Chem. 31 – 9/23 Lecture

Guest LectureDr. 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 Testst 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 mis-labeled 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

Conc. of Cl- in Rainwater

(Units = uM)

Sample # Dixon Cl- PNL Cl-

1 9.9 17.0

2 2.3 11.0

3 23.8 28.0

4 8.0 13.0

5 1.7 7.9

6 2.3 11.0

7 1.9 9.9

8 4.2 11.0

9 3.2 13.0

10 3.9 10.0

11 2.7 9.7

12 3.8 8.2

13 2.4 10.0

14 2.2 11.0

7.1

8.7

4.2

5.0

6.2

8.7

8.0

6.8

9.8

6.1

7.0

4.4

7.6

8.8

Calculations

Step 1 – Calculate Difference

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:

nS

dt

d

Calc

tCalc = 11.5

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)

22

21

S

SFCalc

As with t tests, if FCalc > FTable, difference is statistically significant

S1 > S2

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 TestQuestions

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

Mannosan Calibration

y = 541.09x + 6.9673

R2 = 0.9799

0

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5 0.6

Conc. (ppm)

Pe

ak

Are

a

Best Fit Line Equation

Best Fit Line

Deviations from line

Assumptions for Linear Least Squares Analysis to Work Well

• Actual relationship is linear• All uncertainty is associated with the

y-axis• The uncertainty in the y-axis is

constant

Calibration and Least Squares- number of calibration standards (N)

N Conditions1 Must assume 0 response for 0 conc.; standard must

be perfect; linearity must be perfect2 Gives m and b but no information on uncertainty

from calibrationMethods 1 and 2 result in lower accuracy, undefined precision

3 Minimum number of standards to get information on validity of line fit

4 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

Mg Example:An unknown solution

gives an absorbance of 0.621

Use equation to predict unknown conc.

y = mx + bx = (y – b)/mx = (0.621 + 0.0131)/2.03x = 0.312 ppmCan check value graphically

y = 2.0343x - 0.0131

R2 = 0.9966

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.10 0.20 0.30 0.40 0.50

Mg Conc. (ppm)

Ab

sorb

ance

Calibration “Curve”

Use of Calibration Curve- Uncertainty in Unknown Concentration

2

2

)(

)(11

xxm

yy

nkm

SS

i

iyx

Uncertainty given by Sx (see below):

Notes on equation: m = slope, Sy = standard error in yn = #calibration stds k = # analyses of unknown, xi = indiv std conc., yi = unknown responseThe biggest factors are Sy and mTwo 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

• Quality of Results Depends on:– Calibration Results

• R2 value (measure of variability of response due to conc.)

• Reasonable fit– Range of Unknown

Concentrations• next slide

Good Calibration

y = 0.3634x - 0.1009

R20.9998 =

0.0000

2.0000

4.0000

6.0000

8.0000

10.0000

12.0000

0 5 10 15 20 25 30

Conc. (ppm)

Re

lati

ve

Pe

ak

Are

a

Line fit through Curve

y = 262.44x + 37.034R2 = 0.9772

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5

LG Conc. (ppm)

Pea

k A

rea

MN

Linear (MN)

Poor R^2 Value

y = 0.0041x + 0.0107

R2 = 0.9622

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60

Galactose Standard (ug)

Ab

sorb

ance

(49

0 n

m)

Better fit by curve

Use of Calibration Curve- Quality of Results

• Quality of Results Depends on:– Calibration Results

• on last slide

– Range of Unknown Concentrations

• Extrapolation outside of range of standards should be avoided

• Best concentration range

y = 2.0343x - 0.0131

R2 = 0.9966

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.10 0.20 0.30 0.40 0.50

Mg Conc. (ppm)

Ab

sorb

ance

Range of Standards (0.02 to 0.4 ppm)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.00 0.10 0.20 0.30 0.40 0.50

Mg Conc. (ppm)

Un

cert

ain

ty i

n C

on

c. (

pp

m)

Absolute Uncertainty

0

10

20

30

40

50

60

0.00 0.10 0.20 0.30 0.40 0.50

Mg Conc. (ppm)

% U

nce

rtai

nty

Relative Uncertainty

Best Range: upper 2/3rds of standard range

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.) – 47The 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?