Chem. 31 – 9/21 Lecture

Guest LectureDr. Roy Dixon

Announcements I• Due on Wednesday

– Pipet/Buret Calibration Lab Report– Format: Pipet Report Form, Buret Calibration Plot

and data for these measurements (use Lab Manual pages or photocopy your lab notebook pages if neat/organized)

• Last Week’s Additional Problem– returned in labs– remember to put your LAB SECTION NUMBER on all

assignments turned in (grading is by lab section)

Announcements II• Today’s Lecture

– Error and Uncertainty• Finish up Gaussian Distribution Problems• t and Z based Confidence Intervals• Statistical Tests

– Lecture will be posted under my faculty web page (but I will also send them to Dr. Miller-Schulze for his posting method)

Example Problems

Text Problems

4-2 (a), (d), (e)

4-4 (a), (b)

Done already

Chapter 4 – A Little More on Distributions

• Measurements can be a naturally varying quantity (e.g. student heights, student test scores, Hg levels in fish in a lake)

• Additionally, a single quantity measured multiple times typically will give different values each time (example: real distribution of measurements of mass of an ion)

1343.12360 1343.92636 1344.72913 1345.53189 1346.33466 1347.13742

m/z

0

1200.0

0

10

20

30

40

50

60

70

80

90

100

% In

ten

sit

y

(1) Spec #1 * [BP = 234.1, 52707]

1343.9877

1344.9770

Note: to be considered “accurate mass”, an ion’s mass error must be less than 5 ppm (0.007 amu in above spectrum). This is only possible by averaging measurements so that the average mass meets the requirement.

2 ~ 0.2 amu

x axis is mass

Chapter 4 – A Little More on Distributions

• Reasons for making multiple measurements:– So one has information on the variability of

the measurement (e.g. can calculate the standard deviation and uncertainty)

– Average values show less deviation than single measurements

– Mass spectrometer example: standard deviation in single measurement ~0.1 amu, but standard deviation in 4 s averages ~0.005 amu

Chapter 4 – Calculation of Confidence Interval

1. Confidence Interval = x + uncertainty

2. Calculation of uncertainty depends on whether σ is “well known”

3. If is not well known (covered later)

4. When is well known (not in text)

Value + uncertainty =

Normal Distribution

00.05

0.10.15

0.20.25

0.30.35

0.40.45

-3 -2 -1 0 1 2 3

Z value

Freq

uenc

y

n

Zx

Z depends on area or desired probability

At Area = 0.45 (90% both sides),

Z = 1.65At Area = 0.475 (95% both sides), Z = 1.96 => larger confidence interval

Chapter 4 – Calculation of Uncertainty

Example:The concentration of NO3

- in a sample is measured many times. If the mean value and standard deviation (assume as population standard deviation) are 14.81 and 0.62 ppm, respectively, what would be the expected 95% confidence interval in a 4 measurement average value? (Z for 95% CI = 1.96)What is the probability that a new measurement would exceed the upper 95% confidence limit?

Chapter 4 – Calculation of Confidence Interval with Not Known

Value + uncertainty =n

tSx

t = Student’s t value

t depends on:

- the number of samples (more samples => smaller t)- the probability of including the true value (larger probability => larger t)

Chapter 4 – Calculation of Uncertainties Example

• Measurement of lead in drinking water sample:– values = 12.3, 9.8, 11.4, and 13.0 ppb

• What is the 95% confidence interval?

Chapter 4 – Ways to Reduce Uncertainty

1. Decrease standard deviation in measurements (usually requires more skill in analysis or better equipment)

2. Analyze each sample more time (this increases n and decreases t)

Overview of Statistical Tests

• t-Tests: Determine if a systematic error exists in a method or between methods or if a difference exists in sample sets

• F-Test: Determine if there is a significant difference in standard deviations in two methods (which method is more precise)

• Grubbs Test: Determine if a data point can be excluded on a statistical basis

Statistical TestsPossible Outcomes

• Outcome #1 – There is a statistically significant result (e.g. a systematic error)– this is at some probability (e.g. 95%)– can occasionally be wrong (5% of time possible if

test barely valid at 95% confidence)• Outcome #2 – No significant result can be

detected– this doesn’t mean there is no systematic error or

difference in averages– it does mean that the systematic error, if it

exists, is not detectable (e.g. not observable due to larger random errors)

– It is not possible to prove a null hypothesis beyond any doubt

Statistical Testst Tests

• 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 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 Example

• A new method for determining sulfur content in kerosene was tested on a sample known to contain 0.123% S.

• The measured %S were:0.112%, 0.118%, 0.115%, and 0.119%Do the data show a significant bias at

a 95% confidence level?Clearly lower, but is it significant?

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)