A set of spreadsheets that perform random-number driven simulations of widely-used calibration methods, including single
standard, bracket, calibration curve, and standard addition methods. These simulations
include additive and multiplicative interferences (systematic errors)
and random errors in both signal and in volumetric measurements. Students
observe how non-linearity, interferences, and random errors combine and attempt to optimize precision and
accuracy of the measurement. These
simulations are based on the terminology of Ingle
and Crouch, "Spectrochemical Analysis",
Chapter 6.
You can download six different versions of these spreadsheet models:
Note: to run these spreadsheets, you have to first download the OpenOffice installer (download from OpenOffice),
then install it (by double-clicking on the installer file that you just
downloaded), and then download my spreadsheets from this page.
Once OpenOffice is installed, you can run my spreadsheets just by
double-clicking on them. Note:Don't use version 3.1.0.
There is a bug in OpenOffice 3.1.0 that causes bad x-axis scaling on
some of my graphs. The problems does not occur in version 3.0.1.
I recommend that you use version 3.0.1 until they get this issue fixed in a new release; you can get version 3.0.1 from FileHippo.
OpenOffice program is a free download from OpenOffice.org
for either PCs or Macs.
Background. In
analytical chemistry, the accurate quantitative measurement of the
composition of samples, for example by various types of
spectroscopy, usually requires that the method be calibrated
using standard samples of known composition. This is most commonly, but
not necessarily, done with solution samples and standards dissolved in
a suitable solvent, because of the ease of preparing and diluting
accurate and homogeneous mixtures of samples and standards in solution
form. (Note: calibration methods may be contrasted to "absolute
analytical methods", in which the concentrations of samples are
calculated with the aid of previously-measured fundamental data rather
than from standards that are measured along with the samples. Absolute
methods are occasionally performed, especially in the measurement of
atmospheric gases in situ using laser absorption spectroscopy).
Calibration procedures are subject to error
caused by several complications:
a. Analytical curve non-linearity. The analytical
curve is a plot of the signal from the instrument vs the
concentration of
the analyte (the chemical species whose concentration is
sought). This is closely related to the calibration curve,
which is a plot of the signal from the instrument vs the concentration
of
the standard solutions. In
the absence of interferences, the points of the calibration curve should fall along the analytical curve. If the
analytical curve is linear, calibration procedures are much
simpler, both mathematically and procedurally. If the relationship in
non-linear,
a series of standard solution must be prepared and measured to
establish the shape of the curve, which is time-consuming as well as
requiring the use of larger amounts of standard materials (which can be
expensive
and will eventually require safe disposal). Commonly, calibration
curves are observed to be approximately
linear over a certain concentration range, but non-linear above that
range. In some well-defined cases, the shape of the analytical curve can be predicted, for example in absorption and in fluorescence spectrophotometry.
b.
Additive interferences. Ideally, samples
and standards should give
a zero reading when the analyte concentration is zero. Commonly,
the instrument readout is zeroed when a "blank" (a solution containing
zero concentration of analyte in the same solvent and containing
vessel) is measured. But in many cases this is not enough,
because some other unknown components that are present in the samples
(but not in the standards) are contributing their own signals to the
total signal measured. Unless it it possible to resolve (separate) the
signal
generated
by these components from that of the analyte, the signal measured will
be higher than it should be, leading to an error in the analysis. This
is called an "additive
interference", because the signal from the interfering components adds
to that from the analyte. So even if the analyte's concentration is zero, you still get a signal from the sample. In spectroscopy, this is often
called
a "spectral interference".
c. Multiplicative interferences. Ideally, a given concentration of analyte will give the same
signal reading in the
sample as in the standards (in other words, the slope of the
analytical curve is the same in the samples and standards). But sometimes
there are conditions
or components
present in the samples (but not in the standards) that make the
analyte's signal stronger or weaker that it is in the standards; it
might be a difference in temperature, pH, ionic strength, density,
viscosity, surface tension, or a specific chemical component that
reacts with or binds with the analyte. This is called an "multiplicative interference",
because the analyte's
signal is in effect multiplied by some unknown factor.
This is distinct from an additive interference, because with a
multiplicative interference, you still get a zero signal when the
analyte's concentration is zero.
d. Random errors (uncertainty)
and the propagation of random errors. In
addition to the systematic
errors considered above, analytical methods
are also subject to random error
(imperfect repeatability) due to
several sources. For example, the preparation of analyte and
standard
solutions is subject to random errors in volumetric measurements, and
the instrument's reading itself is subject to random variability due to
electronic
noise generated by electronic amplifiers and detectors,
instability of light sources, and photon noise,
especially when the concentration of the analyte is very low. Proper
instrument design and careful technique can reduce, but never completely
eliminate, such errors.
The size of random errors are usually described by the standard deviation, defined as
where N is the number of data points, xi are the individual points, and x is the mean (average) of all the x's. Spreadsheets and most programming languages have a built-in
function for standard deviation. The relative standard deviation, given by s/x, is often also reported, as a percentage.
e. Calculating the propagation of random errors. The way that random errors combine and
influence the final precision of the measurement is called propagation
of error.
When you compute some quantity that is based on two
or more measurements, you need to be able to figure out how
reproducible the calculated quantity will be when the input variables
are subject to random variability. If you know the standard deviation
of each
of those input measurements, you can calculate the expected standard
deviation of the calculated quantity in two different ways:
1. Do the math. By using the rules for mathematical error propagation. In principle the propagation of errors of the entire calibration method can be
described by closed-form algebraic formalism by breaking down the equation into a series of
differences, sums, products, and ratios, and applying the rules for error propagation,
repeatedly. However, there are two problems with this approach. If the
calculation is complicated, the error propagation can be really
complicated and difficult. Secondly, the usual rules for mathematical
error propagation assume that the random errors of the various parts of
the calculation are not correlated:
if they are correlated the calculations become even more complicated.
Correlation between terms is a factor in the prediction of error
propagation of the bracket and standard addition methods. 2.
Crunch the numbers. By repeating all the calculations over and over again (obviously using
a computer) with random number generators employed to add realistic
amounts of random variability ("noise") to the input variables. (This is
sometimes called a "Monte
Carlo"
approach, a reference to the famous gambling casinos in that small
country). This is relatively easily set up using spreadsheets,
which are well suited to performing laborious repetitive calculations
and even have built-in random number and statistical functions. The advantage of this
approach over closed-form algebraic formalism is that it can be applied
to essentially any arbitrarily-complicated procedure and it
automatically takes into account any correlation between variables. The
disadvantage is that it is less elegant and can not be expressed in a
neat formula.
These spreadsheets perform both of these type of calculations, so you can compare them. It's
important to understand that even a perfectly accurate calculation of
error propagation predicts
only the expected standard deviation "on average", for a very large
number of repeats. If you were to repeat an actual experiment a few
times and compute the standard deviation, you'll often get only a very rough approximation
to the theoretical result, perhaps off by 2 or 3-fold. This is a basic
problem of statistics in analytical chemistry; the theoretical
predictions work well for very large number of repeats, but in
analytical chemistry the cost and time of doing even a few repeats is
often prohibitive. For this reason it is not worth obsessing about
small differences in precision; the uncertainty in measuring the
precision of one method is likely to be greater than the differences
between the methods.
Calibration methods. The
simulations described below model the most commonly-used
analytical calibration methods and simulate their accuracy and
precision with respect to all of these sources of error.
Single external standard:
This is the
simplest calibration method, in which the sample and a single separate
standard solution are measured. This method assumes that the a,b, and c
conditions listed above are absent.
Two-standard bracket method:
In this calibration method, the sample is measured along with two
standard solutions that are close in concentration to the sample
(typically one lower than and one higher than the sample
concentration). This method has the advantage of
approximately
compensating for non-linearity in the analytical curve, if the two
standards are close in concentration to the sample. However, this
method still assumes that conditions (b) and (c) are absent. A
disadvantage of this method is that it requires more time and
uses of twice the amount of standard material as the single-standard
method.
Calibration curve with linear least-squares fit:
A
series of external standard solutions is prepared and
measured. A
first-order least-squares fit of the data is computed and the resulting
equation is used to convert readings of the unknown samples into
concentration. An
advantage of this method are that the random errors in
preparing and reading the standard solutions are averaged
over
several standards. Moreover, non-linearity in the calibration curve can
be detected and avoided (by diluting into the linear range) or
compensated (by using non-linear curve fitting methods). An
obvious disadvantage of this method is that it requires much more time
and
uses more standard material than other methods.
Calibration curve with non-linear curve fit:
A
series of external standard solutions is prepared and
measured. A non-linear least-squares fit of the data is computed
and the resulting non-linear equation is used to convert readings
of the unknown samples into
concentration. An
advantage of this method is that non-linearity in the calibration
curve is
compensated at least approximately, depending on the nature and
severity of the non-linearity of the calibration curve and the choice
of non-linear fitting equation.
A
series of aliquots of the sample solution are taken, increasing amounts
of standard material are added to each one, and the signals from the
resulting mixtures are measured and plotted against the concentration
of added standard. If the resulting calibration curve is sufficiently
linear, a
first-order least-squares fit of the data is computed. The sample
concentration is given by the negative of the x-axis intercept (and to
the ratio of the y-intercept to the slope). This method has the
advantage of compensating for multiplicative interferences. Compared to
the single addition method (below), this method reduces the random
errors in
preparing and reading the standard solutions. Moreover, non-linearity
in the calibration curve can
be detected and avoided (by diluting into the linear range) or
compensated (by using non-linear curve fitting methods). An
obvious disadvantage of this method is that it requires much more time
and
uses more standard material than most other methods.
Single Addition method.
In this
method, the sample is divided into two portions: one is measured
unmodified and the other is "doped" with the addition of a small amount
of pure standard and then measured. This method has the
advantage of compensating for multiplicative interferences (c, above),
but it still assumes that the analytical curve is linear and that additive interferences are absent.
Each
of these calibration methods is modeled by a separate simulation, which
includes all of the above-mentioned systematic
errors, plus random errors due to both volumetric measurement
and signal measurement. The simulations allow you to
investigate
how all of these errors combine an propagate to the final
result. All of the simulations have a very similar structure and
layout, so once you learn hoe to work the first one, using the others
will be relatively straightforward.
The concentration of the unknown sample, Cx, and its predicted
standard deviation, is calculated in a different way in each of these
calibration methods:
The
concentration of the sample Cx is given by Cs*Sx/Ss),
where Cs is the
concentration of the standard solution, Ss is the signal given by that
standard solution, and Sx is the signal given by the sample solution. The predicted relative standard deviation of Cx is easy to
compute in this particular case, if you know the standard deviations of Cs,
Sx, and Ss: there are just three variables, all multiplied or divided, so
according to the rules for error propagation,
the relative standard deviation of Cx is the quadratic sum (square root
of the sum of the squares) of the relative standard deviations of Cs,
Sx, and Ss, which in this simulation are Ev, Es, and Es,
respectively (see cell C68).
The concentration of the sample Cx is calculated by
linear interpolation between two standard solutions and is given by
C1s+(C2s-C1s)*(Sx-S1s)/(S2s-S1s), where C1s and C2s are the
concentrations of the two standard solutions, S1x and S2s are the
signal readings given by the two standards, and Sx is the signal given
by the sample solution. The
predicted standard deviation of Cx is more complex to compute in this case, but
it can be done by breaking down the equation into a series of
differences, sums, products, and ratios, and applying the rules for error propagation, repeatedly. (In "BracketOO.ods", these error propagation calculations are performed in cells C98:F103).
The calibration data (Cs vs Ss) are fitted with a first-order
least-squares fit. (The fit is shown as the straight red line in the
graph). The concentration of the sample Cx is calculated by
Cx = (Sx-intercept)/slope, where Sx
is the signal given by the sample solution, and "slope" and "intercept"
are the results of the least-squares fit. (In this
simulation, rather than choosing each standard solution separately, you
choose the number of standards (from 2 to 18) and the concentration of
the
highest one; the other standards are automatically evenly distributed
between zero and the specified highest). The predicted standard
deviation of Cx is computed from the standard deviations of the
slope
and intercept given by the curve fitting procedure and
the standard deviations of Sx, as shown in cells D111:F119.
The
calibration data are fitted in reverse order (measured absorbance as
the independent variable and standard concentration as the dependent
variable) to a third-order cubic polynomial, in order to avoid the need
to solve a cubic equation when the calibration equation is solved for C and used to convert the measured signals of the unknowns into concentration. The fit
is shown as the curved red line in the graph. The concentration of the
sample is calculated by aSx3+bSx2+cSx+d, where a, b, c, and d are the four coefficients from the cubic fit and Sx
is the signal given by the sample solution. (In this
simulation, the number of standards is fixed at 10; you
can choose the concentration of the
highest one and the other 9 standards are automatically evenly distributed
between zero and the specified highest concentration).
The concentration of the sample is given by intercept/slope, where "slope" and "intercept"
are the results of the first-order least-squares fit of the standard
addition calibration
curve, shown as the straight red line in the graph. The predicted
relative standard deviation of Cx is the quadratic sum (square root of
the sum of the squares) of the relative standard deviations of the
slope and intercept computed by the curve fitting procedure. These
error propagation calculations are performed in cells B82:F87.
The
concentration of the sample is given by
(Sx*Vss*Cs)/(Ss*(Vxx+Vss)-Sx*Vxx), where Cs is the
concentration of the standard solution, Sx is the signal given by that
sample solution by itself, and Ss is the signal given by the sample
solution after the addition of standard, and Vxx and Vss are the
volumes of the samples and standard solution. The predicted standard
deviation of Cx is computed by breaking down the equation into a
series of
differences, sums, products, and ratios, and applying the rules for error propagation, repeatedly. These error propagation calculations are performed in cells D111:F118.
When you are using these spreadsheets, you can inspect the equations
that
perform these calculations by clicking on a calculated cell and looking
for the equation that calculates that cell in the
rectangular box at the top of the screen.
Brief operating Instructions.
The screen display of each of the simulations have five similar areas: The yellow table on the top left of the screen are independent variables that
you can change. Click on the dark blue boldface numbers, type a new
value and press the enter key. Some of these variables can be
controlled continuously by the sliders on the top right. (The units of
concentration in these simulations are normalized to
the range of 0 - 10 for convenience in entering and plotting; you can
think of them as mmoles/liter, µmoles/liter, grams/liter or any other
convenient unit (1 mmole = 0.001 moles; 1 µmole = 10-6 moles). Similarly, the signal units are arbitrary
for
similar reasons).
The graph on the lower right shows the actual analytical curve
(blue
line)
over the concentration range from 0 to 10 (arbitrary units), with the
actual concentration of the unknown sample marked as a yellow triangle.
The red triangles are the standards. The green triangle is the
calculated concentration of the unknown sample according to that
calibration method. In the linear calibration curve and multiple standard addition methods, the red
line represents the
linear least-squares fit to the calibration curve. The graph responds
dynamically as you adjust the variables (e.g. with
the sliders).
The table in the middle left of the screen, labeled "Computed results", are
dependent variables that are automatically calculated from
the
independent variables (don't type in those cells or you will delete the
formulae). The most important dependent variable is "result", which is
a single simulated experimental measurement of the analyte
concentration Cx
based on that calibration method.
The "Statistics"
section at the lower left of the screen shows
the mean, standard deviation, and % relative standard deviation (%RSD) of 20
simulated repeat calibrations and measurements of the unknown concentration Cx. Here, a repeat calibration means that a complete set of new standards
are prepared for each repeat. This gives an idea of the reproducibility
if the entire procedure is repeated. The "% RSD" in the table is the
relative standard
deviation of 20 repeated simulated calibrations; it can be
compared to the "Est. RSD" in the Computed results table above, which
is the predicted relative standard. based onpropagation or error rules.
Both should ideally be the same, but for a variety of
reasons will usually not agree exactly. The statistics are
re-calculated each time an input variable is changed or a slider is
moved.
In
addition to these user-interface areas, there are "off-screen" areas,
below and to the right, that are used by the spreadsheet for graphing,
statistics, and error propagation calculations. You don't need to
change anything there, but you may inspect those areas if you are
interested in the technical aspects of how the simulation spreadsheets
work internally.
Assumptions:
1. The only sources of random error are random errors in volume and
signal measurement. They apply equally to all solutions and readings of
the samples
and of the standards. Errors due to interference and blank correction
errors apply only to the sample readings and are systematic (constant
between measurements). 2. Random errors are expressed as a percentage of the quantity measured
(relative error rather than absolute error). 3. Non-linearity of the analytical curve is introduced by a quadratic
term whose coefficient is the variable "n" (controlled by the first
slider). (This is not rigorously realistic in the case of the
non-linearity in absorption spectroscopy caused by polychromaticity and
unabsorbed stray light. See Instrumental Deviation
from Beer's Law for a treatment of non-linearity in that particular case).
Cell definitions and equations (for Bracket method, OpenOffice
version):
Inputs: mo : Analytical curve slope without interference z : Interference factor (zero -> no interference) n : Analytical curve non-linearly (0 = linear) Ev : Random volumetric error (% RSD ) Es : Signal measurement error (% RSD) Cx : True analyte concentration in sample C1s : Concentration of standard solution 1 C2s : Concentration of standard solution 2 blank : (Uncorrected) blank signal
Outputs:
Analytical curve slope in actual sample m = mo+z
Signal given by standard 1 S1s =(mo*C1s-mo*C1s^2*n)*(1+0.01*2.5*Ev*(RAND()-RAND()))*(1+0.01*2.5*Es*(RAND()-RAND()))
Signal given by standard 2 S2s =(mo*C2s-mo*C2s^2*n)*(1+0.01*2.5*Ev*(RAND()-RAND()))*(1+0.01*2.5*Es*(RAND()-RAND()))
Signal given by unknown sample Sx =(blank+m*Cx-m*Cx^2*n)*(1+0.01*2.5*Ev*(RAND()-RAND()))*(1+0.01*2.5*Es*(RAND()-RAND()))
Measured analyte concentration in sample result = C1s+(C2s-C1s)*(Sx-S1s)/(S2s-S1s)
Relative % effect of interference on signal recovery = m/mo
Array calculations for statistics (performed off-screen): Average: mean = AVERAGE(I99:I118) Standard deviation: s = STDEV(I99:I118) Relative standard deviation: RSD = s/mean Accuracy = (mean-Cx)/Cx
Note: The formulation 2.5*(RAND()-RAND()) seen in the above equations is simply a way of generating random numbers with a "haystack" distribution, a mean of zero and a standard deviation of 1.0, using the RAND() function that by itself gives a uniform distribution between 0 and 1.
Suggested activity: OpenOffice versions.
Error propagation in analytical calibration methods
1. OpenSingleStandardOO.ods (view Screen Shot).
This is the simplest calibration method, in which the only two things
measured are the unknown sample and a single separate
standard solution of known concentration. The table in the upper
left lists the variables that you can change in this simulation.
The most important one is Cx, which is the true concentration of
the sample solution. (Of course, in the real world, you wouldn't know
this beforehand, but in these simulations you can set the true sample
concentration as you wish. The simulation "pretends not to know" the
true value and computes the measured sample concentration from the
sample and standard signals, just as you would in the real world, then
compares that calculated value to the true value to determine the
accuracy of the simulated measurement). The other important variable
is Cs, the concentration of the standard solution that you prepare to
calibrate the system. The variable mo controls the slope of the analytical curve, that is, the magnitude of the simulated signals.
The other variables control simulated imperfections and sources of error: z controls multiplicative interferences, blank controls additive interference, n controls the non-linearity of the analytical curve, and Ev and Es
control the random errors in volume measurement and signal measures,
respectively. If these variable are set to zero, the simulated
measurement should be perfect.
Some of these variables can be varied continuously by means of the sliders on the top right.
The table in the center left lists the quantities that are computed by the simulation.
The most important of these is the measured concentration of the sample, "result".
In this calibration method, it is given by Cs*Sx/Ss),
where Cs is the
concentration of the standard solution, Ss is the signal given by that
standard solution, and Sx is the signal given by the sample solution.
You can click on the numbers in this table and look at the input
line at the top to see the equations that the simulation uses to
calculate that number. The measured signals, Sx and Ss, take into
account all the sources of error due interferences, non-linearity, and
random errors. The "Est. RSD" is the estimated relative standard deviation of the result, computed as described above for that calibration method.
The
graph on the lower right shows the actual analytical curve
(blue
line)
over the concentration range from 0 to 10 (arbitrary units), with the
actual concentration of the unknown sample marked as a yellow triangle.
The red triangle represents the standard solution. The green triangle
is the
calculated concentration of the unknown sample (which should ideally
overlay exactly the yellow triangle representing the true sample
concentration). The entire graph responds
dynamically as you adjust the variables (e.g. with
the sliders).
The "Statistics" section at the lower left of the screen shows
the
result of 20 simulated repeat measurements. The statistics are
re-calculated each time an input variable is changed or a slider is
moved.
Start the experiment with mo=2, blank=0, Ev and Es=0 and use the
sliders to set
z=0, n=0, Cx=5, and Cs=10. This represents an ideal case - a perfect
world with no interferences, no random errors and a perfectly linear
analytical curve. Under these conditions, the sample gives a
reading of Sx=10.000 units and the standard gives a reading of
Ss=20.000 units. So of course the calibration works perfectly
and the "result" equals the true Cx = Cs*Sx/Ss = 10*10/20 = 5.000. The error, percent difference between the true and measured
concentrations, is zero. Now use the Cx slider to vary over its whole range and
you'll see that the results remain perfect for any value of Cx.
Now use the Cs slider to vary the concentration of the standard over
its whole range and you'll see that it also have no effect, as long as
it is not zero. Even "mo",
the slope of the analytical curve, also has no effect as long as it is
not zero, because it effect the signals of samples and standards
equally.
Now let's make the simulation a little more realistic by
introducing some random variability. There are two variables
here, the random volumetric error Ev, and the random signal error Es.
The random volumetric error refers to the random error in measuring
volume or weights when preparing the sample and standard solutions. The
magnitude of these errors depends on the technique used and on the
solution volumes involved. Then using accurate quantitative
glassware (volumetric flasks and pipettes) for volumes in the 10 mL - 1
L range, a volumetric precision of 0.1% is achievable, but a very small
volumes below 1 mL a volumetric precision of 1%
is more likely. The signal measurement error refers to the
reproducibility of the signal output of the instrument, that is, the
signal-to-noise ratio. This varies greatly with the analytical
instrument type and the concentration level of the analyte. It may be
as good as 0.1% under optimum conditions, but is more likely to be in
the 1-10% range, especially at lower concentrations. The signal
measurement error, like the random volumetric error, applies to both
the sample and the standard. Both errors are expressed in terms
of the relative standard deviation (ratio of the standard deviation to
the mean).
For
starters, set Ev = 1% and Es = 1%. Now you'll see that the
instrument readings Ss and Sx are no longer exactly 20.000 and 10.000
as before; they are a little off because of the effect of random
signal measurement error. If you click on the numbers for Ss
and Sx, and look at the entry bar at the top, you'll see the
equations for this numbers. Note that they both involve Es
because the same instrument is used to measure both Ss and
Sx. This causes the calculated sample concentration "result"
to be a little off as well. In fact, if you press the f9 function
key at the top of your keyboard, it will cause the spreadsheet to
recalculate with different random errors. You can see the
Ss and Sx and result jumping around slightly each time you press f9. (Also, the little triangles on the graph move ever so slightly).
But actually the spreadsheet does this automatically, in the Statistics
table.
The Statistics
table, in the bottom left, computes the mean, the standard
deviation, and the percent relative standard deviation (% RSD) of 20
repeat measurements (including both signal measurement and volumetric
error). Notice that the predicted % RSD of result (in cell C68) is actually larger that the 1% RSD that you set for the random signal measurement error Es. Why? That's because Cs, Ss and Sx are subject to random errors: Cs is subject to random error Ev and Ss and Sx are subject to random error Es. But the errors do not simply add up linearly.
Theoretically,
according to the rules for mathematical error propagation, the % RSD of Cx is predicted to be =SQRT((Es)^2+ (Es)^2+(Ev)^2)/100,
if the errors are independent and uncorrelated. If Es = Ev = 1.00, as
in this illustration, this works out to about 1.7%. This is reported
as the "Est. RSD". But this
is the predicted standard deviation "on average", for a very large number
of repeats. Cell C72 gives the actual % RSD of 20 simulated repeat experiments, which should turn out to be somewhere around the Est. RSD, but not exactly because 20 repeats is not really a sufficiently "large" number from a statistical point of view. But
from an analytical laboratory of view, doing 20 repeats of an
analytical calibration is a lot of work, time, and expense. Sometimes
you can only afford to do 3 or 4 repeats, in which case you'll get
an experimental RSD even more approximate, possibly differing from the
predicted by a factor of 2 or 3.
This is a basic problem of statistics in analytical
chemistry; the theoretical predictions work well for very large number
of repeats, but in analytical chemistry the cost and time of doing even
a few repeats is often prohibitive).
Now let's introduce a larger random error. Set Es = Ev = 5% and look at
the % RSD of the result. It's much larger than before - theoretically
8.7% - because of the extra effect of Ev. But again the actual results
bounce around quite a bit as you press f9, in this case mostly between 7 and 11%. What if you use
accurate volumetric glassware (which can have an RSD of 0.1%) and a
highly precise measurement technique that also gives a 0.1% RSD signal
measurement precision)? Set Es = Ev = 0.1% and see what you get for the % RSD of result.
Now let's make the simulation even more realistic by
introducing interferences. Use the "Interference factor" slider to set z
to about 0.5, which causes the analytical signal in the samples to be
substantially stronger than that in the standards (this is type of
multiplicative interference). Note that this causes the
calculated Cx to increase about 25% (as indicated by the Accuracy). Clearly, the single standard method can not compensate for this type of interference. Note: interferences are systematic errors that effect the accuracy but not the precision (% RSD).
Return z
to zero and set "blank" = 1. This simulates an additive interference,
such as a spectral overlap or background interference. Note that
this causes the calculated Cx to be too high (Accuracy is about 10%).
Clearly, the single standard method can not compensate for this
type of interference either
Now test the effect of analytical curve linearity. Return "blank" to zero. Drag the "Analytical
curve non-linearity"
slider to the right and watch the analytical curve
(blue line) in the graph. The curve becomes concave down and the
accuracy degrades as the curvature increases, as indicated by the fact
that the green triangle on the graph (representing the calculated
concentration of the unknown sample) is no longer on top of the yellow
triangle (representing the true concentration). Clearly, the single standard method
depends on having a linear analytical curve. But the problem is that, in the real world, you
wouldn't even have a clue that the analytical curve is non-linear if
you used only one standard. For that, you'd need to measure more than a single standard.
The Two-Standard Bracket Method
OpenBracketOO.ods
(view Screen Shot). In this method the sample is measured along with twostandard solutions that are (ideally) close in concentration to the sample
(typically one lower than and one higher than the expected sample
concentration). The concentration of the sample Cx is calculated by
linear interpolation between the two standards (cell C65). The
bottom two sliders allow you to adjust the two standard concentrations.
The closer the the two standards are together, the smaller the error
due to
analytical curve non-linearity. Of course, this assumes that know
the sample concentrations beforehand, at least approximately, in order
to be able to make up appropriate bracketing standards. (For this
reason, the two-standard
bracket method is mostly used when the approximate range of unknown
concentrations is narrow and fairly well known, as in quality control
applications. It is not well suited when there are a large number
of samples of widely and unpredictable varying concentrations). Start the experiment with mo=2, blank=0, Ev and Es=0 and use the
sliders to set
z=0, n=0, Cx=5, and C1s=4,3 and C2s=5.7. In this case (linear calibration curve, zero noise) everything works perfectly.
Now
slide the non-linearity slider up gradually and watch the shape of the analytic curve (blue) change. Note that the error
(cell 66) stays fairly low, even as the calibration curve becomes
noticeably non-linear. Even when the
non-linearity slider is all the way up (n=0.1), the error is less than
1%, compared to a 5% error for the single-standard method with Cs=5.7.
So the the two-standard bracket method is effective in reducing, but not completely eliminating, the non-linearity error. Try moving the sample
concentration slider Cx just outside
the range of the two standards; it still works pretty well as long you
don't get too far off. Cx does not actually have to fall between the two standards, just close to them.
Now set Ev and Es=1. Note that the predicted RSD
(based on error-propagation calculations) is greater than the measured
RSD in the statistics section. This is caused by the correlation
between the terms in the expression for sample concentration; simple
error propagation math won't work well in this case. Comparing the measured RSD of this method with that of the previous (single standard) method, you can see that the two-standard
bracket method is very slightly less precise, because of the random
error in
preparing and measuring two standards rather than one, but this hardly
matters if method suffers from a significant calibration curve
non-linearity that the bracket method can compensate for.
The Calibration Curve Method with Linear Curve Fit
OpenCalCurveOO.ods
(view Screen Shot).
This simulates a calibration curve with 2 to 18 standard solutions and a linear least-squares fit. This is probably the most
common calibration method in general use. It is laid out just like
the previous simulations, with a few additions. You can choose the
number of standards (ns)
by typing into cell C57 or by clicking on the arrowheads of the "spin
button". Cs (controlled by the bottom slider) now controls the
concentration of the highest standard solution. The concentrations of the other standards are spaced out evenly between 0 and Cs.
The slope and intercept
of the linear least-squares fit to the calibration curve (the red line
on the graph) is shown in the computed results section, and the
equation of the fit and the R2
value (the "Coefficient of Determination", sometimes called the
"Correlation Coefficient") is shown in the upper left of the graph. The R2
value is one way to estimate the "goodness-of-fit" of the least-squares
line to the data; it is 1.000 when the fit is perfect and less than
1.000 when the fit is imperfect.
To start with, set mo=2, blank=0, Ev and Es=0, and use the sliders to set
z=0, n=0, Cx=5, and Cs=10. Now move the linearity slider (variable "n") to the right to
introduce non-linearity. As the analytic curve becomes more
curved, you can clearly see that the linear least-squares fit no longer
describes the curve well. Also, you'll see the R2, which is 1.000 for a perfect straight-line, begin to drop gradually, but R2 still
reads 0.99 when the curve is already severely non-linear and the error has already begun to degrade seriously (see error in the
Statistics table) to about 7% error. Even an R2 value
of 0.999 results in an error of 2%. Maybe 2% sounds
pretty good, and in some applications that may be adequate, but
sometimes analytical methods are called upon to make measurements as
accurate as 0.1% or even better. So this tells us thatR2 must be expressed to several (3 or 4) decimal places for analytical calibration purposes.
Test
this simulation also for interference (variables "z" and "blank");
you'll see that it is no better than the single standard method that
that respect.
Set
Ev and Es=1 to introduce a small random error. You'll see some small
random scatter in the calibration points, with some slightly above
and some slightly below the "best fit" line in red, and the R2 value will dropslightly below 1.0. Also the measured
Cx ("result") will no longer be exact. In the Statistics section, the entire
calibration curve and measurement procedure is repeated 20 times (not
just 20 repeat readings of the sample). With the conditions set
the same as before (mo=2, blank=0, Ev and Es=1,
z=0, n=0, Cx=5, and Cs=10, and ns=2)
you'll notice that the %RSD in the statistics section is slightly
higher than Es and Ev (around 1.4%). The increase is caused by the
variability of the
calibration curve.
There is really no way to prepare a
perfect calibration curve without random error. But is is possible to
reduce the reduce the variability of the computed slope and intercept of the calibration curve by using more standards, thereby "averaging
out" some of the random variability. Try setting ns to 2 and then to 18.
Note that the measured Cx ("result") is more more accurate and
that the %RSD is also lower (about 1) with the higher number of
standards. This is what you get in return
for all that extra work of preparing and running a larger number of
standards. Whether it's worth it or not depends on the situation.
Compared to the single-standard method,
the calibration curve method give a slightly lower %RSD as long as the
number of standards is greater than 2, because a calibration based on
several standards is better than one based on a single standard.
However, the difference is not as much as you might think, because the
reading of the unknown signal Sx has the same uncertainty as in the
single standard method, and that uncertainty is not decreased by using more standards.
How can we predict how much random error we can expect in the result (Cx), without performing a series of experiments or
creating a simulation?The standard way to do this is to perform a propagation of error calculation on
the least-squares slope and intercept and on the equation that
calculates the sample concentration: namely Cx = (Sx-intercept)/slope.
This is done in the table D110:F119, and the result of this calculation is shown as "Pred RSD" in
cell C74. The prediction is based only on a single calibration
curve and is good only insofar as that calibration curve is typical of
others that might be obtained in repeated trials. If your random errors happen to be small when you run your calibration curve, you'll get a deceptively good-looking calibration curve, but then your estimates of the random error in the slope and intercept will be too low. If your random errors happen to be large, you'll get a deceptively bad-looking calibration curve, but then your estimates of the random error in the slope and intercept will be too high. (Here are two examples taken from a set of 20 repeats, one "good" and one "bad", that illustrate this point). Some days it just does not pay to be lucky.
One way to help this situation is to use more standards. Try varying the
number of standards, ns; you will also discover that, if the number of standards
is very small, the agreement between the "Predicted % RSD" and the % RSD of 20 repeat calibrations is very poor. As the number of standards increases, then agreement improves and the actual error decreases. What's the minimum
number of standards needed? There is no hard and fast answer to
that question; it all depends on the quality of the data and the
required quality of results.
These simulated experiments demonstrate
two things: first, the predicted
RSD (because it is based on a single calibration curve) is extremely
unreliable when the number of standards is small, and second, the
%RSD of the result improves slightly when the number
of standards is increased greatly. We rightly expect that the precision
of measurement of concentration should improve if more standards are
used, but not so much as you might expect. Looking at the expression
for the sample concentration, Cx = (Sx-intercept)/slope, the precision of the slope and intercept are inversely proportional to the square
root of number of standards, ns,
but the precision of Sx does not depend on the number of standards. For
example, if we go from using 4 standards to using 16 standards (4 times
as many), the RSD the slope and intercept does decrease by half (the square root of 4), but the RSD
of calculated concentrations decreases only from 1.5% to 1.2%. So, using
a larger number of standards has some benefits, but it may or may not
be "worth it" considering the time
and expense of preparing and running more standards.
The Calibration Curve Method with Non-Linear Curve Fit
OpenCalCurveCubicFitOO.ods
(view Screen Shot). This simulates a calibration curve with 10 standards solutions and a reverse cubic least-squares fit. Set
the usual starting conditions: mo=2, blank=0, Ev and Es=0 and use the
sliders to set
z=0, n=0, Cx=5, and Cs=10. Obviously in this perfect linear case
the results are essentially perfect (zero standard deviation, almost
perfect accuracy, and R2 = 1.000).
Now
increase the calibration curve non-linearity with the n slider, about
half-way up (about n=0.05) and compare the error (in cell C66) with
the error of the linear
method (in adjacent cell B67). You will find that the reversed cubic
method is effective at fitting
moderate degrees of non-linearity, and it does so over the entire range
of concentrations (unlike the bracket method). It fails, however,
if the analytical curve is too non-linear, especially if it goes to a
flat plateau or doubles back on
itself. Try increasing n all the way up to 0.1 and note the error is not so low.
But the real problem with non-linear fits is when there is lots of random error (noise). Return n to 0.05 (half-way up) and set Ev and Es=1. With a modest amount of random noise such as this, the cubic fit works pretty well. Compare the error (in cell C66) with the error of the linear
method (in adjacent cell B67). In
this case, the non-linearity is the dominant source of inaccuracy.
Note that the relative standard deviation of 20 repeat
calibrations is about 2%, a little higher than the linear calibration
curve (but that's hardly a deal-breaker if the error due to
non-linearity is greater than that due to random noise).
But now set Ev
and Es=5. Press the f9 key a few times to simulate different
calibration curves. Now the plot shows a good bit of discrepancy
between the actual analytical curve (blue) and the cubic fit to the
data points (red). The curve fit does its best to fit the data
points, even if it has to weave a wavy line through and between the
points. With more random error, the result get truly strange in some
cases.
The bottom line is that, if you know
from previous experience that the true calibration curve is linear,
then a linear fit will be better than a non-linear fit, especially if
the data are noisy, because a
non-linear fit will try to "fit the noise". If the calibration
curve is clearly non-linear, and the potential errors due to linear
curve-fitting are greater than the random errors due to noise, then a
non-linear fit is a good choice. (With really noisy data, a linear fit
may be best even if
the calibration curve is slightly non-linear).
Technical note: If you are interested in the dirty
underbelly of polynomial least-squares calculations with spreadsheets,
this spreadsheet uses the LINEST function (common to Excel and OpenOffice Calc) in cell B136 of CalCurveCubicFitOO.ods to fit a cubic polynomial to the plot of concentration (y-axis) vs measured absorbance (x-axis). This
method reverses the usual order of axes in order to avoid the need
to solve a cubic equation when the calibration equation is solved for C
and used to convert the measured signals of the unknowns into
concentration. The syntax is LINEST(E117:E126;B117:D126;0;0), where E117:E126 are the 10 concentrations of the standards, D117:D126 are the measured absorbances, C117:C126 are the absorbances squared, and B117:B126 are
the absorbances cubed. (Important detail: Because this is an array function, rather than a normal function, when you enter this function into the cell you have to press Ctrl-Shift-Enter rather than just Enter). The function returns the first-order
coefficient (equivalent to the slope) in cell B136 (the variable named
"qa" in the spreadsheet), the second-order coefficient in cell C137
(the variable "qb"), and the third-order coefficient in cell D137
(the variable "qc"). The constant term is zero. These
coefficients are then used to compute the concentrations C of unknown
samples from their measured absorbance A: C = qa*A+qb*A2+qc*A3.
In the Statistics section, this entire cubic calibration
procedure is repeated 20 times, in the 20 bordered blocks of cells that
extend to the right between rows 115 and 140 out to column DP, and the
results for each repeat are collected in the Results table in column
J. Whew!
The Single Standard Addition Method
OpenSingleStandardOO.ods
(view Screen
Shot).
In this
method, the sample is divided into two portions: one is measured
unmodified and the other is "doped" with the addition of a small amount
of pure standard and then measured. This is similar to the single
standard method, in that only the sample and a single standard are
measured, but the difference is that in this case the standard solution
is in the same matrix as the sample,
so it is effected by the same multiplicative interference, no matter
what the origin of that interference might be.
The downside of this method is
that
each separate sample requires the preparation of its own standard,
whereas in the other methods one standard (or one set of standards) can
be used to analyze a whole series of different samples. Also, the
calculations must compensate for the fact that the concentration of the
standard solution now contains an unknown contribution from the unknown
sample, but this is easily taken care of by a little algebra. The
result is only that the equation used to calculate the unknown
concentration is little more complicated, Cx =
Sx*Vxx*Cs)/(Ss*(Vxx+Vss)-Sx*Vxx), than the equation for the single
standard method, Cx = Cs(Sx/Ss).
To test this method, set mo=2, blank=0, Ev and Es=0 and use the sliders to set
z=0, n=0, Cx=5, and Cs=10 as before. Now move the interference slider (variable "z") to the right to
introduce an increasingly severe multiplicative interference. You can
see the analytical curve changes slope as you do this, but that
both the sample signal (yellow triangle) and the standard signal (green
triangle) track this change, and so the calculated sample concentration
(red triangle) remains accurate.
Now
try setting blank to 1 or 2, to test the affect of an additive
interference. Unfortunately, the standard addition method does
not correct for additive interferences, only for multiplicative
interferences. (You have to rely on other methods to compensate
for additive interferences, such asmultiwavelength methods, wavelength modulation, derivative methods, peak fitting, high-resolution spectroscopies, separation methods, etc). Also, a linear analytical curve is a requirement.
Set
Ev and Es=1 to introduce a small random error. The predicted standard
deviation of Cx (Cell C70) is computed by breaking down the equation for Cx into a
series of
differences, sums, products, and ratios, and applying the rules for error propagation, repeatedly. These error propagation calculations are performed in cells D111:F118. Comparing the measured RSD of this method with that of the single standard method,
you can see that the single standard addition method is less precise by
about a factor of 2, which might seem surprising considering that both
methods measure the unknown sample along with a single standard
solution. You can understand what is going on here by looking at the
expressions for Cx for the two methods: for the single standard method,
it is Cx=Cs*Sx/Ss; for the standard addition method, it is Cx
=
Sx*Vss*Cs)/(Ss*(Vxx+Vss)-Sx*Vxx). The extra volume terms Vss and
Vxx, both of which are subject to random volumetric errors, do not
occur in the single standard in the single standard method. Moreover,
the denominator is the difference between two noisy quantities, Ss*(Vxx+Vss) and Sx*Vxx, which increases the relative standard deviation of the difference. The result is that the precision of standard addition is noticeably poorer than the single standard method, but this the price for correcting for multiplicative interference.
The Multiple Standard Addition Method
The standard addition method can also be used with multiple standards: (StandardAdditionOO.ods , view Screen Shot). In this method a
series of aliquots of the sample solution are taken, increasing amounts
of standard material are added to each one, and the signals from the
resulting mixtures are measured and plotted against the concentration
of added standard. If the resulting calibration curve is sufficiently
linear, a
first-order least-squares fit of the data is computed. The sample
concentration is given by the negative of the x-axis intercept (and to
the ratio of the y-intercept to the slope). The advantage over
the single addition method is that you can verify the linearity of the
calibration curve.
To test this method, keep the same conditions as before (mo=2,
blank=0, Ev and Es=0 and use the sliders to set
z=0, n=0, Cx=5, and Cs=10) and set the number of standards ("ns") to 4.
You can see that the calibration curve is linear and that the x-axis intercept is exactly -5 (which agrees with the negative of Cx). Now move the interference slider (variable "z") to the right to
introduce an increasingly severe multiplicative interference. You can
see the analytical curve changes slope as you do this, but that the x-axis
intercept remains unchanged, proving that this method corrects
perfectly for multiplicative interferences (slope changes). If
you move the Cx slider to change the analyte concentration, the whole
curve slides up and down, so that the x-axis intercept tracks the changes in Cx.
Now introduce some random error: set Ev
and Es = 1%. The calibration curve still looks pretty good, but
as you change the interference slider ("z") or press f9 to recalculate,
the x-axis
intercept changes slightly, as does the "result" in cell C65. The predicted relative standard deviation of Cx (cell C68) is
the quadratic sum (square root of the sum of the squares) of the
relative standard deviations of the slope and intercept computed by the
curve fitting procedure. These error propagation calculations are
performed in cells B82:F87. Note that the %RSD of 20 repeats (cell C72)
is about 2.6%, significantly
greater than Ev or Es, and is only roughly predicted by the Est. RSD (cell C68). However, if you increase the number of standards ("ns") to 16, the
%RSD of 20 repeats is about half that with ns=4 and is much better
predicted by the Est. RSD (both about 1.3%). As you saw before, in the linear calibration curve method, the predicted
RSD (because it is based on a single calibration curve) is extremely
unreliable when the number of standards is small, and secondly, the
%RSD of the result improves slightly when the number
of standards is increased greatly.
If you compare the precision of this method to that of the linear calibration curve method, you'll notice that the multiple
standard method is poorer, even though their expressions for Cx are
very similar: Cx = (Sx-intercept)/slope v. Cx = intercept/slope.
Here again, correlation between terms is significant: there is
sufficient negative correlation between the intercept and the slope in the multiple standard method (the intercept goes down when the slope goes up and vice versa) that the relative standard deviation (RSD) of the ratio of the two is poorer than the square root of the sum of the squares of the relative standard deviations of the two terms individually (as would be the case if they were not correlated).
Table 1: Comparison of calibration methods
Method
Complexity (1=low; 5=high)
Detect non- linearity?
Correct non- linearity?
Correct multiplicative interference?
Correct additive interference?
Single standard
1
no
no
no
no
Bracket
2
yes
partial
no
no
Calibration Curve linear
3
yes
no
no
no
Calibration Curve cubic
4
yes
yes
no
no
Single addition
2
no
no
yes
no
Multiple addition
5
yes
no
yes
no
Table 2: Precision of calibration methods (conditions: Cx = 10, highest standard = 10, Ev = Es = 1%, zero non-linearity, zero interferences)
Method
Volumetric error
Signal error
Predicted RSD
Measured RSD
Single standard
0%
1%
1.4%
1.4%
1%
0%
1.0%
1.0%
1%
1%
1.7%
1.7%
Bracket
0%
1%
1.9%
1.4%
1%
0%
1.6%
1.0%
1%
1%
2.5%
1.8%
Cal. Curve linear
0%
1%
1.2%
1.2%
10 standards
1%
0%
.5%
.5%
1%
1%
1.4%
1.4%
10%
10%
14%
14%
Cal. Curve cubic
0%
1%
--
1.5%
10 standards
1%
0%
--
1.5%
1%
1%
--
2.0%
10%
10%
unstable
Single addition
0%
1%
2.4%
2.4%
1%
0%
2.3%
1.0%
1%
1%
3.4%
3.4%
Multiple addition
0%
1%
2.0%
2.0%
10 standards
1%
0%
2.0%
2.0%
1%
1%
2.5%
2.5%
Table 3: Effect of number of standards for Linear Calibration Curve (Same conditions as above)
Number of standards
RSD of slope
SD of intercept
Predicted RSD
Measured RSD
4
1.4
0.1
2
2
16
0.7
0.05
1.7
1.7
Student assignment for Standard Addition Method, WingZ version:
Wingz player application and basic set of simulation modules,
for windows PCs or Macintosh
The Single Standard Addition Method
Our textbook, Ingle and Crouch, Chapter 6, page 179, says "The
standard addition procedure is a powerful technique that is often used
improperly due to a failure to understand the assumptions involved."
This simulation will help you appreciate the capabilities and
limitations of the standard addition procedure.
1. Open StandardAddition.wkz. This model is
based on the text, page 178-179 and Equation 6-16. The same terminology
is used, with the following modifications: Ss is used for the signal
measured after standard addition instead of Sx+s. Cx means the true
analyte concentration (the unknown in the simulated experiment); the
experimental quantity calculated by equation Equation 6-16, which is
supposed to be a measure of Cx, is called "result". The volumes Vx and Vs
mean the actual volumes (including the random volumetric errors); nomVx and nomVs are the "nominal" volumes, that
is, the labeled volumes of the pipettes and flasks.
2. The
simulation includes the effect of a multiplicative interference (Io =
interferent concentration) and additive interference, i.e. blank error
(blank = uncorrected blank signal), and random errors in volume and
signal measurement. Errors are assumed to be a fixed percentage of the
quantity measured (fixed relative error rather than fixed absolute
error). The analytical curve is assumed to be linear.
3. The following are the independent variable that you can change:
mo
Analytical curve slope without interference
z
Interference factor (zero => no interference)
Io
Interferent concentration in original sample
Ev
Random volumetric error (% RSD )
Es
Signal measurement error (% RSD)
Cx
Analyte concentration in original sample solution
Cs
Analyte concentration of standard solution
blank
(Uncorrected) blank signal
nomVx
Nominal volume of sample solution before addition
nomVs
Nominal volume of standard added to sample
To
change any of these, click on the number (not on the symbol) in the
spreadsheet, type a new value, and press the enter key. The other
quantities in the spreadsheet are dependent variables that are
calculated from these independent variables. The most important of
these is result, which is the experimental estimate of Cx calculated by
equation Equation 6-16. In this simulation we will compare result to
the correct value Cx to see how well Equation 6-16 works.
4.
Choose any value of Cx and nomVx you like, then set Cs = ten-fold or so
larger than Cx. Start with the ideal case of no interferenc (Io =
0; blank = 0) and no random errors (Ev = 0 and Es = 0). Verify that
result = Cx for arbitrary Cs, nomVx, and nomVs.
5. Introduce a
multiplicative interference by making Io > 0 and z > 0, keeping
blank = 0. (The recovery expresses by what percent the analytical
signal is changed by the interference). Does result = Cx? Try arbitrary
values of Io, z, Cx, Cs, nomVx, and nomVs and notice the effect on
result.
6. Introduce an additive interference by making blank
> 0. Compare result and Cx. What do you conclude about the ability
of the standard addition method to compensate for additive and
multiplicative interferences?
7.
Introduce random errors into the volumetric measurement (Ev) and the
signal measurement (Es). To start with make both 1% RSD (Ev
= Es =1). Set Io > 0 and z > 0, keeping blank = 0 to simulate a
multiplicative interference only. Click on the 20 repeat runs button to
simulate 20 separate standard addition measurements. (Quick repeat does
the same thing, only faster). The table on the right shows the result
of each measurement, and at the bottom computes the mean, standard
deviation (s), percent relative standard deviation, and the error (%
difference between the mean and Cx). Why is it that if you perform
several successive 20-run simulations under fixed conditions, the
standard deviation is the exactly the same each time? How could the
simulation be designed to make the standard deviation more reproducible?
8.
Vary Cs and nomVs and observe the effect on the percent relative
standard deviation of the 20 repeats. Is there an optimum value of Cs
and nomVs that minimizes this error? On the basis of your observations,
formulate a rule that allows you to predict the optimum value of Cs and
nomVs.
9. Why is it that, even under the best condition, the % RSD of result is greater than Es or Ev?
(c) T.C. O.Haver, 1992 (WingZ versions), 2009 (OpenOffice versions), Prof. Tom O'Haver ,
Professor Emeritus, The University of Maryland at College Park.
Comments, suggestions and questions should be directed to Prof. O'Haver
at toh@umd.edu. Last updated June 26, 2009.
Number of unique visits since May 17, 2008:
The yellow table on the top left of the screen are independent variables that
you can change. Click on
the dark blue boldface numbers, type a new
value and press the enter key. Some of these variables can be
controlled continuously by the sliders on the top right. (The units of
concentration in these simulations are normalized to
the range of 0 - 10 for convenience in entering and plotting; you can
think of them as mmoles/liter, µmoles/liter, grams/liter or any other
convenient unit (1 mmole = 0.001 moles; 1 µmole = 10-6 moles). Similarly, the signal units are arbitrary
for
similar reasons).
The graph on the lower right shows the actual analytical curve
(blue
line)
over the concentration range from 0 to 10 (arbitrary units), with the
actual concentration of the unknown sample marked as a yellow triangle.
The red triangles are the standards. The green triangle is the
calculated concentration of the unknown sample according to that
calibration method. In the linear calibration curve and multiple standard addition methods, the red
line represents the
linear least-squares fit to the calibration curve. The graph responds
dynamically as you adjust the variables (e.g. with
the sliders).
The table in the middle left of the screen, labeled "Computed results", are
dependent variables that are automatically calculated from
the
independent variables (don't type in those cells or you will delete the
formulae). The most important dependent variable is "result", which is
a single simulated experimental measurement of the analyte
concentration Cx
based on that calibration method.
The "Statistics"
section at the lower left of the screen shows
the mean, standard deviation, and % relative standard deviation (%RSD) of 20
simulated repeat calibrations and measurements of the unknown concentration Cx. Here, a repeat calibration means that a complete set of new standards
are prepared for each repeat. This gives an idea of the reproducibility
if the entire procedure is repeated. The "% RSD" in the table is the
relative standard
deviation of 20 repeated simulated calibrations; it can be
compared to the "Est. RSD" in the Computed results table above, which
is the predicted relative standard. based on propagation or error rules.
Both should ideally be the same, but for a variety of
reasons will usually not agree exactly. The statistics are
re-calculated each time an input variable is changed or a slider is
moved. 
exactly 20.000 and 10.000
as before; they are a little off because of the effect of random
signal measurement error. If you click on the numbers for Ss
and Sx, and look at the entry bar at the top, you'll see the
equations for this numbers. Note that they both involve Es
because the same instrument is used to measure both Ss and
Sx. This causes the calculated sample concentration "result"
to be a little off as well. In fact, if you press the 
It's much larger than before - theoretically
8.7% - because of the extra effect of Ev. But again the actual results
bounce around quite a bit as you press f9, in this case mostly between 7 and 11%. What if you use
accurate volumetric glassware (which can have an RSD of 0.1%) and a
highly precise measurement technique that also gives a 0.1% RSD signal
measurement precision)? Set Es = Ev = 0.1% and see what you get for the % RSD of result.
factor" slider to set z
to about 0.5, which causes the analytical signal in the samples to be
substantially stronger than that in the standards (this is type of
multiplicative interference). Note that this causes the
calculated Cx to increase about 25% (as indicated by the Accuracy). Clearly, the single standard method can not compensate for this type of interference. Note: interferences are systematic errors that effect the accuracy but not the precision (% RSD).
Now test the effect of analytical curve linearity. Return "blank" to zero. 
Drag the "Analytical
curve non-linearity"
slider to the right and watch the analytical curve
(blue line) in the graph. The curve becomes concave down and the
accuracy degrades as the curvature increases, as indicated by the fact
that the green triangle on the graph (representing the calculated
concentration of the unknown sample) is no longer on top of the yellow
triangle (representing the true concentration). Clearly, the single standard method
depends on having a linear analytical curve. But the problem is that, in the real world, you
wouldn't even have a clue that the analytical curve is non-linear if
you used only one standard. For that, you'd need to measure more than a single standard.
solutions that are (ideally) close in concentration to the sample
(typically one lower than and one higher than the expected sample
concentration). The concentration of the sample Cx is calculated by
linear interpolation between the two standards (cell C65). The
bottom two sliders allow you to adjust the two standard concentrations.
The closer the the two standards are together, the smaller the error
due to
analytical curve non-linearity. Of course, this assumes that know
the sample concentrations beforehand, at least approximately, in order
to be able to make up appropriate bracketing standards. (For this
reason, the two-standard
bracket method is mostly used when the approximate range of unknown
concentrations is narrow and fairly well known, as in quality control
applications. It is not well suited when there are a large number
of samples of widely and unpredictable varying concentrations).
Now
slide the non-linearity slider up gradually and watch the shape of the
solutions and a linear least-squares fit. This is probably the most
common calibration method in general use. It is laid out just like
the previous simulations, with a few additions. You can choose the
number of standards (ns)
by typing into cell C57 or by clicking on the arrowheads of the "spin
button". Cs (controlled by the bottom slider) now controls the
concentration of the highest standard solution. The concentrations of the other standards are spaced out evenly between 0 and Cs.
The slope and intercept
of the linear least-squares fit to the calibration curve (the red line
on the graph) is shown in the computed results section, and the
equation of the fit and the R2
value (the "Coefficient of Determination", sometimes called the
"Correlation Coefficient") is shown in the upper left of the graph. The R2
value is one way to estimate the "goodness-of-fit" of the least-squares
line to the data; it is 1.000 when the fit is perfect and less than
1.000 when the fit is imperfect. 
the sample concentration: namely Cx = (Sx-intercept)/slope.
This is done in the table D110:F119, and the result of this calculation is shown as "Pred RSD" in
cell C74. The prediction is based only on a single calibration
curve and is good only insofar as that calibration curve is typical of
others that might be obtained in repeated trials. If your random errors happen to be small when you run your calibration curve, you'll get a deceptively good-looking calibration curve, but then your estimates of the random error in the slope and intercept will be too low. If your random errors happen to be large, you'll get a deceptively bad-looking calibration curve, but then your estimates of the random error in the slope and intercept will be too high. (Here are two examples taken from a set of 20 repeats, one "good" and one "bad", that illustrate this point). Some days it just does not pay to be lucky.
solutions and a reverse cubic least-squares fit. Set
the usual starting conditions: mo=2, blank=0, Ev and Es=0 and use the
sliders to set
z=0, n=0, Cx=5, and Cs=10. Obviously in this perfect linear case
the results are essentially perfect (zero standard deviation, almost
perfect accuracy, and R2 = 1.000). 
In this
method, the sample is divided into two portions: one is measured
unmodified and the other is "doped" with the addition of a small amount
of pure standard and then measured. This is similar to the single
standard method, in that only the sample and a single standard are
measured, but the difference is that in this case the standard solution
is in the same matrix as the sample,
so it is effected by the same multiplicative interference, no matter
what the origin of that interference might be. 
You can
see the analytical curve changes slope as you do this, but that
both the sample signal (yellow triangle) and the standard signal (green
triangle) track this change, and so the calculated sample concentration
(red triangle) remains accurate. 
The standard addition method can also be used with multiple standards: (StandardAdditionOO.ods , view Screen Shot). In this method a
series of aliquots of the sample solution are taken, increasing amounts
of standard material are added to each one, and the signals from the
resulting mixtures are measured and plotted against the concentration
of added standard. If the resulting calibration curve is sufficiently
linear, a
first-order least-squares fit of the data is computed. The sample
concentration is given by the negative of the x-axis intercept (and to
the ratio of the y-intercept to the slope). The advantage over
the single addition method is that you can verify the linearity of the
calibration curve.
To test this method, keep the same conditions as before (mo=2,
blank=0, Ev and Es=0 and use the sliders to set
z=0, n=0, Cx=5, and Cs=10) and set the number of standards ("ns") to 4.
You can see that the calibration curve is linear and that the x-axis intercept is exactly -5 (which agrees with the negative of Cx). Now move the interference slider (variable "z") to the right to
introduce an increasingly severe multiplicative interference. You can
see the analytical curve changes slope as you do this, but that the x-axis
intercept remains unchanged, proving that this method corrects
perfectly for multiplicative interferences (slope changes). If
you move the Cx slider to change the analyte concentration, the whole
curve slides up and down, so that the x-axis intercept tracks the changes in Cx. 
Now introduce some random error: set Ev
and Es = 1%. The calibration curve still looks pretty good, but
as you change the interference slider ("z") or press f9 to recalculate,
the x-axis
intercept changes slightly, as does the "result" in cell C65. The predicted relative standard deviation of Cx (cell C68) is
the quadratic sum (square root of the sum of the squares) of the
relative standard deviations of the slope and intercept computed by the
curve fitting procedure. These error propagation calculations are
performed in cells B82:F87. Note that the %RSD of 20 repeats (cell C72)
is about 2.6%, significantly
greater than Ev or Es, and is only roughly predicted by the Est. RSD (cell C68). However, if you increase the number of standards ("ns") to 16, the
%RSD of 20 repeats is about half that with ns=4 and is much better
predicted by the Est. RSD (both about 1.3%). As you saw before, in the linear calibration curve method, the predicted
RSD (because it is based on a single calibration curve) is extremely
unreliable when the number of standards is small, and secondly, the
%RSD of the result improves slightly when the number
of standards is increased greatly.