Error Analysis. Experimental Error –The uncertainty obtained in a measurement of an experiment...

Error Analysis

Experimental Error

• Experimental Error – The uncertainty obtained in a

measurement of an experiment– Results can from systematic and/or random

errors• Blunders• Human Error• Instrument Limitations

– Relates to the degree of confidence in an answer

– Propagation of uncertainties must be calculated and taken into account

Experimental Error

It is impossible to make an exact measurement. Therefore, all experimental results are wrong. Just how wrong they are depends on the kinds of errors that were made in the experiment.

As a science student you must be careful to learn how good your results are, and to report them in a way that indicates your confidence in your answers.

Types of Errors

• Systematic Errors– These are errors caused by the way in

which the experiment was conducted. In other words, they are caused by the design of the system or arise from flaws in equipment or experimental design or observer

– Sometimes referred to as determinate errors

– Reproducible with precision– Can be discovered and corrected

Systematic Error

The cloth tape measure that you use to measure the length of an object had been stretched out from years of use. (As a result, all of your length measurements were too small)

The electronic scale you use reads 0.05 g too high for all your mass measurements (because it was improperly zeroed at the beginning of your experiment).

Examples:

Detection of Systematic Errors

• Analyze samples of known composition– Use standard Reference material– Develop a calibration curve

• Analyze “blank” samples– Verify that the instrument will give a zero

result• Obtain results for a sample using

multiple instruments– Verifies the accuracy of the instrument

How to Eliminate Systematic Errors

How would you measure the distance between two parallel vertical lines

- most would pull out a ruler, align one end with one bar, read of the distance.

- You should put ruler down randomly (as perpendicular as you can). Note where each mark hits the ruler, then subtract the two readings. Repeat a number of times and average the result.

- Minimize the number of human operations you can

Elimination of systematic error can best be accomplished by a well planned and well executed experimental procedure

Types of Errors• Random Errors

– Sometimes referred to as indeterminate errors or noise errors

– Arises from things that cannot be controlled

• Variations in how an individual or individuals read the measurements

• Instrumentation noise– Always present and cannot always be

corrected for, but can be treated statistically

– The important property of random error is that it adds variability to the data but does not affect average performance for the data

Random Errors

Examples:

You measure the mass of a ring three times using the same balance and get slightly different values: 12.74 g, 12.72 g, 12.75 g

The meter stick that is used for measuring, slips a little when measuring the object

Accuracy or Precision

• PrecisionReproducibility of resultsSeveral measurements afford the same

results Is a measure of exactness

• AccuracyHow close a result is to the “true”

value“True” values contain errors since they

too were measured Is a measure of rightness

Accuracy vs Precision

= Accuracy Precision

3 NO NO

7.18281828 NO YES

3.14 YES NO

3.1415926 YES YES

Calculating Errors

Terminology Significant Figures – minimum number of

digits required to express a value in scientific notation without loss of accuracy

Absolute Uncertainty – margin of uncertainty associated with “a” measurement

Relative Uncertainty – compares the size of the absolute uncertainty with the size of its associated measurement (a percent)

Propagation of Uncertainty – The calculation to determine the uncertainty that results from multiple measurements

Significant Figures

How to determine which digits are Significant Write the number as a power of 10 Zero’s are significant and must be included when

they occur• In the middle of a number• At the end of a number on the right hand side of

the decimal point– This implies that you know the value of a measurement

accurately to a specific decimal point

The significant figures (digits) in a measurement include all digits that can be known precisely, plus a last digit that is an estimate.

Significant Figures

Let’s look at 123.45

1.2345x102

Scientific Notation

We have 5 significant digits

Let’s look at 0.000123

1.23x10-4

We have 3 significant digits

Significant Figures

Determine the number of significant digits in the following numbers:

Significant Figures

142.7 1.427x102

4 significant digits

142.70 1.4270x102


Significant Figures

0.000006302 6.302x10-6


0.003050 3.050x10-3


Significant Figures

9.250x104 9.250x104


10.003x104 1.0003x105


Significant Figures

9000 9x103

1 significant digit

9000. 9.000x103


Significant Figures

The last significant digit in a measured quantity is the first digit of uncertainty

Significant Figures

Certain values 1 degree of uncertainty

True expression

Absorbance 0.23 0.234 0.234 ± 0.001

% Transmittance 58 58.3 58.3 ± 0.1

Determine the significant figures from the diagram below

Significant Figures

When adding or subtracting, the last digit retained is set by the first doubtful number.

When multiplying or dividing, the number of significant digits you use is simply the number of significant figures as is in the term with the fewest significant digits.

Adding Significant Digits

4503+34.90+550= 5090

3 is the first doubtful number



The 87.9 are the doubtful numbers


Via Calculator: 5087.9

Adding Significant Digits

2456.2345+23.21=

23400.00+111.49=

23400+111.49=

234000-2340=

2479.44

23511.49

23500

232000

2479.4445

23511.49

23511.49

231660

Multiplying Significant Digits

2.7812x1.7= 4.72804

Rounded to 4.7 because 1.7 only has 2 significant digits

4.7

Multiplying Significant Digits

14.200x3.2400=

1.00x150.03=

1200x1.234=

45.35.2345=

48.008

150.03

1480.8

8.654121…

48.008

150

1500

8.65

Rounding

Rounding is the process of reducing the number of significant digits in a number. The result of rounding is a "shorter" number having fewer non-zero digits yet similar in magnitude. The result is less precise but easier to use. There are several slightly different rules for rounding.

Rounding

Common method• This method is commonly used, for example

in accounting.• Decide which is the last digit to keep. • Increase it by 1 if the next digit is 5 or more

(this is called rounding up) • Leave it the same if the next digit is 4 or less

(this is called rounding down) • Example: 7.146 rounded to hundredths is

7.15 (because the next digit [6] is 5 or more).

RoundingThis method is also known as statistician's rounding . It is identical to the common method of rounding except when the digit(s) following to rounding digit start with a five and have no non-zero digits after it. The new algorithm is:

Decide which is the last digit to keep. Increase it by 1 if the next digit is 6 or more, or a 5 followed by one or

more non-zero digits. Leave it the same if the next digit is 4 or less Round up or down to the nearest even digit if the next digit is a five

followed (if followed at all) only by zeroes. That is, increase the rounded digit if it is currently odd; leave it if it is already even.

Examples:

7.016 rounded to hundredths is 7.02 (because the next digit (6) is 6 or more)

7.013 rounded to hundredths is 7.01 (because the next digit (3) is 4 or less)

7.015 rounded to hundredths is 7.02 (because the next digit is 5, and the hundredths digit (1) is odd)

7.045 rounded to hundredths is 7.04 (because the next digit is 5, and the hundredths digit (4) is even)

7.04501 rounded to hundredths is 7.05 (because the next digit is 5, but it is followed by non-zero digits)

Increasing Precision with Multiple Measurements

One way to increase your confidence in experimental data is to repeat the same experiment many times.

When dealing with repeated measurements, there are three important statistical quantities

Mean (or average)

Standard Deviation

Standard Error

Mean

What is it:

An estimate of the true value of the measurement

Statistical Interpretation:

The central value

Symbol:

x

Standard Deviation

What is it:

A measure of the spread in the data


You can be reasonably sure (about 70% sure) that if you repeat the same experiment one more time, that the next measurement will be less than one standard deviation away from the average

Symbol:

Use you calculator or computer to determine the Standard Deviation.

Standard Error

What is it:

An estimate in the uncertainty in the average of the measurements


You can be reasonably sure (about 70% sure) that if you repeat the entire experiment again with the same number of repetitions, the average value from the new experiment will be less than one standard deviation away from the average value of this experiment

Symbol:M

N

Standard Error Example

Measurements: 0.32, 0.54, 0.44, 0.29, 0.48

Calculate the Mean:

0.41

Calculate the Standard Deviation:

0.09

Calculate the Standard Error:

0.04

MN

Therefore: 0.41±0.04

Use this technique to determine the uncertainty if you do not know the uncertainty of a measurement, but have multiple measurements of the value.

Propagation of Uncertainty

• Since measurements commonly will contain random errors that lead to a degree of uncertainty, arithmetic operations that are performed using multiple measurements must take into account this propagation of errors when reporting uncertainty values

Systematic Errors

Errors calculated from data are Random Errors

Errors from the instrument are called System Errors (usually labeled on instrument or told by instructor as a percent)

15.23 0.05 0.17

random systemk k k

22

15.23 0.18

15.2 0.2

ran systemk k k

or

Error Propagation

There are 3 different ways of calculating or estimating the uncertainty in calculated results

Significant digits (The easy way out)Useful when a more extensive uncertainty analysis is not needed.

Error Propagation (Not as bad as it looks)Useful for limited number or single measurements

Statistical Methods (When you have lots of numbersUseful for many measurements

Dependent Error Propagation

Adding and Subtracting

2 2 2...e x y z

Multiplying and Dividing

2 2 2

...x y z

e Ex y z

Average

v v

Dependent (approx)

(121)+(52)-(73)

(121)*(52)*(73)

2 2 2

1 2 312 5 7 12 5 7

12 5 7

420 249

If the Average is 25, then 25 5

2 2 212 5 7 1 2 3

10 4

Propagation of Errors Basic Rule

If x and y have independent random errors and , then error in z=x+y is

2 2z x y

x y

3 0.1

4 0.2

x

y

3 4

7

z

2 20.1 0.2

0.223

z

7 0.2Therefore we have



1.76 (0.03) + 1.89 (0.02) – 0.59 (0.02) =

Z=1.76+1.89-0.59=3.06

2 2 2

3

0.03 0.02 0.02

1.7 10

0.041231056

z

Therefore Z=3.06 0.04

Propagation of Errors Basic Rule

If x and y have independent random errors and , then error in z=xy is 22

x yz z

x y

x y

3 0.1

4 0.2

x

y

3 4

12

z

2 20.1 0.2

123 4

0.72211102555

z

12 0.7Therefore we have



[1.76(0.03) x 1.89(0.02)] / 0.59(0.02) =

Z=

1.76 1.89

5.6379661020.59

2 2 21.76 1.89 0.03 0.02 0.02

0.59 1.76 1.89 0.59

0.222083034

z

Therefore z=5.6 0.2

Putting it Together

x=200 2Y=50 2z=40 2

xq

y z

x, y, z are independent, find q

Let d=y-z 2 250 40 2 2

10 2 2

10 3

d

200

1020

xq

d

2 22 3

20200 10

20 0.0901

6

q

Therefore q=20 6

What about Functions of 1 Variable

Find error for with s=20.023V s

We cannot use because

s, s, s are not independent

2 2 2s s s

z zs s s

What to the rescue???

Calculus

V=s3

Let’s take the derivative of V with respect to s 23

dVs

ds

2

2

3

3 2 0.02

0.24

dV s ds

Therefore the value for V is

V=80.2

32

8

V

Think of dV and ds as a small change (error) in V and s

x=100 6 then find V when V x

A function of one variable… CALCULUS

1

2

1

21

2

26

2 1000.3

V x

x

dVx

dxdx

dVx

Therefore V=10.0 0.3

100

10

V x

What about a Function with a Constant?

You measure the diameter of a circle to be 20.02

Determine the area of the circle2

2

2

2

1

4

A r

d

d

Calculus

1

21

21

2 0.0220.06

dAd

dd

dA d dd

2

21

A r

The area is 3.14 0.06

If q=f(x1, x2, x3, …xn)

22 2

1 21 2

... nn

q q qq x x x

x x x

then

Let q=x1+x2

2 2

1 21 2

2 2

1 2

2 2

1 1

1 1

q qq x x

x x

x x

x x

2 2

1 1q x x Previous rule

PROOF

If q=f(x1, x2, x3, …xn)

Let q=x1*x2

2 2

1 1

1 2

x xq q

x x

Previous rule

PROOF

2 2

1 21 2

2 2

2 1 1 2

2 22 22 1 1 2

2 22 2

1 22 21 2

2 2

1 2

1 2

q qq x x

x x

x x x x

x x x x

q qx x

x x

x xq

x x

21

qx

x

The Atwood Machine consists of two masses M and m attached to the ends of a light, frictionless pulley. When the masses are released, the mass M is show to accelerate down with an acceleration:

M ma g

M m

Suppose the M and m are measured as M=100 1g and m=50 1 g. Find the uncertainty in a

2

2

1 1

2

M m M mag

M M m

mg

M m

2

2

1 1

2

M m M mag

m M m

Mg

M m

The Partial Derivatives are:

2 2

2 2

2 2

2 22 22

2 2 2 2

2

2 2

2

2 9.850 1 100 1

100 50

0.1

a aq M m

M m

mg MgM m

M m M m

gm M M m

M m

100 509.8

100 503.3

M ma g

M m

Therefore a=(3.3 0.1) m/s2

Uncertainty

Focal Length

pqf

p q

Determine the focal length plus uncertainty when p=100±2 cm and q=30±1 cm

2

2

2

1q p q pqf

p p q

q

p q

2

2

2

1p p q pqf

q p q

p

p q

Focal Length

2 2

2 24 4

2

2 24 4

2

30 2 100 1

100 30

0.6112

f ff p q

p q

q p p q

p q

100 30

100 3023.076

23.1

f

The focal length is (23.1±0.6) cm or (23±1) cm

Ugly Trig Problem

2

cos 4

xq

x y

Determine q and error is x=10±2, y=7±1, Ø=400±30

2

cos 4 2

cos 4

yq

x x y

2

2 cos 4

cos 4

xq

y x y

2

4 2 sin 4

cos 4

x yq

x y

=-0.732

=0.963

=9.813

Still the Ugly Trig Problem

2

2 20.732 2 0.963 1 9.813 3

180

3.3

1.8165

q

Therefore q=3.5±2

Trig should be in

radians

Max-Min Technique

If you do not have a calculus background, then you can use this technique to determine the uncertainties in a complicated equation.

1) Determine the actual value.2) Make the largest possible value.3) Make the smallest possible value.4) Average the difference between the actual and the largest

value and the actual and the smallest value.5) This average is the uncertainty.

Ugly Trig Problem again

2

cos

x yq

Determine q and error is x=10±2, y=7±1, Ø=400±30

2

cos

10 2 7

cos 40

24.8027

x yq

max

1

2

cos

2

cos

30.0812

2 8

43

x yq

min

2

cos

2

cos

20.034

8 6

37

x yq

max

min

24.8028 30.0812 5.2784

24.8028 20.034 4.7688

q q

q q

Ave=5.0236

Therefore: 25±5

Using Percent Errors

Two Simple Rules:

1. When Adding or Subtracting add the Absolute Errors like you would normally do, then convert to Percent (Relative) Error.

2. When Multiplying or Dividing add the Percentage Errors

Observed Value - True ValuPerc

e

Tru =

e Ve

ant Er

lueror 100

Percent Error Exmple

4.52 0.02 , 2.0 0.2 , 3.0 0.6w x y

2znd yF wxi

For wx: 0.02 0.20.004425 0.1 0.104425 10.44%

4.52 2.0

For y2:0.6 0.6

0.2 0.2 0.4 40%3.0 3.0

To find Error for z, we need to convert Percent Errors to Absolute

2

0.910.4 444%

40%

4.52 2.0

3.63.0

2 218 0.9 3.6 18 3.79.0 180.9 9.0 3.6 4

Therefore:

Error Analysis. Experimental Error –The uncertainty obtained in a measurement of an experiment...

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