Transcript ME 3031 Lecture Notes Week 1

Uncertainty & Error
“Science is what we have learned
about how to keep from fooling
ourselves.”
― Richard P. Feynman
Types of Errors
http://ibguides.com/physics/notes/measurement-and-uncertainties
Error
 Uncertainty is NOT error!
 Difference between measured result and true value.
 Illegitimate errors




Blunders resulting from mistakes in procedure. You must be careful.
Computational or calculational errors after the experiment.
Not paying attention!
Bias or Systematic errors

An offset error; one that remains with repeated measurements (i.e. a
change of indicated pressure with the difference in temperature from
calibration to use).




Systematic errors can be reduced through calibration
Faulty equipment--such as an instrument which always reads 3% high
Consistent or recurring - observer bias
This type of error cannot be evaluated directly from the data but can be
determined by comparison to theory or other experiments.
“Human Error”
 Giraffes
don’t do science!!
 Of course you’re a human.
 You probably mean “systematic error”
 OR . . . You mean uncertainty, which
isn’t an error at all. It is you being
honest.
 Explain what you mean! (Example:
Parallax on Meter Stick)
 Letting it fall isn’t human error, it is not
following the procedure.
Types of Uncertainties
AKA- “Plus/Minuses” , +/- , Tolerance , Standard
Deviations
Random Uncertainties: result from the randomness of
measuring instruments. They can be dealt with by making
repeated measurements and averaging. One can calculate the
standard deviation of the data to estimate the uncertainty.
Systematic Uncertainties: result from a flaw or limitation in the
instrument or measurement technique. Systematic uncertainties
will always have the same sign. For example, if a meter stick is
too short, it will always produce results that are too long.
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Accuracy vs. Precision
Accurate:
•
How close a measurement is to an accepted / “true” value.
•
An accurate measurement correctly reflects the size of the
thing being measured.
•
Must know the correct answer beforehand!
Precise:
• How close a measurement is to another.
• repeatable, reliable, getting the same
measurement each time. A measurement can be
precise but not accurate.
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Bias, Precision, and Total Error
Total Error
Bias Error
Precision
Error
X True
X measured
Percent Difference: It’s Accuracy!
Calculating the percent difference is a useful way to compare
experimental results with the accepted value, but it is not a
substitute for a real uncertainty estimate.
 accepted value - experiment al value
% diff  
accepted value


 100%

Example: Calculate the percent difference if a measurement
of g resulted in 9.4 m / s2 .
 9.8 m 2  9.4 m 2 

s
s  100 %   4%
% diff  

9.8 m 2


s


Uncertainty
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Absolute and Percent Uncertainties
If x = 99 m ± 5 m then the 5 m is referred to as an absolute
uncertainty and the symbol σx (sigma) is used to refer to it. You
may also need to calculate a percent uncertainty/fractional
uncertainty ( %σx):
 5m 
  100%  5%
% x  
 99 m 
NO UNITS!
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Uncertainty Analysis


The estimate of the error is called the uncertainty.
 It includes both bias and precision errors.
 We need to identify all the potential significant errors for the
instrument(s).
 All measurements should be given in three parts
 Mean value
 Uncertainty
 Confidence interval on which that uncertainty is based
(typically 95% C.I.)
Uncertainty can be expressed in either
absolute terms (i.e., 5 Volts ±0.5 Volts)
or in
percentage terms (i.e., 5 Volts ±10%)
(relative uncertainty = DV / V x 100)
 We will use a 95 % confidence interval throughout this course
(20:1 odds).
Use Statistics to Estimate Random Uncert.

Mean: the sum of measurement values divided by
the number of measurements.
1 N
x   xi
N i 1

Deviation: the difference between a single result and
the mean of many results.
d i  xi  x

Standard Deviation: the smaller the standard
1
deviation the more precise the data
1
2 2

 Large sample size

   xi  x
n


Small sample size (n<30)

Slightly larger value

1
2
1
2 

x

x

n  1  i

 s  
The Population

Population: The collection of all items
(measurements) of the group. Represented by a large
number of measurements.



Gaussian distribution*
3
- 2
-
x
x i  x  1
n
68.3% of the time
xi  x  2
n
95.4% of the time
x i  x  3
n
99.7% of the time

2
3
Sample: A portion of (or limited number of items in)
a population.
*Data do not always abide by the Gaussian distribution. If
not, you must use another method!!
Standard Deviation
Uncertainty
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Standard Deviation
The average or mean of a set of data is
1
x
N
N
x
i 1
i
The formula for the standard deviation given below is the one
used by Microsoft Excel. It is best when there is a small set of
measurements. The version in the book divides by N instead of
N-1.

1 N
( xi  x ) 2

N  1 i 1
Unless you are told to use the above function, you may use the
Excel function ‘=stdev(B2:B10)’
Uncertainty
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Expressing Results in terms of
the number of σ
•In this course we will use σ to represent the uncertainty in a
measurement no matter how that uncertainty is determined
•You are expected to express agreement or disagreement
between experiment and the accepted value in terms of a
multiple of σ.
•For example if a laboratory measurement the acceleration
due to gravity resulted in
g = 9.2 ± 0.2 m / s2 you
would say that the results differed by 3σ from the accepted
value and this is a major disagreement
•To calculate Nσ
Uncertainty
N 
accepted  exp erimental

9.8  9.2

3
0.2
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Uncertainty resulting from
averaging N measurements
If the uncertainty in a single measurement of x is statistical, then
you can reduce this uncertainty by making N measurements and
averaging.
x
 avg 
N
Example: A single measurement of x yields
x = 12.0 ± 1.0, so you decide to make 10 measurements and
average. In this case N = 10 and σx = 1.0, so the uncertainty
in the average is

1.0
 avg  x 
 0.3
N
10
This is not true for systematic uncertainties- if your meter stick
is too short, you don’t gain anything by repeated
Uncertainty
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measurements.
Propagation of Error
 Used
to determine uncertainty of a
quantity that requires measurement of
several independent variables.
 Volume
of a cylinder = f(D,L)
 Volume of a block = f(L,W,H)
 Density of an ideal gas = f(P,T)
 IB
Does this on a worst case scenario!
Uncertainty when a number is raised to
a power
If z = xn then %σz = n ( % σx )
Example: If z = 12 ± 1.0 = 12.0 ± 8.3 % then
z 3  (12  8.3%) 3 1728  3(8.3%)  1728  25%
z 3 1728  430  1700  400
z
1
2
1
1
 (12  8.3%) 2  3.46  (8.3%)  3.46  4.1%
2
z
Uncertainty
1
2
 3.46  .14
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Uncertainty when calculation involves a
special function
For a special function, you add and subtract the uncertainties
from the value and calculate the function for each case.
Then plug these numbers into the function.
Example: If θ = 120 ± 2.00
sin(140) = 0.242
sin(120)
sin(100)
= 0.208
= 0.174
0.034
0.034
And thus sin(120 ± 20 ) = 0.208 ± 0.034
Uncertainty
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