1. Data Analysis and Presentation
Chapter 5- Calibration Methods and Quality Assurance
EXCEL – How To Do
1- least squares and linear calibration
curve/function (chapter 4)
2- standard addition (chapter 5)
3-internal standard (addition) (chapter 5)

2. External standard 1- least squares and linear calibration
curve/function (chapter 4)

3. CHAPTER 05: Opener B

4. CHAPTER 05: Table 5.1

5. Calibration function
Q: How the selected instrument responds to the change in quantity (concentration, M ppm ) of the measured analyte?
A: Measure the instrument responses (peak hights, peak areas etc.) from known but different concentrations of the analyte (say 0, 5,10, ppm standards). Then construct the response diagram figure (response vs. concentration) with the response or calibration function through the points.

6. Calibration Data:
Answer 2
GO TO EXCELL or your calculator

7. MFYI: Method of Least Squares/Linear regression
The method of least squares assumes that the errors in the y values are substantially greater than the errors in the x values. A second assumption is that the uncertainties in all of the y values are similar.
Derivation of the Least Squares Method
(Equation for a straight line) y = mx + b
Vvertical deviation = dii = yi–y = yi–(mxi+b)
SSome of the deviations are positive and some are negative.
To minimize the magnitude of the deviations irrespective of
their signs, we square the deviations to create positive numbers.

8. OLD SLIDE: Caffeine again: first step calibration
If points are out of line: error
If one assumes that the calibration curve is linear – there are physical reasons for that. There is a linear function that can best represent the response of the instrument.
There are methods based on the propagation of error that can help us calculate the best fit: linear regression of the straight-line calibration curves

9. CALIBRATION: Instrument used to measure signals must be calibrated, must have the calibration relation. Example (linear):
y = m [x ]+ b (general algebraic form for linear equation)
Often given in other similar forms:
Smeasured(total signal) =k nA (concentration) + Sreag ( signal from reagents, from blank)
Calibration is the determination of that relation: it is used for the determination of analyte by using standards and blanks to determine a relationship (function) between concentrations and assay responses.  
Validation implies that other labs have approved the analytical method of analysis and produced similar results.   
Standards are materials containing accurately known concentrations of the desired analyte.
Standardization is the process of determining relationship between the measured signal and the amount of analyte (fittting for k in linear relations).
Blanks are solutions containing all added reagents except the sample analyte.(method, reagent and field blanks)
Controls can be an alternate sample, in which the contents are well known.
Primary and Secondary reagents

10. Calibration Data:
Answer 2
GO TO EXCELL or your calculator

11. CHAPTER 04: Unnumbered Table 4.3

12. CHAPTER 04: Unnumbered Table 4.4

13. CHAPTER 04: Figure 4.12

14. CHAPTER 04: Figure 4.14

15. IMPORTANT : BLANK !!
Before you start measuring standards, you need to see the signal
from a sample without analyze.
BLANK: everything but analyte

16. CHAPTER 04: Table 4.7

17. EXAMPLE: CORRECTION FOR nonzero BLANK
Your blank is (for zero concentration of analyte)
Your signals are as follows
Signal Corrected signal for blank (!) concentration
0.001
0.003
0.0287
0.006
0.0601
0.009
0.009—0.001
0.0889

18. -prepare standards to cover region of interest (linear response)
Concentration Signal
Signal-B C A
blank=0.001
0.001
0.0287
0.003
0.002
0.0601
0.006
0.005
0.0889
0.009
0.008
blank for analyte =0.003
0.01
0.007
X= ( )/0.0906
-prepare standards to cover region of interest (linear response)
-correct for blank (signal-blank)
-correct sample analyte signal for its own blank
-if analyte signal is NOT in linear calibrated region: DILUTE !!!!

19. 2- standard addition (chapter 5)

20. Is such external standardization the only way to calibrate and acquire the accurate and precise values?
No, there are also other methods that include addition of analyte or some other material into the sample (aliquot)

21. Possible problems with external standardization

22. Spike
Sometimes (often) the response to analyte is affected by something else in the sample, which we call MATRIX.
SPIKE is a known quantity of analyte added directly to the sample to verify if the response to analyte is the same as that expected from pure sample observed in the calibration curve.

23. 2. Standard addition
SStandard addition: Known quantities of analyte are added to the unknown, and the increased signal lets us deduce how much analyte was in the original unknown. Typically we use the method of standard addition when unknown sample matrix is sufficiently complex. The blank and standards are not representative of the unknown sample and lead to analysis error. This method requires a linear response to analyte.
 For a single spike, one trial
[X]i / [X]f + [S]f = Ix / I s+x
 [X]i = unknown initial concentration of analyte
Ix = signal from first solution
[S]f = concentration of standard in second solution
[X]f = diluted concentration of analyte
I s+x = signal from second solution

24. CHAPTER 05: Equation 5.7

25. Example 2 A blood serum sample containing sodium ion gives a signal of 4.27 mV on a light intensity meter in an atomic emission experiment. The Na+ concentration in the serum is then increased by M by a standard addition, without significantly diluting the sample. This "spiked" serum sample gives a signal of 7.98 mV in atomic emission. Find the original concentration of Na+ in the serum. 
[X]i = unknown initial concentration of Na+
Ix = 4.27 mV
[S]f = M
[X]f = diluted concentration of analyte = [X]i
I s+x = 7.98 mV [X]i = M

26. x / ( x+0.104 ) = 4.27/7.98 Solve it using EXCEL, Matlab Mathematica……
Solve[ x / ( x ) == 4.27/7.98,x]
{{x® }}

27. FYI: (2) Sampledirect addition
S sample = S spike
a b a b

28. FYI: Accurate description
(1)Sampleseparate aliquots (portions)
S sample = S spike
a b
Ssample = signal sample
CS= concentration spiked
CA= concentration analyte a b

29. Example 3.
We use standard addition methods when our sample has matrix effects which are hard to duplicate in a blank. Let's again find the amount of Pb+2 in a blood sample. A 5.00 mL blood sample containing lead yielded a signal of units. Afterwards, the sample was spiked with 5.00 L of 1560 ppb Pb+2 standard. (note: a small addition) This spiked solution gave a reading of units. Find the concentration of Pb+2.
[X]i / [X]f + [S]f = Ix / I s+x or [C]i / [C]f + [Cs]f = Sx / S s+x
Be careful with volume!
CA=1.33ppb

30. Solve[x/((x*5/5.005)+ (0.005/5.005)*1560)==0.712/1.546,{x}]
NOTE –Drawing can HELP!!!!!!!!
Draw two volumetric flask, fill with all components :

31. 3. Internal Standard
In addition to the analyte you measure add another similar compound (not analyte ) that has similar response ( sensitivity ) as the analyte.

32. Internal Standard is sometimes added to an unknown sample
Internal Standard is sometimes added to an unknown sample. The reason may be to verify the signal response in situations where instrument response varies slightly from run to run. For example, an analysis is preformed on different days or different instruments or under different operating conditions. Typically, the internal standard resembles the analyte. Let's say we are separating isomers of octane and determining their concentration on a gas chromatograph (GC). To verify how much is present we might add a known amount of cyclohexane as an internal standard

33. Internal Standard
An internal standard is a known amount of a compound, different from the analyte, that is added to the unknown sample. A Standard mixture of analyte and standard is prepared before hand.
Internal standards are desirable whenever losses of sample are likely to occur during handling or analysis. They are also used to calibrate the instrument when same analysis is done on different days. Use the following relation:

34. Internal Standard SA(signal due to analyte) =k A CA
SIS(signal due to int. stand) =kIS CIS

35. CHAPTER 05: Equation 5.11

36. = K CA= K K=3.05 CA=1.33ppb Pb 2+ SA(signal due to analyte) = k A CA
Example Example on internal standard: Note we are not diluting the sample significantly!
A lab uses Cu+2 as a internal standard for a Pb+2 analysis in blood. Known solution standards for Pb+2 (1.75 ppm) and Cu+2 (2.25 ppm) give readings such that the signal ratio of Pb to Cu is A blood sample was spiked with the Cu standard (2.25 ppm). We assume that the blood sample volume does not change significantly. The ratio of the readings for Pb to Cu in the unknown spiked solution is Based on the response for the known amount of Cu+2 added to a blood sample determine, the amount of Pb+2 present.
First measurement
SA(signal due to analyte) = k A CA CA
= K
SIS(signal due to int. stand) = kIS CIS
CIS
Second measurement
CIS SA
CA= K
SIS
K=3.05
CA=1.33ppb Pb 2+

37. 1) And 2) use the same formula !!!
Example Example on internal standard: Note we are not diluting the sample significantly!
A lab uses Cu+2 as a internal standard for a Pb+2 analysis in blood. Known solution standards for Pb+2 (1.75 ppm) and Cu+2 (2.25 ppm) give readings such that the signal ratio of Pb to Cu is A blood sample was spiked with the Cu standard (2.25 ppm). We assume that the blood sample volume does not change significantly. The ratio of the readings for Pb to Cu in the unknown spiked solution is Based on the response for the known amount of Cu+2 added to a blood sample determine, the amount of Pb+2 present.
Sx =Ax (in book) SIS = AS (in book) CA=X (in book) CIS=S (in book)
Finding F (K=F (in book)
AX/AS =F (X/A ) if AX/AS = 2.37 and X=1.75 and S =2.25, F=3.05
2) Finding [X]: , F=3.05 , if AX/AS =1.80 , and S = X=1.33ppb
1) And 2) use the same formula !!!

38. Uncertainty and instruments: how to use information and make decisions

39. Accuracy, precision Accuracy vs. precision

40. -Scale of operation (analytes classification
Microanalysis - limits of trace analysis
There are detection limits associated with every analytical method. As samples and their analyte content become smaller, there is a real concern how good is the analysis. This is especially true when we get in the ~ parts per million range and beyond.
Here are several factors that affect trace analysis:
1) the nature of the analyte
2) matrix effect of the component, interferences
3) reagent quality and sample preparation
4) type of assay ( instrumental or lab bench)
5) pre-concentration of analyte

41. Scale of operation

42. Sensitivity, selectivity, detection limits, noise
The detection limit is the concentration level to which a method can reliably give a signal statistically different from the background signal (noise).
Often for an instrumental method, it is the ratio of the average signal amplitude / the average noise amplitude. In this instance the detection limit is defined as when the ratio (S/N = 2).
For a wet analysis it may be defined as the Analyte Concentration which produces a signal three times the Std. Dev. of the blank.
Sensitivity is the method's ability to discriminate the differences between two trace concentration levels. Stated a little different:
Sensitivity is ratio for the change in signal output to the change in concentration of analyte.
Sensitivity =  I (signal) /  C (analyte concentration)
Selectivity: A method is selective if its signal is a function of only the amount of
analyte present in the sample, not the interferents etc..

43. Noise
TThree types of noise in electrical instruments: White noise, Drift noise, and Line noise.
 i) White noise (thermal) is always present. It involves the motion of electrons in the circuits.
iii)    Drift noise (sometimes called flicker or 1/f ) is not well understood but is known to be inversely proportional to frequency of the signal. It is significant at frequencies lower than ~100 mHz.
iii) Line noise arises from environmental noise (power lines, TV & radio signals, lighting, etc.).
Line and drift noise can be reduced by using frequency modulation, filters, and insulation.

44. Signal Averaging can be used to improve signal to noise ratios in certain instrumental techniques. This is true due to the digital storage of signal data.
To improve the signal-to-noise ratio by a factor of n requires averaging n2 spectra.
(S/N ratio improvement, Fellgett, 1958)
The S/N ratio improvement through signal averaging stems from the fact that the noise, N, will grow by a factor of n (where n is number of times the spectrum is recorded) where as the signal, S, should grow by a factor of n.
As seen in the figure above; 4 scans vs. 256 scans, the signal becomes enhanced due to S/N improvement.

45. There is a working range for an analysis. It should be such that the analyte concentration is at least ~ 10 times the Detectin .Limit and small enough to produce a linear signal response.
If performing an established method of analysis, the information about D.L. and linear range for calibration curves can be found.
(*if the analyte concentration is too high(concentrated) such that the signal response is not linear to concentration we can DILUTE. If opposite we can concentrate
Constant sample error stays the same regardless of sample size.
Proportional sample error varies with variation in sample size.

46. CHAPTER 05: Equation 5.3

47. CHAPTER 05: Equation 5.4

48. CHAPTER 05: Equation 5.5

49. CHAPTER 05: Figure 5.2