Calibration methods
Chemistry 243
Figures of merit: Performance
characteristics of instruments
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Precision
Accuracy
Selectivity
Sensitivity
Limit of Detection
Limit of Quantitation
Dynamic Range
Precision vs. Accuracy in the
common verbiage (Webster’s)

Precision:
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Accuracy:
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The quality or state of being precise; exactness;
accuracy; strict conformity to a rule or a standard;
definiteness.
The state of being accurate; exact conformity to
truth, or to a rule or model; precision.
These are not synonymous when describing
instrumental measurements!
Precision and accuracy in this
course

Precision: Degree of mutual agreement among data
obtained in the same way.
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Accuracy: Measure of closeness to accepted value
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Absolute and relative standard deviation, standard error of
the mean, coefficient of variation, variance.
Extends in between various methods of measuring the
same value
Absolute or relative error
Not known for unknown samples
Can be precise without being accurate!!!
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Precisely wrong!
Precision - Metrics
Most important
Often seen as %
Handy, common
Sensitivity vs.
Limit of Detection
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NOT THE SAME THING!!!!!
Sensitivity: Ability to discriminate between small
differences in analyte concentration at a
particular concentration.
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calibration sensitivity—the slope of the calibration
curve at the concentration of interest
Limit of detection: Minimum concentration that
can be detected at a known confidence limit

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Typically three times the standard deviation of the
noise from the blank measurement (3s or 3s is
equivalent to 99.7% confidence limit)
Such a signal is very probably not merely noise
Calibration Curve, Limit of
Detection, Sensitivity
Signal
Sensitivity* = Slope
*Same as Working Curve
**Not improved by amplification alone
S/N = 3
0
0
LOD
Analyte Mass or Concentration
Selectivity

Degree to which a method is free from
interference from other contaminating signals
in matrix
S  mAc A  mB cB  mC cC

No measurement is completely free of
interferences
mB
 Selectivity coefficient:
kB, A 
mA
Calibration Curves:
Sensitivity and LOD

Signal

For a given sample standard
deviation, s, steeper calibration
curve means better sensitivity
Insensitive to amplification
S/N = 3
0
0
LOD
LOD
Analyte Mass or Concentration
Dynamic range

The maximum range over which an accurate
measurement can be made

From limit of quantitation to limit of linearity
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LOQ: 10 s of blank
LOL: 5% deviation from linear
Ideally a few logs
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Absorbance: 1-2
MS, Fluorescence: 4-5
NMR: 6
Calibration Curves:
Dynamic Range and Noise
Regions
Signal
Calibration
Curve
becomes
poor above
this amount
of analyte
Poor
Quant
Noise
Region
Dynamic Range
S/N = 3
0
0
LOD
LOQ
LOL
Analyte Mass or Concentration
Types of Errors
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Random or indeterminate errors
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Systematic errors
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Handled with statistical probability as already shown
Instrumental errors
Personal errors
Method errors
Gross errors

Human error
 Careless mistake, or mistake in understanding
 Often seen as an outlier in the statistical distribution
 “Exactly backwards” error quite common
Systematic errors
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Present in all measurements made in the same way
and introduce bias.
Instrumental errors
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Personal errors
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Wacky instrument behavior, bad calibrations, poor
conditions for use
 Electronic drift, temperature effects, 60Hz line noise,
batteries dying, problems with calibration equipment.
Originate from judgment calls
 Reading a scale or graduated pipette, titration end points
Method errors
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Non-ideal chemical or physical behavior
 Evaporation, adsorption to surfaces, reagent degradation,
chemical interferences
Instrument calibration
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Determine the relationship between response
and concentration
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Calibration curve or working curve
Calibration methods typically involve standards
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Comparison techniques
External standard*
Standard addition*
Internal standard*
* calibration curve is required
External standard calibration
(ideal)
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External Standard – standards are not in the
sample and are run separately
Generate calibration curve (like PS1, #1)
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Run known standards and measure signals
Plot vs. known standard amount (conc., mass, or mol)
Linear regression via least squares analysis
Compare response of sample unknown and
solve for unknown concentration

All well and good if the standards are just like the
sample unknown
External standard calibration
(ideal)
Sample
Unknown
Signal
External
Calibration
Standards
including
a blank
Sample
Unknown
Amount
S/N = 3
0
0
LOD
Analyte Mass or Concentration
In class example of external
standard calibration
Skoog, Fig. 13-13
b  pathlength
c  concentration
Sample
Unknown
Signal
P0
A  log
  bc
P
  molar absorptivity
External
Calibration
Standards
including
a blank
Sample
Unknown
Amount
S/N = 3
0
0
LOD
Analyte Mass or Concentration
Real-life calibration

Subject to matrix interferences

Matrix = what the real sample is in
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pH, salts, contaminants, particulates
Glucose in blood, oil in shrimp
Concomitant species in real sample lead to different
detector or sensor responses for standards at same
concentration or mass (or moles)
Several clever schemes are typically employed
to solve real-world calibration problems:
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Internal Standard
Standard Additions
Internal standard
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A substance different from the analyte added in a
constant amount to all samples, blanks, and
standards or a major component of a sample at
sufficiently high concentration so that it can be
assumed to be constant.
Plotting the ratio of analyte to internal-standard as a
function of analyte concentration gives the
calibration curve.
Accounts for random and systematic errors.
Difficult to apply because of challenges associated
with identifying and introducing an appropriate
internal standard substance.

Similar but not identical; can’t be present in sample
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Lithium good for sodium and potassium in blood; not in blood
Standard additions
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Classic method for reducing (or simply
accommodating) matrix effects
 Especially for complex samples; biosamples
 Often the only way to do it right
You spike the sample by adding known amounts of
standard solution to the sample
 Have to know your analyte in advance
Assumes that matrix is nearly identical after standard
addition (you add a small amount of standard to the
actual sample)
As with “Internal Standard” this approach accounts for
random and systematic errors; more widely applicable
Must have a linear calibration curve
How to use standard additions

To multiple sample volumes of an unknown,
different volumes of a standard are added and
diluted to the same volume.
Fixed parameters:
cs = Conc. of std. – fixed
Vt = Total volume – fixed
Vx = Volume of unk. – fixed
cx = Conc. of unk. - seeking
Calibration Standard
(Fixed cs)
Vx
Vx
Vx
Vx
Vs1
Vs2
Vs3
Vs4
Vt
Vt
Vt
Vt
Non-Fixed Parameter:
Vs = Volume of std. – variable
Volume top-off step:
Vx diluted to Vt
Vs diluted to Vt
How to use standard additions
To multiple sample volumes of an unknown,
different volumes of a standard are added
and diluted to the same volume.
𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑆𝑠𝑡𝑑 + 𝑆𝑥
Combined Signal

S1
S2
S3
S4
0
0
Concentration
How to use standard additions
𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑆𝑠𝑡𝑑 + 𝑆𝑥
𝑆𝑠𝑡𝑑 = 𝑘 ∙ 𝑓𝑑𝑖𝑙 ∙ 𝑐𝑠𝑡𝑑
𝑘𝑉𝑠𝑡𝑑 𝑐𝑠𝑡𝑑
=
𝑉𝑡𝑜𝑡𝑎𝑙
𝑘𝑉𝑥 𝑐𝑥
𝑆𝑥 = 𝑘 ∙ 𝑓𝑑𝑖𝑙′ ∙ 𝑐𝑥 =
𝑉𝑡𝑜𝑡𝑎𝑙
Combined Signal
k = slope or sensitivity
0
0
Concentration
How to use standard additions
𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑆𝑠𝑡𝑑 + 𝑆𝑥
𝑆𝑡𝑜𝑡𝑎𝑙
𝑘𝑉𝑠𝑡𝑑 𝑐𝑠𝑡𝑑 𝑘𝑉𝑥 𝑐𝑥
=
+
𝑉𝑡𝑜𝑡𝑎𝑙
𝑉𝑡𝑜𝑡𝑎𝑙
How to use standard additions
𝑆𝑡𝑜𝑡𝑎𝑙
𝑘𝑉𝑠𝑡𝑑 𝑐𝑠𝑡𝑑 𝑘𝑉𝑥 𝑐𝑥
=
+
𝑉𝑡𝑜𝑡𝑎𝑙
𝑉𝑡𝑜𝑡𝑎𝑙
𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑚𝑉𝑠𝑡𝑑 + 𝑏
𝑘𝑐𝑠𝑡𝑑
𝑚=
𝑉𝑡𝑜𝑡𝑎𝑙
𝑘𝑉𝑥 𝑐𝑥
𝑏=
𝑉𝑡𝑜𝑡𝑎𝑙
Remember,
Vstd is the
variable.
Knowns:
cstd
Vtotal
Vx
How to use standard additions
𝑘𝑐𝑠𝑡𝑑
𝑚=
𝑉𝑡𝑜𝑡𝑎𝑙
𝑘𝑉𝑥 𝑐𝑥
𝑏=
𝑉𝑡𝑜𝑡𝑎𝑙
S, Combined Signal
𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑚𝑉𝑠𝑡𝑑 + 𝑏
Get m (slope) and
b (intercept) from
linear least squares
0
0
How do I handle k ?
Vs
Determine cx via standard
curve extrapolation …
𝑆𝑡𝑜𝑡𝑎𝑙
𝑘𝑉𝑠𝑡𝑑 𝑐𝑠𝑡𝑑 𝑘𝑉𝑥 𝑐𝑥
=
+
𝑉𝑡𝑜𝑡𝑎𝑙
𝑉𝑡𝑜𝑡𝑎𝑙
At the x-intercept, S = 0
𝑘𝑉𝑠𝑡𝑑 𝑐𝑠𝑡𝑑
𝑘𝑉𝑥 𝑐𝑥
=−
𝑉𝑡𝑜𝑡𝑎𝑙
𝑉𝑡𝑜𝑡𝑎𝑙
𝑉𝑠𝑡𝑑 𝑐𝑠𝑡𝑑 = −𝑉𝑥 𝑐𝑥
Skoog, Fig. 1-10
𝑉𝑠𝑡𝑑 0 𝑐𝑠𝑡𝑑
𝑐𝑥 = −
𝑉𝑥
Seeking
[analyte]
known
known
Vstd when S = 0
… or determine cx by directly
using fit parameters
𝑉𝑠𝑡𝑑 𝑐𝑠𝑡𝑑
𝑐𝑥 = −
𝑉𝑥
𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑚𝑉𝑠𝑡𝑑 + 𝑏 = 0
𝑚𝑉𝑠𝑡𝑑 = −𝑏
Final calculation:
𝑉𝑠𝑡𝑑
𝑏
=−
𝑚
All knowns
𝑏
−
𝑐
𝑚 𝑠𝑡𝑑 𝑏𝑐𝑠𝑡𝑑
𝑐𝑥 = −
=
𝑉𝑥
𝑚𝑉𝑥
… in conclusion, an easy procedure to perform and interpret;
you take values you know and do a linear least squares fit to get m and b