Treatment of Uncertainties

1
Treatment of Uncertainties

• PHYS 244 and
• PHYS 246

2
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.
4
Standard Deviation
The average or mean of a set of data is
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.
Unless you are told to use the above function,
you may use the Excel function stdev(B2B10)
5
Absolute and Percent Uncertainties
If x 99 m 5 m then the 5 m is referred to as
an absolute uncertainty and the symbol sx (sigma)
is used to refer to it. You may also need to
calculate a percent uncertainty ( sx)
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
7
Standard Deviation
8
Expressing Results in terms of the number of s

• In this course we will use s 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 s.
• 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 3s from the accepted
  value and this is a major disagreement
• To calculate Ns

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Propagation of Uncertainties withAddition or
Subtraction
If z x y or z x y then the absolute
uncertainty in z is given by
Example
10
Propagation of Uncertainties withMultiplication
or Division
If z x y or z x / y then the percent
uncertainty in z is given by
Example
11
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.
12
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.
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 sx 1.0,
so the uncertainty in the average is
This is not true for systematic uncertainties- if
your meter stick is too short, you dont gain
anything by repeated measurements.
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Special RuleUncertainty 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.
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Uncertainty when a number is raised to a power
If z xn then sz n ( sx )
Example If z 12 1.0 12.0 8.3 then
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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 ? 120 2.00 sin(140) 0.242
sin(120) 0.208 sin(100) 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.
Example Calculate the percent difference if a
measurement of g resulted in 9.4 m / s2 .