Chem. 31 – 2/16 Lecture
Announcements
• Turn in Pipet/Buret Calibration Report
• Wednesday
– AP1.2 due + quiz
• Today’s Lecture
– Chapter 4 Material
• Statistical Tests
• Calibration and Least Square’s Analysis
Chapter 4 – Gaussian Distributions
Now for a “real” limit problem example:
A man wants to get life insurance. If his measured
cholesterol level is over 240 mg/dL (2,400 mg/L), his
premium will be 25% higher. His level is measured and
found to be 249 mg/dL. His uncle, a biochemist who
developed the test, tells him that a typical standard
deviation on the measurement is 25 mg/dL. What is the
chance that a second measurement (with no crash diet
or extra exercise) will result in a value under 240 mg/dL
(e.g. beat the test)?
Statistical Tests
t Tests
• Case 1
– used to determine if there is a significant bias by measuring a
test standard (concentration known) 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 two samples) by measuring one sample
multiple time 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 sample
once by each method (or each sample in each set once)
Case 1 t test
• Methylmannopyranoside (MMP) example
• Added as an internal standard at 5 ppm
• Analysis will tell if sample causes a bias
compared to standard
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. do fish caught in a lake near a power
plant and far from the plant have the Hg
concentration)
– 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:
– not required to know math to determine m and b
– 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