Calibration methods
Chemistry 243
Figures of merit: Performance characteristics of instruments Precision Accuracy Selectivity Sensitivity Limit of Detection Limit of Quantitation Dynamic Range
Precision vs. Accuracy in the common verbiage (Webster’s) Precision:
The quality or state of being precise; exactness; accuracy; strict conformity to a rule or a standard; definiteness.
Accuracy: 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. Absolute and relative standard deviation, standard error of
the mean, coefficient of variation, variance. Accuracy: Measure of closeness to accepted value
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!!! Precisely wrong!
Precision - Metrics
Most important
Often seen as %
Handy, common
Sensitivity vs.Limit of Detection NOT THE SAME THING!!!!! Sensitivity: Ability to discriminate between small
differences in analyte concentration at a particular concentration. 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 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
Sign
al
00
Analyte Mass or Concentration
S/N = 3
Calibration Curve*
LOD
Sensitivity* = Slope
*Same as Working Curve**Not improved by amplification alone
Selectivity Degree to which a method is free from
interference from other contaminating signals in matrix
No measurement is completely free of interferences Selectivity coefficient:
A A B B C CS m c m c m c
,B
B AA
mk
m
Calibration Curves:Sensitivity and LOD
Sign
al
0
Analyte Mass or Concentration
S/N = 3
LOD
Mor
e S
ensi
tive
Less Sensitive
LOD0
For a given sample standard deviation, s, steeper calibration curve means better sensitivity
Insensitive to amplification
Dynamic range The maximum range over which an accurate
measurement can be made From limit of quantitation to limit of linearity
LOQ: 10 s of blank LOL: 5% deviation from linear
Ideally a few logs Absorbance: 1-2 MS, Fluorescence: 4-5 NMR: 6
Calibration Curves:Dynamic Range and Noise Regions
Sign
al
00
Analyte Mass or Concentration
S/N = 3
LOD
Dynamic Range
Calibration Curve
CalibrationCurvebecomespoor abovethis amountof analyte
NoiseRegion
LOQ
PoorQuant
LOL
Types of Errors Random or indeterminate errors
Handled with statistical probability as already shown Systematic errors
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 Present in all measurements made in the same way
and introduce bias. Instrumental errors
Wacky instrument behavior, bad calibrations, poor conditions for use Electronic drift, temperature effects, 60Hz line noise,
batteries dying, problems with calibration equipment. Personal errors
Originate from judgment calls Reading a scale or graduated pipette, titration end points
Method errors Non-ideal chemical or physical behavior
Evaporation, adsorption to surfaces, reagent degradation, chemical interferences
Instrument calibration Determine the relationship between response
and concentration Calibration curve or working curve
Calibration methods typically involve standards Comparison techniques External standard* Standard addition* Internal standard*
* calibration curve is required
External standard calibration (ideal)
External Standard – standards are not in the sample and are run separately
Generate calibration curve (like PS1, #1) 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
Sign
al
00
Analyte Mass or Concentration
S/N = 3
LOD
External standard calibration(ideal)
SampleUnknown
SampleUnknownAmount
ExternalCalibrationStandardsincludinga blank
In class example of external standard calibration
Skoog, Fig. 13-13
0log
molar absorptivitypathlengthconcentration
PA bc
P
bc
Sign
al
00 Analyte Mass or Concentration
S/N = 3
LOD
SampleUnknown
SampleUnknownAmount
ExternalCalibrationStandardsincludinga blank
Real-life calibration Subject to matrix interferences
Matrix = what the real sample is in 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: Internal Standard Standard Additions
Internal standard 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
Lithium good for sodium and potassium in blood; not in blood
Standard additions 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
Non-Fixed Parameter: Vs = Volume of std. – variable
Calibration Standard(Fixed cs)
Vx Vx Vx Vx
Vs3Vs2 Vs4Vs1
Vt VtVt Vt
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.
Com
bine
d Si
gnal
Concentration00S1 S2 S4S3
𝑆𝑡𝑜𝑡𝑎𝑙=𝑆𝑠𝑡𝑑+𝑆𝑥
How to use standard additions
00
Concentration
Com
bine
d Si
gnal
k = slope or sensitivity
𝑆𝑡𝑜𝑡𝑎𝑙=𝑆𝑠𝑡𝑑+𝑆𝑥
𝑆𝑠𝑡𝑑=𝑘 ∙ 𝑓 𝑑𝑖𝑙 ∙𝑐𝑠𝑡𝑑=𝑘𝑉 𝑠𝑡𝑑𝑐𝑠𝑡𝑑
𝑉 𝑡𝑜𝑡𝑎𝑙
𝑆𝑥=𝑘 ∙ 𝑓 𝑑𝑖𝑙′ ∙𝑐 𝑥=𝑘𝑉 𝑥𝑐𝑥
𝑉 𝑡𝑜𝑡𝑎𝑙
How to use standard additions
Signal from st
andard
Signal from unkn
own
𝑆𝑡𝑜𝑡𝑎𝑙=𝑆𝑠𝑡𝑑+𝑆𝑥
𝑆𝑡𝑜𝑡𝑎𝑙=𝑘𝑉 𝑠𝑡𝑑𝑐𝑠𝑡𝑑
𝑉 𝑡𝑜𝑡𝑎𝑙+𝑘𝑉 𝑥𝑐 𝑥
𝑉 𝑡𝑜𝑡𝑎𝑙
How to use standard additions
Remember,Vstd is thevariable.
Knowns:cstd Vtotal
Vx
𝑆𝑡𝑜𝑡𝑎𝑙=𝑚𝑉 𝑠𝑡𝑑+𝑏
𝑚=𝑘𝑐𝑠𝑡𝑑
𝑉 𝑡𝑜𝑡𝑎𝑙𝑏=
𝑘𝑉 𝑥𝑐𝑥
𝑉 𝑡𝑜𝑡𝑎𝑙
𝑆𝑡𝑜𝑡𝑎𝑙=𝑘𝑉 𝑠𝑡𝑑𝑐𝑠𝑡𝑑
𝑉 𝑡𝑜𝑡𝑎𝑙+𝑘𝑉 𝑥𝑐 𝑥
𝑉 𝑡𝑜𝑡𝑎𝑙
How to use standard additions
00
Vs
S, C
ombi
ned
Sign
alGet m (slope) and b (intercept) from
linear least squares
How do I handle k ?
𝑆𝑡𝑜𝑡𝑎𝑙=𝑚𝑉 𝑠𝑡𝑑+𝑏
𝑚=𝑘𝑐𝑠𝑡𝑑
𝑉 𝑡𝑜𝑡𝑎𝑙
𝑏=𝑘𝑉 𝑥𝑐𝑥
𝑉 𝑡𝑜𝑡𝑎𝑙
Determine cx via standard curve extrapolation …
At the x-intercept, S = 0
Seeking[analyte]
known
knownSkoog, Fig. 1-10
Vstd when S = 0
𝑆𝑡𝑜𝑡𝑎𝑙=𝑘𝑉 𝑠𝑡𝑑𝑐𝑠𝑡𝑑
𝑉 𝑡𝑜𝑡𝑎𝑙+𝑘𝑉 𝑥𝑐 𝑥
𝑉 𝑡𝑜𝑡𝑎𝑙
𝑘𝑉 𝑠𝑡𝑑𝑐𝑠𝑡𝑑
𝑉 𝑡𝑜𝑡𝑎𝑙=−
𝑘𝑉 𝑥𝑐𝑥
𝑉 𝑡𝑜𝑡𝑎𝑙
𝑉 𝑠𝑡𝑑𝑐𝑠𝑡𝑑=−𝑉 𝑥𝑐𝑥
𝑐𝑥=−(𝑉 𝑠𝑡𝑑 )0𝑐𝑠𝑡𝑑
𝑉 𝑥
… or determine cx by directly using fit parameters
Final calculation: All knowns
… 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
𝑐𝑥=−𝑉 𝑠𝑡𝑑𝑐𝑠𝑡𝑑
𝑉 𝑥
𝑆𝑡𝑜𝑡𝑎𝑙=𝑚𝑉 𝑠𝑡𝑑+𝑏=0
𝑚𝑉 𝑠𝑡𝑑=−𝑏 𝑉 𝑠𝑡𝑑=−𝑏𝑚
𝑐𝑥=−(− 𝑏
𝑚 )𝑐 𝑠𝑡𝑑
𝑉 𝑥=𝑏𝑐𝑠𝑡𝑑
𝑚𝑉 𝑥