Transcript Measurement Uncertainties PowerPoint

Treatment of Uncertainties
PHYS 244, 246
© 2003
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Types of Uncertainties
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: means correct. An accurate measurement
correctly reflects the size of the thing being
measured.
Precise: repeatable, reliable, getting the same
measurement each time. A measurement can be
precise but not accurate.
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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)’
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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 ( %σx):
 5m 
 100%  5%
% x  
 99 m 
Please do not write a percent uncertainty as a decimal ( 0.05)
because the reader will not be able to distinguish it from an
absolute uncertainty.
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Standard Deviation
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Standard Deviation
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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
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N 
accepted  exp erimental

9.8  9.2

3
0.2
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Propagation of Uncertainties with
Addition or Subtraction
If z = x + y or z = x – y then the absolute uncertainty in z is
given by
 z   x2   y2
Example:
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Propagation of Uncertainties with
Multiplication or Division
If z = x y or z = x / y then the percent uncertainty in z is given
by
%  % 2  % 2
z
x
y
Example:
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Propagation of Uncertainties in mixed
calculations
If a calculation is a mixture of operations, you propagate
uncertainties in the same order that you perform the calculations.
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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
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measurements.
Special Rule:
Uncertainty when a number is multiplied by a
constant
Example: If x = 12 ± 1.0 = 12.0 ± 8.3 % and z = 2 x, then z =
24.0 ± 8.3 % or z = 24 ± 2. It should be noted that you would
get the same result by multiplying 2 (12 ± 1.0)= 24 ± 2.
This is actually a special case of the rule for multiplication and
division. You can simply assume that the uncertainty in the
constant is just zero and get the result given above.
% z  (0%) 2  (8.3%) 2  8.3%
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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
 (12  8.3%)
z
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2
1
2
1
 3.46  (8.3%)  3.46  4.1%
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
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Percent Difference
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


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