1. Calibration methods
Chemistry 243

2. Figures of merit: Performance characteristics of instruments
Precision
Accuracy
Selectivity
Sensitivity
Limit of Detection
Limit of Quantitation
Dynamic Range

3. Precision vs. Accuracy in the common verbiage (Webster’s)
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!

4. 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!

5. Precision - Metrics
Most important
Often seen as %
Handy, common

6. 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

7. Calibration Curve, Limit of Detection, Sensitivity
Sensitivity* = Slope
Signal
Calibration Curve*
*Same as Working Curve
**Not improved by amplification alone
S/N = 3
LOD
Analyte Mass or Concentration

8. 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:

9. Calibration Curves: Sensitivity and LOD
For a given sample standard deviation, s, steeper calibration curve means better sensitivity
Insensitive to amplification
More Sensitive
Signal
Less Sensitive
S/N = 3
LOD
LOD
Analyte Mass or Concentration

10. 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

11. Calibration Curves: Dynamic Range and Noise Regions
becomes
poor above
this amount
of analyte
Signal
Calibration Curve
Poor
Quant
Noise
Region
Dynamic Range
S/N = 3
LOD
LOQ
LOL
Analyte Mass or Concentration

12. Types of Errors Random or indeterminate errors Systematic 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

13. 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

14. 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

15. 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

16. External standard calibration (ideal)
Sample
Unknown
External
Calibration
Standards
including
a blank
Signal
Sample
Unknown
Amount
S/N = 3
LOD
Analyte Mass or Concentration

17. In class example of external standard calibration
Skoog, Fig
Signal
Analyte Mass or Concentration
S/N = 3
LOD
Sample
Unknown
Amount
External
Calibration
Standards
including
a blank

18. 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

19. 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

20. 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

21. 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
Vs1
Vs2
Vs3
Vs4
Volume top-off step:
Vx diluted to Vt
Vs diluted to Vt Vt Vt Vt Vt

22. 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.
𝑆 𝑡𝑜𝑡𝑎𝑙 = 𝑆 𝑠𝑡𝑑 + 𝑆 𝑥
Combined Signal S1 S2 S3 S4
Concentration

23. How to use standard additions
𝑆 𝑡𝑜𝑡𝑎𝑙 = 𝑆 𝑠𝑡𝑑 + 𝑆 𝑥
k = slope or sensitivity
𝑆 𝑠𝑡𝑑 =𝑘∙ 𝑓 𝑑𝑖𝑙 ∙ 𝑐 𝑠𝑡𝑑 = 𝑘 𝑉 𝑠𝑡𝑑 𝑐 𝑠𝑡𝑑 𝑉 𝑡𝑜𝑡𝑎𝑙
Combined Signal
𝑆 𝑥 =𝑘∙ 𝑓 𝑑𝑖𝑙 ′ ∙ 𝑐 𝑥 = 𝑘 𝑉 𝑥 𝑐 𝑥 𝑉 𝑡𝑜𝑡𝑎𝑙
Concentration

24. How to use standard additions
𝑆 𝑡𝑜𝑡𝑎𝑙 = 𝑆 𝑠𝑡𝑑 + 𝑆 𝑥
𝑆 𝑡𝑜𝑡𝑎𝑙 = 𝑘 𝑉 𝑠𝑡𝑑 𝑐 𝑠𝑡𝑑 𝑉 𝑡𝑜𝑡𝑎𝑙 + 𝑘 𝑉 𝑥 𝑐 𝑥 𝑉 𝑡𝑜𝑡𝑎𝑙
Signal from standard
Signal from unknown

25. How to use standard additions
𝑆 𝑡𝑜𝑡𝑎𝑙 = 𝑘 𝑉 𝑠𝑡𝑑 𝑐 𝑠𝑡𝑑 𝑉 𝑡𝑜𝑡𝑎𝑙 + 𝑘 𝑉 𝑥 𝑐 𝑥 𝑉 𝑡𝑜𝑡𝑎𝑙
Remember,
Vstd is the
variable.
Knowns:
cstd
Vtotal Vx
𝑆 𝑡𝑜𝑡𝑎𝑙 = 𝑚𝑉 𝑠𝑡𝑑 +𝑏
𝑚= 𝑘 𝑐 𝑠𝑡𝑑 𝑉 𝑡𝑜𝑡𝑎𝑙
𝑏= 𝑘 𝑉 𝑥 𝑐 𝑥 𝑉 𝑡𝑜𝑡𝑎𝑙

26. How to use standard additions
𝑆 𝑡𝑜𝑡𝑎𝑙 = 𝑚𝑉 𝑠𝑡𝑑 +𝑏
𝑚= 𝑘 𝑐 𝑠𝑡𝑑 𝑉 𝑡𝑜𝑡𝑎𝑙
S, Combined Signal
Get m (slope) and
b (intercept) from
linear least squares
𝑏= 𝑘 𝑉 𝑥 𝑐 𝑥 𝑉 𝑡𝑜𝑡𝑎𝑙 Vs
How do I handle k ?

27. Determine cx via standard curve extrapolation …
𝑆 𝑡𝑜𝑡𝑎𝑙 = 𝑘 𝑉 𝑠𝑡𝑑 𝑐 𝑠𝑡𝑑 𝑉 𝑡𝑜𝑡𝑎𝑙 + 𝑘 𝑉 𝑥 𝑐 𝑥 𝑉 𝑡𝑜𝑡𝑎𝑙
At the x-intercept, S = 0
𝑘 𝑉 𝑠𝑡𝑑 𝑐 𝑠𝑡𝑑 𝑉 𝑡𝑜𝑡𝑎𝑙 =− 𝑘 𝑉 𝑥 𝑐 𝑥 𝑉 𝑡𝑜𝑡𝑎𝑙
𝑉 𝑠𝑡𝑑 𝑐 𝑠𝑡𝑑 =− 𝑉 𝑥 𝑐 𝑥
Skoog, Fig. 1-10
𝑐 𝑥 =− 𝑉 𝑠𝑡𝑑 0 𝑐 𝑠𝑡𝑑 𝑉 𝑥
known
known
Seeking
[analyte]
Vstd when S = 0

28. … or determine cx by directly using fit parameters
𝑐 𝑥 =− 𝑉 𝑠𝑡𝑑 𝑐 𝑠𝑡𝑑 𝑉 𝑥
𝑆 𝑡𝑜𝑡𝑎𝑙 =𝑚 𝑉 𝑠𝑡𝑑 +𝑏=0
𝑉 𝑠𝑡𝑑 =− 𝑏 𝑚
𝑚 𝑉 𝑠𝑡𝑑 =−𝑏
Final calculation:
All knowns
𝑐 𝑥 =− − 𝑏 𝑚 𝑐 𝑠𝑡𝑑 𝑉 𝑥 = 𝑏 𝑐 𝑠𝑡𝑑 𝑚 𝑉 𝑥
… 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