Uncertainty Analysis

Dr. Sarah Wang

Duane Long

Department of Chemical Engineering

University of Minnesota Duluth

Reference:

Textbook: An introduction to error

analysis: the study of uncertainties in

physical measurements by J. R.

Taylor (QA275.T38 1982)

3

Presentation Outline

Objective

Why Worry About Uncertainty?

How to Calculate and Express Uncertainty

Error Presentation

• Significant Figure, Rounding

Error Origination

• Human, Fixed, Random

• Fixed Error: Propagation

• Random Error: Calibration

• Experimental Procedure

3

4

Why Worry About Uncertainty?

No measurement is free from error

Engineers are responsible for

reporting reliability of data

Ethical consequences

4

5

How?

1. What is a rational way of estimating

uncertainty in measurements?

2. How do you calculate propagation of

this uncertainty into the measurand?

3. How do you present the results?

Answer Three Questions

5

6

Expressing Uncertainty

Significant Figures

Rounding

6

7

Significant Figures

Significant Figures

6.02, 0.596, 0.000610

Multiplication & Division

Retain the significant figures of the lesser

value

Addition + Subtraction

Retain the precision of the lesser value

7

8

Rounding

Significant figures limited by the

resolution or precision of the

measurement

When the figure ends in ‘5’ choose the

even round value (up or down)

Fixed uncertainty estimated as ½ the

precision

8

9

Least Count vs Readability

Least Count?

10 psi

Readability?

1 to 2 psi

10

Systematic (Fixed) Uncertainty

Input Precision

P(x)

x

12

xzu

10

readability For an analog readout,

unless other data is available,

uncertainty is the readability

divided by square root of 12

11

Analog vs Digital Meters

Readability does not apply to

digital meters.

Unless other data is available,

precision is the least count.

Uncertainty is half of precision.

0.05 psi

12

Estimating Uncertainty

1. Human Error

2. Systematic or Fixed Error

3. Random Error

Origin of Uncertainty

12

13

Output Precision

Output = Y, Inputs are: A, B, C

e.g.: Output, Re = Duρ/µ

= (Q/t)Dρ/µ

13

14

3 Cases of Output Precision

CBAY 222CBAY

ABCY 222

C

C

B

B

A

A

Y

Y

Y = Aα

A

A

Y

Y

14

Propagation of error

15

Combination Cases of Output Precision

22 ; BCACABY

22

C

C

A

A

Y

Y

B

B

C

C2

22

4

B

B

A

A

Y

Y

15

16

Uncertainty for Complex

Equations

Previous examples are derived from

this

Work by hand or

MathCAD can help solve 16

BAfY ,

22

B

B

fA

A

fY

17

Combined uncertainties

Random uncertainty with systematic

uncertainty

17

2 2

i ir isu u u

18

Calibration and

Experimental Uncertainty

Single Point Calibration

Multi-point Calibration

Linear Regression

18

19

Standard Error Single-point Calibration

Sample mean

Standard Deviation

Standard error of mean

Estimate of x =

1

1 n

i

i

x xn

2/1nx

2/1

2

1

1

i i xx

n

xxy 19

i.e., measuring a single point

multiple times

20

Multi-point Line Fit

cxmy ii ˆ

i i

i ii

xx

yyxxm

2

xmyc

2/122

i ii i

i ii

yyxx

yyxxR

20

Linear Regression

21

Multi-point Line Fit, cont.

Estimates for uncertainty

(Standard Error of regression)

for the fit of y upon x:

the slope:

and the intercept:

21

Linear Regression

2/1

2ˆ

2

1

i iiyx yy

nS

2/12

i i

yx

m

xx

SS

2/1

2

2

i i

i i

yxcxxn

xSS

22

Multi-point Calibration

Uncertainty in average measurement

determined from linear regression

calibration

Where n is the number of calibration

readings, k is the number of

measurements, m is the slope, 𝑦 0 is the

average of the measurements, 𝑦 is the

average of the calibration readings, and

𝑠2 is the sample variance of the x-

variable.

22

𝑈𝑦 =𝑆𝑦𝑥

𝑚

1

𝑘+

1

𝑛+

𝑦 0 − 𝑦 2

𝑚2𝑠2(𝑛 − 1)

23

Weighted Multi-point Line Fit

with Uncertainty

22

iiiii

iiiiiiii

xwxww

ywxwyxwwm

22

2

iiiii

iiiiiiiii

xwxww

yxwxwywxwc

222 /11 fri uuuw 23

Linear Regression when the error distribution changes with y -error distribution is weighted

24

Example

Experimental results for the specific heat of air.

T/K 500 600 700

cp / (J g-1 K-1) 1.0296 1.0507 1.0743

s / (J g-1 K-1) 0.0022 0.0032 0.0046

ur / (J g-1 K-1) 0.0013 0.0018 0.0027

uf / (J g-1 K-1) 0.0001 0.0001 0.0001

u / (J g-1 K-1) 0.0013 0.0019 0.0027

w 620000 290000 140000

sn

x 2/1

24

25

25

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

400 500 600 700 800 900 1000 1100

Cp/(J/g-K)

T/K

Specific heat of air with temperature: ∙ ♦ experimental data; — best fit with

weighting: --- best fit without weighting

26

Experimental Procedure

1. Identify the data reduction equation or

sequence of calculation steps.

2. Collect (or estimate) a data point: x1, x2, x3,

... xn.

3. Estimate the uncertainties in each variable:

∆x1, ∆x2, ∆x3, … ∆xn.

4. Calculate the predicted mean value, y.

26

27

Experimental Procedure

5. Calculate the uncertainty of the predicted

mean value according to the error

propagation equation, ∆y.

6. Identify the variables with the largest

contribution to uncertainty

7. Modify your experimental technique to

reduce the larger uncertainties.

8. Take the rest of the data and repeat these

steps if necessary.

27

28

Present the Results

Predicted value Uncertainty (odds)

y y (95% confidence)

Significant figures

Report uncertainty with only one significant

figure .

Precision in the predicted mean value is

determined from the uncertainty

Example, compare

• 123.456 0.123 should be reported as follows

• 123.4 0.1

28

29

Summary

How to Calculate and Express Uncertainty

Error Presentation

• Significant Figures, Rounding

Error Origination

• Human, Fixed, Random

• Fixed Error: Propagation

• Random Error: Calibration

29

30

The End

Next week’s lecture

Technical Writing

Peer Proofreading

Each lab partner brings a hard copy of

a draft report, i.e., two for each group.

30