Transcript introduction

Uncertainties in Measurement
Laboratory investigations involve taking measurements
of physical quantities. All measurements will involve
some degree of experimental uncertainty.
QUESTIONS
1. How does one express the uncertainty in an experimental
measurement ?
2. How does one determine the uncertainty in an experimental
measurement ?
3. How does one compare an experimental measurement with an
accepted (or published) value ?
4. How does one determine the uncertainty in a quantity that is
computed from uncertain measurements ?
Expressing Uncertainty
We will express the results of measurements in this
laboratory as
(measured value  uncertainty) units
For example
2
g  Dg = (9.803  0.008) m/s
Types of Experimental Uncertainty
Random, Indeterminate or Statistical
– Results from unknown and unpredictable variations that
arise in all experimental situations.
– Repeated measurements will give slightly different values
each time.
– You cannot determine the magnitude (size) or sign of
random uncertainty from a single measurement.
– Random errors can be estimated by taking several
measurements.
– Random errors can be reduced by refining experimental
techniques.
Types of Experimental Uncertainty
Systematic or Determinate
– Associated with particular measurement instruments or
techniques.
– The same sign and nearly the same magnitude of the error
is obtained on repeated measurements.
– Commonly caused by improperly “calibrated” or “zeroed”
instrument or by experimenter bias.
Accuracy and Precision
Accuracy
Is a measure of how close an experimental result is
to the “true” (or published or accepted) value.
Precision
Is a measure of the degree of closeness of repeated
measurements.
Accuracy and Precision
Consider the two measurements:
A = (2.52 ± 0.02) cm
B = (2.58 ± 0.05) cm
Which is more precise ?
Which is more accurate ?
Accuracy and Precision
Answer with GOOD or POOR ...
_____ accuracy
_____ accuracy
_____ accuracy
_____ precision
_____ precision
_____ precision
Implied Uncertainty
The uncertainty in a measurement can sometimes be
implied by the way the result is written. Suppose the
mass of an object is measured using two different
balances.
Balance 1 Reading = 1.25 g
Balance 2 Reading = 1.248 kg
Significant Figures
In a measured quantity, all digits are significant except any
zeros whose sole purpose is to show the location of the
decimal place.
123 g
_________
1.23 x 102 g
123.0 g
_________
1.230 x 102 g
0.0012 m
_________
1.2 x 10-3 m
0.0001203 cm
_________
1.203 x 10-4 s
0.001230 s
_________
1.230 x 10-4 s
1000 cm
_________
1 x 103 cm
1000. cm
_________
1.000 x 103 cm
_________
150
150
Rounding
• If the digit to the right of the position you wish to round to is <
5 then leave the digit alone.
• If the digit to the right of the position you wish to round to is
>= 5 then round the digit up by one.
• For multiple arithmetic operations you should keep one or two
extra significant digits until the final result is obtained and
then round appropriately.
• Proper rounding of your final result will not introduce
uncertainty into your answer. ROUNDING DURING
CALCULATIONS IS NOT A VALID SOURCE OF ERROR.
Expressing Uncertainty
When expressing a measurement and its associated uncertainty as
(measured value  uncertainty) units
• Round the uncertainty to one significant digit, then
• round the measurement to the same precision as the
uncertainty.
For example, round 9.802562  0.007916 m/s2 to
2
g  Dg = (9.803  0.008) m/s
Significant Figures in Calculations
Multiplication and Division
When multiplying or dividing physical quantities, the number of
significant digits in the final result is the same as the factor (or
divisor…) with the fewest number of significant digits.
6.273 N
0.0204 mm
* 5.5 m
 21 C°
34.5015 N·m
0.00097142857 mm/C°
________ N·m
_________ mm/C°
Significant Figures in Calculations
Addition and Subtraction
When adding or subtracting physical quantities, the precision of the
final result is the same as the precision of the least precise term.
132.45 cm
0.823 cm
+ 5.6
cm
138.873 cm --> _______ cm
Comparing Experimental and Accepted Values
E ± DE = An experimental value and its uncertainty.
A = An accepted (published) value.
Percent Discrepanc y 
EA
100%
A
Percent Discrepancy quantifies the __________ of a measurement.
DE
Percent Uncertaint y 
100%
E
Percent Uncertainty quantifies the __________ of a measurement.
Comparing Two Experimental Values
E1 and E2 = Two different experimental values.
E2  E1
Percent Difference 
100%
 E1  E2 


 2 
Average (Mean) Value
Let x1, x2,… xN represent a set of N measurements of a
quantity x.
The average or mean value of this set of measurements is
given by
1 N
1
x   xi   x1  x2  ...  x N 
N i 1
N
Frequency Distribution (N=10)
hist.d
0
0
0
0
1
0
0
0
2
2
2
0
1
2
0
0
0
0
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0
Mean = 19.6
2.0
1.5
Frequency
data
14
22
20
18
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23
hist.x
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1.0
0.5
0.0
10
15
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Value
25
30
Frequency Distribution (N=100)
Mean = 19.89
hist.d
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Frequency
hist.x
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Value
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Frequency Distribution (N=1,000)
Mean = 19.884
hist.d
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Frequency
hist.x
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Value
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Frequency Distribution (N=10,000)
Mean = 19.9879
hist.d
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1184
1435
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Frequency
hist.x
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Value
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Expressing Uncertainty
The Standard Deviation of a set of N measurements of x is given by:
sx 
1 N
2
(
x

x
)
 i
N  1 i 1
The Standard Deviation of the Mean (or Standard Error of the Mean)
of a set of N measurements of x is given by:
sx
s x  SEM 
N
Expressing Uncertainty
N
Mean
S.D.
SEM
Result
10
19.6
2.71
0.857
19.6  0.9
100
19.89
2.26
0.226
19.9  0.2
1,000
19.884
2.48
0.0784
19.88  0.08
10,000
19.9879
2.52
0.0252
19.99  0.03
Combining Uncertainties:
Propagation of Uncertainty
Let A ± DA and B ± DB represent two measured quantities.
The uncertainty in the sum S = A + B is
DS = DA + DB
The uncertainty in the difference D = A - B is ALSO
DD = DA + DB
Combining Uncertainties:
Propagation of Uncertainty
Let A ± DA and B ± DB represent two measured quantities.
The uncertainty in the product P = A * B is
DA DB 

DP  P 


B 
 A
The uncertainty in the quotient Q = A / B is ALSO
DA DB 
DQ  Q 


B 
 A