Chem. 31 – 9/23 Lecture
Guest Lecture
Dr. Roy Dixon
Announcements
• Small renewable resources company looking for
interns for chemical analysis work (see bulletin
board outside 446)
• Additional Problem with no name on it
• Due Today
– Pipet/Buret Calibration Lab Report
• Today’s Lecture
– Error and Uncertainty
• Finish up Statistical Tests
– Least Squares Calibration (last part of Chapter 4)
Statistical Tests
t Tests - Review
• Case 1
– used to determine if there is a significant bias by measuring a
test standard and determining if there is a significant difference
between the known and measured concentration
• Case 2
– used to determine if there is a significant differences between
two methods (or samples) by measuring one sample multiple
times by each method (or each sample multiple times)
• Case 3
– used to determine if there is a significant difference between
two methods (or sample sets) by measuring multiple samples
once by each method (or each sample in each set once)
Case 2 t test Example
• A winemaker found a barrel of wine that was labeled as
a merlot, but was suspected of being part of a
chardonnay wine batch and was obviously mis-labeled.
To see if it was part of the chardonnay batch, the mislabeled barrel wine and the chardonnay batch were
analzyed for alcohol content. The results were as
follows:
– Mislabeled wine: n = 6, mean = 12.61%, S = 0.52%
– Chardonnay wine: n = 4, mean = 12.53%, S = 0.48%
• Determine if there is a statistically significant difference
in the ethanol content.
Case 3 t Test Example
• Case 3 t Test used when multiple
samples are analyzed by two different
methods (only once each method)
• Useful for establishing if there is a
constant systematic error
• Example: Cl- in Ohio rainwater measured
by Dixon and PNL (14 samples)
Case 3 t Test Example –
Data Set and Calculations
Calculations
Conc. of Cl- in Rainwater
(Units = uM)
Step 1 –
Calculate
Difference
Sample #
Dixon Cl-
PNL Cl-
1
9.9
17.0
7.1
2
2.3
11.0
8.7
3
23.8
28.0
4.2
4
8.0
13.0
5.0
5
1.7
7.9
6.2
6
2.3
11.0
8.7
7
1.9
9.9
8.0
8
4.2
11.0
6.8
9
3.2
13.0
9.8
10
3.9
10.0
6.1
11
2.7
9.7
7.0
12
3.8
8.2
4.4
13
2.4
10.0
7.6
14
2.2
11.0
8.8
Step 2 - Calculate
mean and standard
deviation in differences
ave d = (7.1 + 8.7 + ...)/14
ave d = 7.49
Sd = 2.44
Step 3 – Calculate t value:
tCalc 
d
Sd
tCalc = 11.5
n
Case 3 t Test Example –
Rest of Calculations
• Step 4 – look up tTable
– (t(95%, 13 degrees of freedom) = 2.17)
• Step 5 – Compare tCalc with tTable, draw
conclusion
– tCalc >> tTable so difference is significant
t- Tests
• Note: These (case 2 and 3) can be applied to
two different senarios:
– samples (e.g. sample A and sample B, do they have
the same % Ca?)
– methods (analysis method A vs. analysis method B)
F - Test
• Similar methodology as t tests but to compare
standard deviations between two methods to
determine if there is a statistical difference in
precision between the two methods (or
variability between two sample sets)
FCalc
S1 > S2
S12
 2
S2
As with t tests, if FCalc > FTable,
difference is statistically significant
Grubbs Test Example
• Purpose: To determine if an “outlier” data point
can be removed from a data set
• Data points can be removed if observations
suggest systematic errors
•Example:
•Cl lab – 4 trials with values of 30.98%, 30.87%, 31.05%, and 31.00%.
•Student would like less variability (to get full points for precision)
•Data point farthest from others is most suspicious (so 30.87%)
•Demonstrate calculations
Dealing with Poor Quality Data
• If Grubbs test fails, what can be done to
improve precision?
– design study to reduce standard deviations
(e.g. use more precise tools)
– make more measurements (this may make an
outlier more extreme and should decrease
confidence interval)
Statistical Test
Questions
1. A chemist has developed a new test to
measure gamma hydroxybutyrate that is
expected to be faster and more precise than a
standard method. What test should be used
to test for improved precision? Are multiple
samples needed or multiple analyses of a
single sample?
2. The chemist now wants to compare the
accuracy for measuring gamma
hydroxybutyrate in alcoholic beverages.
Describe a test to determine if the method is
accurate.
Calibration
•
•
•
•
For many classical methods direct
measurements are used (mass or volume
delivered)
Balances and Burets need calibration, but
then reading is correct (or corrected)
For many instruments, signal is only
empirically related to concentration
Example Atomic Absorption Spectroscopy
– Measure is light absorbed by “free” metal
atoms in flame
– Conc. of atoms depends on flame
conditions, nebulization rate, many
parameters
– It is not possible to measure light
absorbance and directly determine conc.
of metal in solution
– Instead, standards (known conc.) are
used and response is measured
Light
beam
To light
Detector
Method of Least Squares
• Purpose of least squares method:
– determine the best fit curve through the data
– for linear model, y = mx + b, least squares determines best m
and b values to fit the x, y data set
– note: y = measurement or response, x = concentration, mass or
moles
• How method works:
– the principle is to select m and b values that minimize the sum
of the square of the deviations from the line (minimize Σ[yi –
(mxi + b)]2)
– in lab we will use Excel to perform linear least squares method
Example of Calibration Plot
300
Best Fit Line
Equation
Mannosan Calibration
Best Fit Line
y = 541.09x + 6.9673
2
R = 0.9799
250
Peak Area
200
150
Deviations from line
100
50
0
0
0.1
0.2
0.3
Conc. (ppm)
0.4
0.5
0.6
Assumptions for Linear Least
Squares Analysis to Work Well
• Actual relationship is linear
• All uncertainty is associated with the yaxis
• The uncertainty in the y-axis is constant
Calibration and Least Squares
- number of calibration standards (N)
N
1
2
3
4
Conditions
Must assume 0 response for 0 conc.; standard must be
perfect; linearity must be perfect
Gives m and b but no information on uncertainty from
calibration
Methods 1 and 2 result in lower accuracy, undefined
precision
Minimum number of standards to get information on validity
of line fit
Good number of standards for linear equation (if standards
made o.k.)
More standards may be needed for non-linear curves, or
samples with large ranges of concentrations
Use of Calibration Curve
An unknown solution gives
an absorbance of 0.621
Use equation to predict
unknown conc.
y = mx + b
x = (y – b)/m
x = (0.621 + 0.0131)/2.03
x = 0.312 ppm
Can check value graphically
Calibration “Curve”
1.0
0.8
Absorbance
Mg Example:
y = 2.0343x - 0.0131
R2 = 0.9966
0.6
0.4
0.2
0.0
0.00
0.10
0.20
0.30
Mg Conc. (ppm)
0.40
0.50
Use of Calibration Curve
- Uncertainty in Unknown Concentration
Uncertainty given by Sx (see below):
Sy
Sx 
m
1 1
( yi  y )2
 
k n m( xi  x )2
Notes on equation: m = slope, Sy = standard error in y
n = #calibration stds k = # analyses of unknown, xi = indiv std conc., yi
= unknown response
The biggest factors are Sy and m
Two other parameters that often indicate calibration quality are R2 and b. R2
should be close to 1 (good is generally >0.999); b should be small relative to y of
lowest standard.
Use of Calibration Curve
- Quality of Results
– Calibration Results
• R2 value (measure of
variability of response due
to conc.)
• Reasonable fit
PoorCalibration
R^2 Value
Good
0.25
12.0000
Relative Peak
Absorbance
(490Area
nm)
• Quality of Results
Depends on:
y = 0.3634x - 0.1009
R20.9998
=
y = 0.0041x
+ 0.0107
10.0000
0.2
8.0000
R2 = 0.9622
0.15
6.0000
4.0000
0.1
2.0000
0.05
0.0000
0
0
0
5
10
10
20
15
20
30
40
Conc. (ppm)
25
30
50
60
Galactose Standard (ug)
– Range of Unknown
Concentrations
Line fit through Curve
600
• next slide
y = 262.44x + 37.034
R2 = 0.9772
500
Peak Area
Better fit by
curve
400
MN
300
Linear (MN)
200
100
0
0
0.5
1
1.5
LG Conc. (ppm )
2
2.5
Use of Calibration Curve
- Quality of Results
0.8
Absorbance
– Calibration Results
1.0
0.0
0.00
0.10
0.20
0.30
0.40
0.50
Mg Conc. (ppm)
Relative
Uncertainty
Absolute
Uncertainty
60 0.014
50
% Uncertainty
• Extrapolation outside of
range of standards
should be avoided
• Best concentration
range
0.4
0.2
• on last slide
– Range of Unknown
Concentrations
y = 2.0343x - 0.0131
R2 = 0.9966
0.6
Uncertainty in Conc. (ppm)
• Quality of Results
Depends on:
Range of Standards
(0.02 to 0.4 ppm)
0.012
0.010
40
0.008
30 0.006
20 0.004
10
0.002
Best Range: upper
2/3rds of standard range
0.000
0
0.00
0.10
0.20
0.30
0.40
0.00
0.10
0.20
0.30
Mg Conc.
(ppm) 0.40
Mg Conc. (ppm)
0.50
0.50
Calibration Question
• A student is measuring the concentrations of
caffeine in drinks using an instrument. She
calibrates the instruments using standards
ranging from 25 to 500 mg/L. The calibration
line is:
Response = 7.21*(Conc.) – 47
The response for caffeine in tea and in espresso
are 1288 and 9841, respectively. What are
the caffeine concentrations? Are these values
reliable? If not reliable, how could the
measurement be improved?