl (mean blank signal) and
stdbl (std. dev. of blank signals). Use these to obtain Sm value.
o Using slope (m) from calibration curve. Detection limit (Cm) is calculated
by:
Sm—Sbl
cm =
m
(Rearranged from Sm = mc + Sbl)
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Detection Limit
Example 1:
An analysis of the calibration data for the determination of lead based upon its
flame emission spectrum yielded an equation: S = 1.12C + 0.312 where C is the
Pb concn in ppm and S is a measure of the relative emission intensity. The
following replicate data were obtained:
No. of
Concn
replicate
(C), ppm
s
10.0
10
1.00
10
0.000
24
Mean value
of S
Std.
dev.
11.62
1.12
0.0296
0.15
0.025
0.0082
Calculate (a) the calibration sensitivity, (b) the analytical sensitivity at 1 and
10 ppm of Pb, and (c) the detection limit.
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Detection Limit
Example Solution:
a) calibration sensitivity is the slope of the calibration curve = 1.12
b) Using γ= m/Ss :
at 10 ppm
at 1 ppm
γ= 1.12 / 0.15 = 7.5
γ= 1.12 / 0.025 = 45
c) Using: Sm = Sbl + k(stdbl)
Sm = 0.0296 + 3(0.0082) = 0.054
Therefore from:
Sm—Sbl
cm =
m
cm
0.054 - 0.0296
1.12
= 0.022 ppm Pb
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Dynamic Range
¾ [Range] over which measurements can be made. Extends from LOQ to LOL.
¾ LOQ (limit of quantitation): [lowest] at which quantitative measurements can
reliably be made. Equal to 10 x Average Signal for blank i.e. 10Sbl.
¾ LOL (limit of linearity): point where signal is no longer proportional to
concentration.
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Selectivity
¾ Degree to which a measurement is free from interferences by other species
contained in the matrix.
¾ Analytical Signal Detected is a sum of the Analyte signal plus interference
signals
S = maCa + mbCb + mcCc + Sblank
¾ Selectivity is a measure of how easy it is to distinguish between the analyte
signal and the interference signal.
¾ Selectivity of an analytical method can be described using a figure of merit
called selectivity coefficient.
kb,a = mb/ma : kc,a = mc/ma
S = ma (Ca + kb,aCb + kc,aCc) + Sblank
¾ Selectivity coefficients range from 0 to >> 1. Can be negative if interference
reduces the observed signal.
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Instrument Calibration Methods
Limited number of techniques that we can use to find the relationship
between analyte concentration and instrument response or signal. The
most common approaches are;
• External Calibration Method (Calibration curve)
• Standard Addition Method
• Internal Standard Method
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Calibration of Instrumental Methods
Calibration curve (also analytical curve or working curve)
Basis of quantitative analysis is that the magnitude of the measured property is
proportional to concentration of analyte.
It is technically possible to determine the proportionality constant (k) between the
[analyte] and resulting signal using a single standard;
¾ k = (SStandard)/(CStandard) some times called k factor.
However, both the systematic and indeterminate errors would also be included.
Instead, a plot of the analytical signal (of the measured property) versus the
analyte concentration is used to determine a linear calibration relationship.
This is typically obtained by measuring the analytical signals for a series of
standards i.e. analyte solutions of known concentration.
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Method of Least Squares:
Ideally the analyte content and the measured signal should exhibit a perfectly
linear relationship of formula: Y = mX + b or Signal = m (Conc.) + Sblank
In reality such perfectly linear relationships don’t exist when real samples and
standards are used.
Therefore the “best” line relationship among the experimental points is used
finding the best-fit line of the data.
Visual estimation of the “best” line through a set of points is subject to error, both
in plotting the points and in fitting the line.
A preferable approach for finding such a line is the method of least squares or
sometimes referred to as linear regression.
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Method of Least Squares - Assumptions
¾ That the values plotted on x-axis are known with much higher precision and
accuracy than those on the y-axis.
¾ The uncertainties in the y values are greater than those in the x values.
¾ The line representing the data should be drawn so that deviations of the y
values are minimized.
Thus the best fit line or the least squares line is the straight line drawn that
minimizes the vertical deviations between the points and the line.
¾ These vertical deviations are called residuals
¾ Since some of the deviations are positive and some are negative we
preferably minimize the sum of the squares of the deviations. Hence its
name the method of least squares.
¾ That is we draw the straight line that has the least value for the sum of
squares of the deviations
¾ In practice, for a set of data points, it is possible to calculate the values of
the slope (m) of line and the intercept (b) which minimizes the sum of
squares.
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Method of Least Squares - Reliability
¾ Notice: the least-squares line need not pass right through the data points.
¾ This is the consequence of indeterminate errors affecting the signal.
¾ The reliability of the least squares method can be estimated by calculating
the uncertainties in the slope (m) and the intercept (b) of the line.
Estimation of uncertainty in a result from a calibration curve
¾ Frequently in instrumental analysis a calibration curve is prepared, and
values for unknown samples are read from it.
¾ If the calibration is a straight line, the standard deviation of the unknown, Sc,
can be estimated from the measurements from which the straight line was
obtained and the measurements made on the sample.
¾ Note that Sc is smallest when yc = y.
o Measurements of the unknown should be near the center of the
calibration plot for minimum error.
o Since the uncertainty increases as yc moves away from y, results
obtained by extrapolation are prone to increased error.
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Calibration of Instrumental Methods - Regression
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Method of Least Squares - Calculations using Excel
Manual Calculations of Least Squares can be done by manually using calculation
techniques outlined in Appendix 1
In practice, spreadsheets such as Excel provide the most practical method for
calculating Least Squares.
Excel contains three main functions that can be used to fit Least Squares
¾ Trendline
o Operative only in Graph mode
o Very limited stats available
¾ Regression
o Available in Tools/Data Analysis/Regression
o Requires to be added on
o Numerous Parameters Reported. Many not needed.
¾ LINEST
o Most useful set of data
o Difficult to access and unlabelled data
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Calibration of Instrumental Methods
[Analyte] # Replicates Mean Signal Std Dev Predicted Residuals
0.00
25
0.031 0.0079
0.030
0.001
2.00
5
0.173 0.0094
0.164
0.009
6.00
5
0.422 0.0084
0.432
-0.010
10.00
5
0.702 0.0084
0.700
0.002
14.00
5
0.956 0.0085
0.969
-0.013
18.00
5
1.248 0.0110
1.237
0.011
1.4
1.2
y = 0.0670x + 0.0300
R2 = 0.9996
1
Series1
0.8
Linear
(TRENDLINE)
0.6
0.4
0.2
0
0.00
5.00
10.00
15.00
20.00
Functions
Intercept
Slope
0.0300
0.0670
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
[Analyte]
0.00
2.00
6.00
10.00
14.00
18.00
# Replicates
25
5
5
5
5
5
Mean Signal Std Dev
0.031
0.0079
0.173
0.0094
0.422
0.0084
0.702
0.0084
0.956
0.0085
1.248
0.0110
Results Produced Using Tools/Data Analysis/Regression
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.999783818
R Square
0.999567683
Adjusted R Square
0.999459604
Standard Error
0.010873998
Observations
6
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
1
4
5
Coefficients
0.030013699
0.067038356
RESIDUAL OUTPUT
Observation
Predicted Y
1
0.030013699
2
0.164090411
3
0.432243836
4
0.70039726
5
0.968550685
6
1.23670411
SS
MS
F
Significance F
1.093574358 1.09357436 9248.468238
7.00969E-08
0.000472975 0.00011824
1.094047333
Standard Error
t Stat
P-value
0.007311135 4.10520381 0.014789982
0.000697089 96.1689567 7.00969E-08
Residuals
0.000986301
0.008909589
-0.010243836
0.00160274
-0.012550685
0.01129589
Upper 95.0%
Lower 95% Upper 95%Lower 95.0%
0.009714692 0.050313 0.009715 0.050313
0.065102922 0.068974 0.065103 0.068974
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Calibration of Instrumental Methods - Regression
Plot Generated by Regression
1.4
1.2
Y
1
0.8
Y
Predicted Y
0.6
0.4
0.2
0
0.00
5.00
10.00
15.00
X Variable 1
20.00
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Calibration of Instrumental Methods - Regression
X Variable 1 Residual Plot
Residuals
0.015
0.01
0.005
0
-0.0050.00
5.00
10.00
-0.01
-0.015
X Variable 1
15.00
20.00
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Calibration of Instrumental Methods - LINEST
X
Y
[Standard] Mean Signal
Predicted Y
Residual Y
0.00
0
0.209
0.209
0.10
12.36
12.279
-0.081
0.20
24.83
24.350
-0.480
0.30
35.91
36.420
0.510
0.40
48.79
48.491
-0.299
0.50
60.42
60.561
0.141
@LINEST(Y-Values, X-Values,Intercept true or False, Stats true or False)
Select 2 x 5 Region, Insert Function LINEST, ctrl-shift-return
Values Generated using @LINEST Function
Col. 1-Linest ArraCol 2-Linest Array
Slope =>
120.7057143
0.208571429 <= Intercept
Std Dev m=>
0.964064525
0.29188503 <= Std Dev b
R^2=>
0.999744903
0.403297125 <= Variance =
Std Dev Regression
F function=>
15676.29604
4 <= # degree of freedom
Sum Res.^2
2549.727156
0.650594286 <=Sum Res.Y^2
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Calibration of Instrumental Methods - LINEST
X
Y
[Standard] Mean Signal
Predicted Y
Residual Y
0.00
0
0.209
0.209
0.10
12.36
12.279
-0.081
0.20
24.83
24.350
-0.480
0.30
35.91
36.420
0.510
0.40
48.79
48.491
-0.299
0.50
60.42
60.561
0.141
70
y = 120.71x + 0.2086
R2 = 0.9997
60
50
40
30
20
10
0
0.00
0.10
0.20
0.30
0.40
Values Generated using @LINEST Function
Slope =>
120.7057143
0.208571429
Std Dev m=>
0.964064525
0.29188503
R^2=>
0.999744903
0.403297125
F function=>
15676.29604
4
Sum Res.^2 =
2549.727156
0.650594286
0.50
0.60
<= Intercept
<= Std Dev b
<= Std Dev reg
<= # DoF
<=Sum Res.Y^2
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Example Problem for Practice:
Do linear regression analysis on each of the following data sets. Indicate which is
the “best fit”. What parameter was most indicative of your conclusion?
Data Set 1
X
Y
4
5
6
7
8
9
10
11
12
13
14
4.26
5.68
7.24
4.82
6.95
8.81
8.04
8.33
10.84
7.58
9.66
Data Set 2
X
Y
4
5
6
7
8
9
10
11
12
13
14
Data Set 3
X
Y
3.1
4.74
6.13
7.26
8.14
8.77
9.14
9.26
9.13
8.74
8.1
4
5
6
7
8
9
10
11
12
13
14
5.39
5.73
6.08
6.42
6.77
7.11
7.64
7.81
8.15
12.74
8.84
Data Set 4
X
Y
8
8
8
8
8
8
8
8
8
8
19
6.58
5.76
7.71
8.84
8.47
7.04
5.25
5.56
7.91
6.89
12.5
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Standard Addition: If the sample matrix is both complex and problematic an
external calibration curve may not be acceptable. In standard addition, the
sample is used as the matrix for the calibration.
[Initial Analyte]
[Analyte + Standard]
=
Initial Signal from Analyte alone
Signal from Analyte & Standard
If the volume of standard added is large enough to dilute the solution, then the
dilution factor must also be considered.
For Example: A volume, V0 sample is diluted to a final volume, Vf, and the
signal, Ssamp is measured. A second identical aliquot of sample is spiked with a
volume, Vs, of a standard solution for which the analyte's concentration, Cs, is
known. The spiked sample is diluted to the same final volume and its signal,
Sspike, is recorded.
Ssamp = kCA
V0
Vf
(
Sspike = k CA
V0
Vf
+ Cs
Vs
Vf
)
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
In other words;
Ssamp = kCA
(
Sspike = k CA
V0
Vf
V0
Vf
+ Cs
Vs
Vf
)
A one step standard addition is
effectively equilivent to a one point
calibration curve
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
A multi-point Standard addition calibration can also be
used and this has similar advantages to the normal
external calibration curve.
The y-intercept will correspond to the signal of the
“unspiked” sample
Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques
Internal Standard Method:
An internal standard is a compound other than the analyte that is added to the
unknown. The signal response of the internal standard is compared to the signal
response of the analyte to determine the amount of analyte present. In order for
this method to work the relative response factor of the analyte to the internal
standard must be known.
Response Factor (RF) for Analyte (A) and Internal Standard (IS):
RFA=
SignalA
RFIS=
[A]
Relative Response Factor (RRF):
SignalA
RRF =
RFA
RFIS
=
[A]
SignalIS
[IS]
SignalIS
[IS]