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1
Error AnalysisExperimental 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 ananswer
– Propagation of uncertainties must becalculated and taken into account
Experimental Error
It is impossible to make an exactmeasurement. Therefore, all experimentalresults are wrong. Just how wrong they aredepends on the kinds of errors that were madein the experiment.
As a science student you must be careful tolearn how good your results are, and to reportthem in a way that indicates your confidence inyour answers.
Types of Errors
• Systematic Errors– These are errors caused by the way in which
the experiment was conducted. In otherwords, they are caused by the design of thesystem or arise from flaws in equipment orexperimental design or observer
– Sometimes referred to as determinate errors– Reproducible with precision– Can be discovered and corrected
2
Systematic Error
The cloth tape measure that you use to measurethe length of an object had been stretched outfrom years of use. (As a result, all of your lengthmeasurements were too small)
The electronic scale you use reads 0.05 g toohigh for all your mass measurements (because itwas improperly zeroed at the beginning of yourexperiment).
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 twoparallel vertical lines
-most would pull out a ruler, align one end withone bar, read of the distance.
-You should put ruler down randomly (asperpendicular as you can). Note where each markhits the ruler, then subtract the two readings.Repeat a number of times and average theresult.
-Minimize the number of human operations youcan
Elimination of systematic error can best beaccomplished by a well planned and well executedexperimental procedure
Types of Errors• Random Errors
– Sometimes referred to as indeterminateerrors 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 becorrected for, but can be treatedstatistically
– The important property of random error isthat it adds variability to the data but doesnot affect average performance for the data
3
Random Errors
Examples:
You measure the mass of a ring three timesusing the same balance and get slightly differentvalues: 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
resultsIs a measure of exactness
• AccuracyHow close a result is to the “true” value“True” values contain errors since they too
were measuredIs a measure of rightness
Accuracy vs Precision
YESYES3.1415926
NOYES3.14
YESNO7.18281828
NONO3
PrecisionAccuracy=
Calculating Errors
TerminologySignificant Figures – minimum number of digits
required to express a value in scientific notation withoutloss of accuracy
Absolute Uncertainty – margin of uncertaintyassociated with “a” measurement
Relative Uncertainty – compares the size of theabsolute uncertainty with the size of its associatedmeasurement (a percent)
Propagation of Uncertainty – The calculationto determine the uncertainty that results from multiplemeasurements
4
Significant Figures
How to determine which digits are SignificantWrite the number as a power of 10Zero’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 measurementinclude all digits that can be known precisely, plus alast 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 significantdigits in the following numbers
– 142.7– 142.70– 0.000006302– 0.003050– 10.003 x 104
– 9.250 x 104
– 9000– 9000.
Significant Figures
142.7 1.427x102
4 significant digits
142.70 1.4270x102
5 significant digits
5
Significant Figures
0.000006302 6.302x10-6
4 significant digits
0.003050 3.050x10-3
4 significant digits
Significant Figures
9.250x104 9.250x104
4 significant digits
10.003x104 1.0003x105
5 significant digits
Significant Figures
9000 9x103
1 significant digit
9000. 9.000x103
4 significant digits
Significant Figures
The last significant digit in ameasured quantity is the first digitof uncertainty
6
Significant Figures
58.3 ± 0.158.358% Transmittance
0.234 ± 0.0010.2340.23Absorbance
True expression1 degree ofuncertainty
Certain values
Determine the significant figures from thediagram below
Significant Figures
When adding or subtracting, the last digitretained is set by the first doubtfulnumber.
When multiplying or dividing, the numberof significant digits you use is simply thenumber of significant figures as is in theterm with the fewest significant digits.
Adding Significant Digits
4503+34.90+550= 5090
3 is the first doubtful number
0 is the first doubtful number
5 is the first doubtful number
The 87.9 are the doubtful numbers
3 significant digits
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
7
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
Significant Figures in Logarithms andAntilogarithms
• Logarithm of n– n = 10a or log n = a
• 2 parts to a logarithm– Characteristic – integer part– Mantissa – decimal part
• Logarithm – the number of significant digits found in n = thenumber of significant digits in the mantissa
• Antilogarithm – the number of significant digits in the mantissa =the number of significant digits expressed in the answer
1.23451.2345
Express the answer of each of the following withthe correct # of Significant Figures
• Logarithms and Antilogarithms– log 339 = log 1237 =– log (3.39 x 10-5) = log 3.2 =– antilog (-3.42) = antilog 4.37 =– Log 0.001237 = 104.37 =– 10-2.600 = log (2.2 x 10-18) =– antilog (-2.224) = 10-4.555 =
8
Logarithms
log 339 = 2.530 log 1237 =3.0924
log (3.39 x 10-5) =-4.470 log 3.2 =0.51
antilog (-3.42) =3.8x10-4 antilog 4.37 =23000
Log 0.001237 =-2.9076 104.37 =23000
10 -2.600 =2.51 log (2.2 x 10-18) =-17.66
antilog (-2.224) = 5.97x10-310 -4.555 =2.79x10-5
Rounding
Rounding is the process of reducing the numberof significant digits in a number. The result ofrounding is a "shorter" number having fewernon-zero digits yet similar in magnitude. Theresult is less precise but easier to use. Thereare several slightly different rules forrounding.
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 identicalto the common method of rounding except when the digit(s)following to rounding digit start with a five and have no non-zerodigits 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 iscurrently 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)
9
Increasing Precision withMultiple Measurements
One way to increase your confidence in experimentaldata is to repeat the same experiment many times.
When dealing with repeated measurements, there arethree 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
Statistical Interpretation:
You can be reasonably sure (about 70% sure)that if you repeat the same experiment onemore time, that the next measurement will beless than one standard deviation away fromthe average
Symbol:
Standard Error
What is it:
An estimate in the uncertainty in the averageof the measurements
Statistical Interpretation:
You can be reasonably sure (about 70% sure)that if you repeat the entire experiment againwith the same number of repetitions, theaverage value from the new experiment will beless than one standard deviation away fromthe average value of this experiment
Symbol:M N
10
Example
Measurements: 0.32, 0.54, 0.44, 0.29, 0.48
Calculate the Mean: 0.41
Calculate the Standard Deviation: 0.11
Calculate the Standard Error: 0.05
M N
Therefore: 0.41±0.05
Relative Compared to AbsoluteUncertainty
Absolute uncertainty illustrates theuncertainty in a measurement
6.3302± 0.001
Relative uncertainty illustrates themagnitude of uncertainty with regardto the measurement.
Relative Compared to AbsoluteUncertainty
Relative uncertainty – compares theabsolute uncertainty with the size ofthe associated measurement
Relative uncertainty = absolute uncertainty / measurement
Percentage Relative Uncertainty
% relative uncertainty = relative uncertainty x 100
Propagation of Uncertainty
• Since measurements commonly willcontain random errors that lead to adegree of uncertainty, arithmeticoperations that are performed usingmultiple measurements must takeinto account this propagation oferrors when reporting uncertaintyvalues
11
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 asa percent)
15.23 0.05 0.17random systemk k k
22
15.23 0.1815.2 0.2
ran systemk k k
or
Error Propagation
There are 3 different ways of calculating orestimating the uncertainty in calculated results
Significant digits (The easy way out)Useful when a more extensive uncertainty analysis is notneeded.
Error Propagation (Not as bad as it looks)Useful for limited number or single measurements
Statistical Methods (When you have lots ofnumbersUseful for many measurements
Dependent Error Propagation
Adding and Subtracting
...e x y z
Multiplying and Dividing
...x y z
e Ex y z
Average
v v
Dependent (approx)
(121)+(52)-(73) =(12+5-7) (1+2+3)=106
(121)*(52)*(73)
1 2 312 5 7 12 5 7
12 5 7420 383
Average is 25, then 25 5
12
Propagation of ErrorsBasic Rule
If x and y have independent random errorsand , then error in z=x+y is
2 2z x y
x y
3 0.14 0.2
xy
3 47
z
2 2
0.1 0.2
0.223
z
7 0.2Therefore we have
Adding and Subtracting
Adding and Subtracting
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
1.7 100.041231056
z
Therefore Z=3.06 0.04
Propagation of ErrorsBasic Rule
If x and y have independent randomerrors and , then error in z=xy is
22x yz z
x y
x y
3 0.1
4 0.2
x
y
3 412
z
2 20.1 0.212
3 40.72211102555
z
12 0.7Therefore we have
Multiplying and Dividing
Multiplying and Dividing
[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 0.020.59 1.76 1.89 0.59
0.222083034
z
Therefore z=5.6 0.2
13
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
20010
20
xq
d
2 22 320
200 10
20 0.901
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 withrespect 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
12
121
2
26
2 1000.3
V x
x
dV xdx
dxdV
x
Therefore V=10.0 0.3
10010
V x
14
What about a Function witha Constant?
You measure the diameter of a circle to be 20.02
Determine the area of the circle2
2
2
214
A r
d
d
Calculus
12121
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 xx x x
then
Let q=x1+x2
2 2
1 21 2
2 21 2
2 2
1 1
1 1
q qq x xx x
x x
x x
2 21 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 22 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 xx x
x x x x
x x x x
q qx x
x x
x xq
x x
21
qxx
The Atwood Machine consists of two masses M and mattached to the ends of a light, frictionless pulley. Whenthe masses are released, the mass M is show to acceleratedown 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
mgM m
2
2
1 1
2
M m M ma gm M m
MgM m
The Partial Derivatives are:
15
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 mM 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±2cm and q=30±1 cm
2
2
2
1q p q pqfp p q
qp q
2
2
2
1p p q pqfq p q
pp 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 30100 30
23.07623.1
f
The focal length is 23.1±0.6 cm or 23±1 cm
Ugly Trig Problem
2
cos 4xq
x y
Determine q and error is x=10±2, y=7±1, Ø=400±30
2
cos 4 2
cos 4
yqx x y
2
2 cos 4
cos 4
xqy x y
2
4 2 sin 4
cos 4
x yq
x y
=-0.732
=0.963
=9.813