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Data Analysis
Since analytical chemistry is the science of making quantitative measurements,
understanding the difference between accuracy and precision is vital. Also, it is important
that raw data is manipulated and reported correctly to give a realistic estimate of the
uncertainty in a result. Presentation of data in the form of graphs is extremely useful.
Simple data manipulations may only require keeping track of significant figures. More
complicated calculations require propagation-of-error methods.
The uncertainity in a result can be categorized into random error and systematic error.
See the statistical formula document for more quantitative descriptions of describing and
testing data sets.
See the statistics of sampling document for information about uncertainty involved in
Accuracy and Precision
The accuracy of an analytical measurement is how close a result comes to the true value.
Determining the accuracy of a measurement usually requires calibration of the analytical
method with a known standard.
Precision is the reproducibility of multiple measurements and is usually described by the
standard deviation, standard error, or confidence interval.
Making measurements with any analytical method or instrument requires calibration to
ensure the accuracy of the measurement. There are two common calibration procedures:
using a working curve, and the standard-addition method. Both of these methods require
one or more standards of known composition to calibrate the measurement.
Instrumental methods are usually calibrated with standards that are prepared (or
purchased) using a non-instrumental analysis. There are two direct analytical methods:
gravimetry and coulometry. Titration is similar but requires preparation of a primary
The chief advantage of the working curve method is that it is rapid in that a single set of
standards can be used for the measurement of multiple samples. The standard-addition
method requires multiple measurements for each sample, but can reduce inaccuracies due
to interferences and matrix effects.
Graphs and Graphical Methods
By a graph we mean a representation of numerical values or functions by the positions of
points and lines on a two-dimensional surface. A graph is inherently more limited in
precision than a table of numerical values or an analytic equation, but it can contribute a
"feel'' for the behavior of data and functions that numerical tables and equations cannot.
A graph reveals much more clearly such features as linearity or nonlinearity, maxima and
minima, points of inflection, etc. Graphical methods of smoothing data and of
differentiation and integration are sometimes easier than numerical methods. Graphs and
graphical methods suffer, of course, from the limitations of a two-dimensional surface.
Thus, normally, a plotted point has only two degrees of freedom, which we assume here
to be represented by one independent variable x (increasing from left to right) and one
dependent variable y (increasing from bottom to top). The word "normally" is required in
the above sentence because three-dimensional plots, representing, for example, points
with coordinates x, y, and z, can be represented in oblique projection on a twodimensional surface by computer techniques.
Wherever possible, the data from an experiment should be plotted at an early stage, even
if numerical methods will be used subsequently for greater accuracy. This is particu-larly
true when functions are to be fitted by least-squares or other methods, since a graph may
make evident special problems or requirements that might otherwise be missed.
Following are sections on:
A. Manual preparation of graphs
B. Computer generated plots
C. Graphical uncertainties
A. Manual preparation of graphs
For constructing a graph by hand, the following six step procedure is recommended.
1. Design your graph first
2. Choose your graph paper
3. Define the plotting area
4. Plot the points; Draw the lines or curves
5. Add a legend
6. Check the overall appearance
1. Design Your Graph First.
Plan it so as to make best use of the area available and present the data in a manner that
will be clearest to the viewer. In most cases the entire useful area of a normal-sized sheet
of graph paper should be dedicated to the graph. In some cases, the paper should be
turned sideways (i.e., with the binding margin on top) so that the independent variable, x,
runs longways on the page, to provide an aspect ratio more favorable for displaying the
desired features.
Determine the maximum and minimum values of x and y (remember, it is not necessary
that they both be zero at the lower left corner!) so as to fit the points, lines, and curves
into the available area. Be sure to make adequate allowance of space for borders all
around, scale numbers and scale labels, at least at the bottom and at the left, and a legend
telling what the graph is about.
Return to top.
2. Choose Your Graph Paper.
The graph paper should be of good quality, accurately ruled with thin, lightweight lines.
In most cases these will constitute a grid of lines equally spaced in two perpendicular
directions; special papers are available for plotting logarithmic functions in one direction
("semilog paper") or in both directions ('log-log paper"), or for plotting in polar
coordinates, etc. Most good graph papers are ruled in colors that are soft enough that they
do not distract the viewer from the plotted lines and points, which are usually in black.
The most commonly used graph papers are ruled with a 1-mm spacing, the lines at 5 and
10 mm being heavier. The smallest divisions on the paper should preferably represent
multiples of 1 or 2 or perhaps 5 in the plotted variable.
Return to top.
3. Define the Plotting Area.
Draw, with heavy pencil lines, a rectangle within which all points and lines will be
plotted, leaving sufficient margins all around, extra width for the binding margin, and
also adequate allowance for the scale numbers and legends mentioned above. Indicate the
major scale divisions with short, heavy pencil marks and label them with appropriate
numerical values. Under the scale numbers for the independent variable enter the scale
label, stating the quantity that is varying, and its units, e.g.,
for absolute temperature in kelvin. To avoid scale numbers that are too large or too small
for convenient use, multiply the quantity by a power of 10, e.g.,
 x l04 (g cm-3)
for the density of a gas. This is equivalent to multiplying all the scale numbers by 10-4 but
is much more convenient. Similarly, enter the scale label for the dependent variable along
the side scale on the left; if necessary, turn the paper sideways to write the label so that it
will run vertically.
Return to top.
4. Plot the Points; Draw the Lines and Curves.
Using a sharp, hard pencil, plot the experimental points as small dots, as accurately as
possible. Try to plot with an accuracy of one-fifth of the smallest grid division. Draw
small circles (or squares, triangles, etc.) of uniform size (2 or 3 mm) around the points in
ink to give them greater visibility.
If it is desired to draw a straight line to provide a linear fit to the plotted points, draw a
faint pencil line with a good straightedge such as a transparent ruler or draftsman's
triangle, and do not hesitate to erase the line and try again. When you are satisfied that
you have made the best possible visual fit, go over the line again with a softer pencil or a
pen to make it sufficiently heavy; this time, do not allow the line to cross the circles or
other symbols drawn around the experimental points. Smooth nonlinear curves may be
drawn either freehand or with ships curves, splines, or other devices, with as much trial
and error and erasing as required.
If a "theoretical curve" is to be drawn based on an analytic function representing
predicted behavior, plot the function at as many points, closely spaced, as reasonably
possible; do not draw symbols around them. Carefully draw a smooth curve through the
points, lightly at first, and then "heavy it up" as desired.
Return to top.
5. Add a Legend.
Somewhere on the graph (if possible, at the bottom) enter a legend with an identifying
figure number. The legend should state what the graph is about, identify symbols .and
line types used, and provide all needed information that will not be provided in the
document to which the graph will be attached.
Return to top.
6. Check the Overall Appearance.
After all points, lines, curves, scales, and legends are in place, check the overall clarity
and legibility of the graph. If points or lines are too faint, make them heavier; if the graph
is messy or confusing, attempt to improve it by erasing and redrawing the features
concerned. If the end result falls seriously short in the expected qualities, start over. Your
second attempt will probably take much less time to prepare, and it will look much better
and will better suit its intended purpose.
Return to top.
B. Computer generated plots
In most cases, you should take advantage of the ability of computer spreadsheet programs
to generate graphs. These may be generated on the computer's monitor screen for
inspection and editing and then printed out on ink-jet or laser printers. Much of the above
discussion of manual preparation of graphs applies also to computer graphs. Since
computer graphs can be produced quickly, it may be worth while to make several tries to
obtain the best results. The first plot may be a "default plot'' to see the overall appearance
of the plotted data; then user-chosen scales, symbols, aspect ratio, and labels may be
introduced as needed. If the computer output uses generic labels such as x and y, be sure
to relabel with appropriate variables.
Return to top.
C. Graphical uncertainties
It frequently happens that an intermediate or final result of the calculations in a given
experiment is obtained from the slope or intercept of a straight line on a graph - for
example, a plot of y against x. In such a case it is desirable to evaluate the uncertainty in
the slope or in the position of the intercept. A rough procedure for doing this is based on
drawing a rectangle with width 2(xi) and height 2(yi) around each experimental point
(xi, yi), with the point at its center.
The significance of the rectangle is that any point contained in it represents a possible
position of the ''true'' point (xi, yi) and all points outside are ruled out as possible
positions. Having already drawn the best straight line through the experimental points,
and having derived from this line the slope or intercept, now draw two other (dashed)
lines representing maximum and minimum values of the slope or intercept, consistent
with the requirement that both lines pass through every rectangle. as shown in the figure
below. Where there are a dozen or more experimental points, it may be justifiable to
neglect partially or completely one or more obviously "bad'' points in drawing the
original straight line and the limiting lines, provided that good judgment is exercised. The
difference between the two slopes or intercepts of the limiting lines can be taken as an
estimate of twice the limit of error in the slope or intercept of the best straight line.
Figure Caption: A graphical method of determining the limit of error in a slope.
Where the number of points is sufficiently large, the limits of error of the position of
plotted points can be inferred from their scatter. Thus, an upper bound and a lower bound
can be drawn, and the lines of limiting slope drawn so as to lie within these bounds. Such
graphical methods are justifiable only for rough estimates. In either case, the possibility
of systematic error should be kept in mind.
General rules for Significant Figures
reporting results
Results are generally reported with one uncertain significant figure (usually +/-1).
Example: reporting a mass of 12.554 g indicates that the experimenter is certain that the
true mass lies in the range 12.553 - 12.555 g. If the uncertainty in the last significant
figure is greater than +/- 1 it should be written explicitly: 12.554 g +/- 0.003 g.
addition and subtraction
The answer must be truncated to the same least significant decimal place as the number
with the least significant figure from the right.
8.9444 g
multiplication and division
The answer has the same number of significant figures as the factor with the smallest
number of significant figures.
8.9 g / 12.01 g/mol = 0.74 mol
It is common to carry one additional significant figure through extended calculations and
to round off the final answer at the end.
Standard Addition
Due to matrix effects the analytical response for an analyte in a complex sample may not
be the same as for the analyte in a simple standard. In this case, calibration with a
working curve would require standards that closely match the composition of the sample.
For routine analyses it is feasible to prepare or purchase realistic standards, e.g. NIST
standard reference materials.
For diverse and one-of-a-kind samples, this procedure is time consuming and often
An alternative calibration procedure is the standard addition method. An analyst usually
divides the unknown sample into two portions, so that a known amount of the analyte (a
spike) can be added to one portion. These two samples, the original and the original plus
spike, are then analyzed. The sample with the spike will show a larger analytical response
than the original sample due to the additional amount of analyte added to it.
The difference in analytical response between the spiked and unspiked samples is due to
the amount of analyte in the spike. This provides a calibration point to determine the
analyte concentration in the original sample.
Example Calculation
(F--selective electrode)
Two 10.0 ml aliquots are taken from a sample of wastewater effluent and one is spiked
with 1.00 ml of 0.0207 M NaF standard solution. The response from a F--selective
electrode is 66.2 mV for the unspiked sample and 28.6 mV for the spiked sample. The Fconcentration (Note: we use [F-] for clarity but the electrode actually measures activity.)
in the original sample is:
E = constant - 2.303 * (RT/nF) * log[F-]
Measurement of the unspiked sample:
0.0662 V = constant - (0.0591 V) * log[F-]
Measurement of the spiked sample:
0.0286 V = constant - (0.0591 V) * log{([F-]0.0100 L + 0.0207 M * 0.00100 L)/0.0110
0.0286 V = constant - (0.0591 V) * log{0.909[F-] + 0.00188}
Combining equations:
0.0662 V - 0.0286 V = -(0.0591 V) * log{[F-]/(0.909[F-] + 0.00188)}
-0.636 = log{[F-]/(0.909[F-] + 0.00188)}
0.231 = [F-]/(0.909[F-] + 0.00188)
[F-] = 5.50 x 10-4
Statistical Methods
The following methods are used to describe, analyze, or test sets of data.
Statistical Formulas
standard deviation
A measure of the uncertainty due to random error in a set of data (also: precision
of a set of measurements).
A test to determine if one data point in a set of data is significantly different
enough to be discarded.
linear regression
A method to find the best straight line through a set of data.
Working Curve or Calibration Curve
A working curve is a plot of the analytical signal (the instrument or detector response) as
a function of analyte concentration. These working curves are obtained by measuring the
signal from a series of standards of known concentration. The working curves are then
used to determine the concentration of an unknown sample, or to calibrate the linearity of
an analytical instrument.
Example of a working curve
Determining the best line for calibration data is done using linear regression.
Electrochemistry can be broadly defined as the study of charge-transfer phenomena. As
such, the field of electrochemistry includes a wide range of different chemical and
physical phenomena. These areas include (but are not limited to): battery chemistry,
photosynthesis, ion-selective electrodes, coulometry, and many biochemical processes.
Although wide ranging, electrochemistry has found many practical applications in
analytical measurements.
Electroanalytical chemistry
A good working definition of the field of electroanalytical chemistry would be that it is
the field of electrochemistry that utilizes the relationship between chemical phenomena
which involve charge transfer (e.g. redox reactions, ion separation, etc.) and the electrical
properties that accompany these phenomena for some analytical determination. This
relationship is further broken down into fields based on the type of measurement that is
Potentiometry involves the measurement of potential for quantitative analysis, and
electrolytic electrochemical phenomena involve the application of a potential or current
to drive a chemical phenomenon, resulting in some measurable signal which may be used
in an analytical determination.
Electrolytic Methods
Unlike potentiometry, where the free energy contained within the system generates the
analytical signal, electrolytic methods are an area of electroanalytical chemistry in which
an external source of energy is supplied to drive an electrochemical reaction which would
not normally occur. The externally applied driving force is either an applied potential or
current. When potential is applied, the resultant current is the analytical signal; and when
current is applied, the resultant potential is the analytical signal. Techniques which utilize
applied potential are typically referred to as voltammetric methods while those with
applied current are referred to as galvanostatic methods.
Voltammetric Methods
Voltammetry refers to the measurementof current that results from the application of
potential. Unlike potentiometry measurements, which employ only two electrodes,
voltammetric measurements utilize a three electrode electrochemical cell. The use of the
three electrodes (working, auxillary, and reference) along with the potentiostat
instrument allow accurate application of potential functions and the measurement of the
resultant current. The different voltammetric techniques that are used are distinguished
from each other primarily by the potential function that is applied to the working
electrode to drive the reaction, and by the material used as the working electrode.
Common techniques to be discussed here include:
Hydrodynamic Voltammetry
Normal-pulse polarography (NPP)
Differential-pulse polarography (DPP)
Cyclic voltammetry
Anodic-stripping voltammetry
Time Based Techniques
Chromatography is a separations method that relies on differences in partitioning
behavior between a flowing mobile phase and a stationary phase to separate the the
components in a mixture.
A column (or other support for TLC, see below) holds the stationary phase and the
mobile phase carries the sample through it. Sample components that partition strongly
into the stationary phase spend a greater amount of time in the column and are separated
from components that stay predominantly in the mobile phase and pass through the
column faster.
As the components elute from the column they can be quantified by a detector and/or
collected for further analysis. An analytical instrument can be combined with a separation
method for on-line analysis. Examples of such "hyphenated techniques" include gas and
liquid chromatography with mass spectrometry (GC-MS and LC-MS), Fourier-transform
infrared spectroscopy (GC-FTIR), and diode-array UV-VIS absorption spectroscopy
Specific chromatographic methods:
Gas chromatography (GC)
Applied to volatile organic compounds. The mobile phase is a gas and the
stationary phase is usually a liquid on a solid support or sometimes a solid
High-performance liquid chromatography (HPLC)
A variation of liquid chromatography that utilizes high-pressure pumps to
increase the efficiency of the separation.
Liquid chromatography (LC)
Used to separate analytes in solution including metal ions and organic
compounds. The mobile phase is a solvent and the stationary phase is a liquid on a
solid support, a solid, or an ion-exchange resin.
Size-exclusion chromatography (SEC)
Also called gel-permeation chromatography (GPC), the mobile phase is a solvent
and the stationary phase is a packing of porous particles.
Thin-layer chromatography (TLC)
A simple and rapid method to monitor the extent of a reaction or to check the
purity of organic compounds. The mobile phase is a solvent and the stationary
phase is a solid adsorbent on a flat support.
Chromatography Theory
The underlying principles that determine chromatographic separations are dynamic
behaviors that depends on partitioning and mass transport. These phenomena are too
complex to model directly, and chromatography theory consists of empirical relationships
to describe chromatographic columns and the separation of peaks in chromatograms.
Description of Chromatograms
The retention of an analyte by a column is described by the capacity factor, k', where:
k' = tr - tr
where tr is the time for the analyte to pass through the column, and tm is the time for
mobile phase to pass through the column.
More info:
Resolution of peaks in a chromatogram
Description of chromatographic columns
The resolution of chromatographic columns is described by the theoretical plate height,
H, or the number of theoretical plates, N. These two quantities are related by:
where L is the length of the column.
H and N provide useful measures to compare the performance of different columns for a
given analyte. Useful expressions are:
H = L W2 / 16 tr2
N = 16 (tr / W)2
where W is the width of the peak at its base.
Gas Chromatography (GC)
Gas chromatography is a chromatographic technique that can be used to separate volatile
organic compounds. A gas chromatograph consists of a flowing mobile phase, an
injection port, a separation column containing the stationary phase, and a detector. The
organic compounds are separated due to differences in their partitioning behavior
between the mobile gas phase and the stationary phase in the column.
Mobile phases are generally inert gases such as helium, argon, or nitrogen. The injection
port consists of a rubber septum through which a syringe needle is inserted to inject the
sample. The injection port is maintained at a higher temperture than the boiling point of
the least volatile component in the sample mixture. Since the partitioning behavior is
dependant on temperture, the separation column is usually contained in a thermostatcontrolled oven. Separating components with a wide range of boiling points is
accomplished by starting at a low oven temperture and increasing the temperture over
time to elute the high-boiling point components. Most columns contain a liquid stationary
phase on a solid support. Separation of low-molecular weight gases is accomplished with
solid adsorbents. Separate documents describe some specific GC Columns and GC
Schematic of a gas chromatograph
Pictures of some gas chromatographs
Gas Chromatography Columns
Gas chromatography columns are of two designs: packed or capillary. Packed columns
are typically a glass or stainless steel coil (typically 1-5 m total length and 5 mm inner
diameter) that is filled with the stationary phase, or a packing coated with the stationary
phase. Capillary columns are a thin fused-silica (purified silicate glass) capillary
(typically 10-100 m in length and 250 µm inner diameter) that has the stationary phase
coated on the inner surface. Capillary columns provide much higher separation efficiency
than packed columns but are more easily overloaded by too much sample.
Picture of a packed GC column, Picture of capillary GC column
Stationary Phases
The most common stationary phases in gas-chromatography columns are polysiloxanes,
which contain various substituent groups to change the polarity of the phase. The
nonpolar end of the spectrum is polydimethyl siloxane, which can be made more polar by
increasing the percentage of phenyl groups on the polymer. For very polar analytes,
polyethylene glycol (a.k.a. carbowax) is commonly used as the stationary phase. After the
polymer coats the column wall or packing material, it is often cross-linked to increase the
thermal stability of the stationary phase and prevent it from gradually bleeding out of the
Small gaseous species can be separated by gas-solid chromatography. Gas-solid
chromatography uses packed columns containing high-surface-area inorganic or polymer
packing. The gaseous species are separated by their size, and retention due to adsorption
on the packing material.
Gas Chromatography (GC) Detectors
After the components of a mixture are separated using gas chromatography, they must be
detected as they exit the GC column. The links listed below provide the details of some
specific GC detectors. The thermal-conductivity (TCD) and flame-ionization (FID)
detectors are the two most common detectors on commercial gas chromatographs. The
requirements of a GC detector depends on the separation application. For example, one
analysis might require a detector that is selective for chlorine-containing molecules,
another analysis might require a detector that is non-destructive so that the analyte can be
recovered for further spectroscopic analysis.
Specific GC detectors
Atomic-emmision detector (AED)
Chemiluminescence detector
Electron-capture detector (ECD)
Electron Capture Detector (ECD)
The ECD uses a radioactive Beta emitter (electrons) to ionize some of the carrier gas and
produce a current between a biased pair of electrodes. When organic molecules that
contain electronegative functional groups, such as halogens, phosphorous, and nitro
groups pass by the detector, they capture some of the electrons and reduce the current
measured between the electrodes. The ECD is as sensitive as the FID but has a limited
dynamic range and finds its greatest application in analysis of halogenated compounds.
Schematic of an ECD
The ECD is as sensitive as the FID but has a limited dynamic range and finds its
greatest application in analysis organic molecules that contain electronegative
functional groups, such as halogens, phosphorous, and nitro groups.
Flame-ionization detector (FID)
Flame-Ionization Detector (FID)
An FID consists of a hydrogen/air flame and a collector plate. The effluent from the GC
column passes through the flame, which breaks down organic molecules and produces
ions. The ions are collected on a biased electrode and produce an electrical signal. The
FID is extremely sensitive with a large dynamic range, its only disadvantage is that it
destroys the sample.
Schematic of FID
The FID is extremely sensitive with a large dynamic range, its only disadvantage
is that it destroys the sample.
Flame-photometric detector (FPD)
Mass spectrometer (MS)
Mass spectrometers provide structural information to identify the analyte in a
chromatographic peak.
Photoionization detector (PID)
Photoionization Detector
The reason to use more than one kind of detector for gas chromatography is to achieve
selective and/or highly sensitive detection of specific compounds encountered in
particular chromatographic analyses. The selective determination of aromatic
hydrocarbons or organo-heteroatom species is the job of the photoionization detector
(PID). This device uses ultraviolet light as a means of ionizing an analyte exiting from a
GC column. The ions produced by this process are collected by electrodes. The current
generated is therefore a measure of the analyte concentration.
If the energy of an incoming photon is high enough (and the molecule is quantum
mechanically "allowed" to absorb the photon) photo-excitation can occur to such an
extent that an electron is completely removed from its molecular orbital, i.e. ionization.
A Photoionization Reaction
If the amount of ionization is reproducible for a given compound, pressure, and light
source then the current collected at the PID's reaction cell electrodes is reproducibly
proportional to the amount of that compound entering the cell. The reason why the
compounds that are routinely analyzed are either aromatic hydrocarbons or heteroatom
containing compounds (like organosulfur or organophosphorus species) is because these
species have ionization potentials (IP) that are within reach of commercially available
UV lamps. The available lamp energies range from 8.3 to 11.7 ev, that is, lambda max
ranging from 150 nm to 106 nm. Although most PIDs have only one lamp, lamps in the
PID are exchanged depending on the compound selectivity required in the analysis.
Selective detection using a PID
Here is an example of selective PID detection: Benzene's boiling point is 80.1 degrees C
and its IP is 9.24 ev. (Check the CRC Handbook 56th ed. page E-74 for IPs of common
molecules.) This compound would respond in a PID with a UV lamp of 9.5 ev
(commercially available) because this energy is higher than benzene's IP (9.24).
Isopropyl alcohol has a similar boiling point (82.5 degrees C) and these two compounds
might elute relatively close together in normal temperature programmed gas
chromatography, especially if a fast temperature ramp were used. However, since
isopropyl alcohol's IP is 10.15 ev this compound would be invisible or show very poor
response in that PID, and therefore the detector would respond to one compound but not
the other.
Since only a small (but basically unknown) fraction of the analyte molecules are actually
ionized in the PID chamber, this is considered to be a nondestructive GC detector.
Therefore, the exhaust port of the PID can be connected to another detector in series with
the PID. In this way data from two different detectors can be taken simultaneously, and
selective detection of PID responsive compounds augmented by response from, say, an
FID or ECD. The major challenge here is to make the design of the ionization chamber
and the downstream connections to the second detector as low volume as possible (read
small diameter) so that peaks that have been separated by the GC column do not broaden
out before detection.
Schematic of a gas chromatographic photoionization detector
Thermal conductivity detector (TCD)
The TCD is not as sensitive as other dectectors but it is non-specific and nondestructive.
A TCD detector consists of an electrically-heated wire or thermistor. The temperature of
the sensing element depends on the thermal conductivity of the gas flowing around it.
Changes in thermal conductivity, such as when organic molecules displace some of the
carrier gas, cause a temperature rise in the element which is sensed as a change in
resistance. The TCD is not as sensitive as other dectectors but it is non-specific and nondestructive.
Two pairs of TCDs are used in gas chromatographs. One pair is placed in the column
effluent to detect the separated components as they leave the column, and another pair is
placed before the injector or in a separate reference column. The resistances of the two
sets of pairs are then arranged in a bridge circuit.
Schematic of a bridge circuit for TCD detection
The bridge circuit allows amplification of resistance changes due to analytes passing over
the sample thermoconductors and does not amplify changes in resistance that both sets of
detectors produce due to flow rate fluctuations, etc.