Outline
PC1221/2
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Fundamentals of Physics I/II
Semester I 2007/08
Measurements and Error Analysis
Rules of the Game
Errors in measurements
Accuracy and Precision
Systematic and Random Errors
Statistical languages
9 mean
9 standard deviation
9 standard error
™ Error Propagation
™ Linear Least Squares Fit
Level 1000 Physics Laboratory
Department of Physics
National University of Singapore
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What is measurement?
Uncertainties and/or Errors
™ All measurements have some degree of
Measurement is the process of quantifying
experience of the external world.
“when you can measure what you are
speaking about and express it in numbers,
you know something about it; but, when you
cannot measure it, when you cannot
express it in numbers, your knowledge is of
a meager and unsatisfactory kind; it may be
the beginning of knowledge, but you have
scarcely in your thoughts to the stage of
science.”
uncertainties/errors.
™ Error is the difference between the result of
the measurement and the true value.
™ The study and evaluation of error in
measurement is often called error analysis.
™ The complete statement of a measured value
SHOULD include an estimate of its error.
Lord Kelvin
Pioneer of thermodynamics &
statistical mechanics
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Expressing uncertainties
Absolute and Fractional uncertainty
The result of a measurement is presented as
(best estimate ± uncertainty) units
(x best ± dx )unit s
(best estimate ± uncertainty) units
(x best ± dx )unit s
Absolute uncertainty: dx
It represents the actual amount by which the best estimated
value is uncertain.
We are quite confident that the quantity lies within
x best - dx < x < x best + dx
dx
Fractional uncertainty:
x best
Example: g ± Δg = (9.803 ± 0.008) m/s2
It gives us the significance of the uncertainty with respect to
the best estimated value.
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Accuracy and Precision
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Accuracy versus Precision
Accuracy
- a measure of how close an experimental result
is to the “true” (or published or accepted) value.
Precision
- a measure of the degree of closeness of
repeated measurements.
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POOR accuracy
POOR accuracy
GOOD accuracy
GOOD precision
POOR precision
GOOD precision
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Accuracy and Precision
Types of Uncertainties
Consider the two measurements:
A = (2.52 ± 0.02) cm
Results from unknown and unpredictable variations
that arise in all experimental situations.
9 Repeated measurements will give slightly different
values each time.
9 You cannot determine the magnitude (size) or sign of
random uncertainty from a single measurement.
9 Random errors can be estimated by taking several
measurements.
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B = (2.58 ± 0.05) cm
Which is more precise?
Which is more accurate?
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Types of Uncertainties
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Systematic versus Random Errors
Associated with particular measurement instruments
or techniques.
9 The same sign and nearly the same magnitude of the
error is obtained on repeated measurements.
9 Commonly caused by improperly “calibrated” or
“zeroed” instrument or by experimenter bias.
9 Systematical errors cannot be eliminated by
averaging or treated statistically.
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Systematic: SMALL
Systematic: SMALL
Random: SMALL
Random: LARGE
Systematic: LARGE
Systematic: LARGE
Random: SMALL
Random: LARGE
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True value is generally not known…
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.
… we can still assess to random errors easily
but cannot tell anything about systematic errors.
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123 g
_________
1.23 x 10 g
123.0 g
_________
1.230 x 10 g
0.0012 m
_________
1.2 x 10 m
0.0001203 cm
_________
1.203 x 10 s
0.001230 s
_________
1.230 x 10 s
1000 cm
_________
1 x 10 cm
1000. cm
_________
1.000 x 10 cm
_________
150
150
2
-3
-4
-4
3
3
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Expressing uncertainty
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Significant figures in calculations
When expressing a measurement and its associated uncertainty as
Addition and Subtraction
(measured value ± uncertainty) units
When adding or subtracting physical quantities, the precision
of the final result is the same as the precision of the least
precise term.
™Round the uncertainty to one significant figure, then
™round the measurement to the same precision as the uncertainty.
132.45 cm
0.823 cm
For example, round 9.802562 ± 0.007916 m/s2 to
+ 5.6
g ± Δg = (9.803 ± 0.008) m/s2
cm
138.873 cm --> 138.9 cm
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Mean (Average) value
Significant figures in calculations
Let x1, x2,… xN represent a set of N measurements of
a physical quantity x.
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°
35 N·m
0.00097 mm/C°
The average or mean value of this set of
measurements is given by
1
x =
N
N
å
i= 1
xi =
1
(x 1 + x 2 + ... + x N )
N
The mean is almost entirely free from random errors
and gives the best estimate for the value of the
quantity measured for a large number of readings.
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Standard deviation
Standard deviation as uncertainty
Standard deviation quantifies the spread of the data
about the mean.
Suppose the mean and standard deviation of a set of
N measurements of a physical quantity x have been
determined.
sx =
N
1
( x i - x )2
å
N - 1 i= 1
If one more extra measurement is to be made (under
the same conditions), then the reading xN+1 would
have a probability of 68.3% lying within
Statistical Interpretation:
x - s x < xN +1 < x + s x
9 68.3% within 1σ
9 95.5% within 2σ
9 99.73% within 3σ
The standard deviation is then treated as the
uncertainty for the measurement of a single reading.
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Standard deviation of the mean
x1
x1
x1
x1
x2
x2
x2
x2
x3
x3
x3
….
….
….
….
xN
xN
xN
xN
x
x
x
x
..…
x3
The overall mean for all means
can then be determined.
Expressing uncertainty
(best estimate ± uncertainty) units
Standard deviation of the
mean measures the spread
of all means about the
overall mean.
sx =
(x best ± dx )unit s
sx
N
Best estimate: x best = x
Uncertainty:
It is also called the standard
error and is the error usually
quoted for a measurement
in the literature.
dx = s x
Statistically, the true value would have a probability
of 68.3% lying within
x - s x < x t rue < x + s x
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Standard error as uncertainty
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Example 1
N
Mean
S.D.
S.E.
Result
10
19.6
2.71
0.857
19.6 ± 0.9
100
19.89
2.26
0.226
19.9 ± 0.2
1000
19.884
2.48
0.0784
19.88 ± 0.08
10000
19.9879
2.52
0.0252
19.99 ± 0.03
A student measures the resistance of a coil eight
times and obtains the results as follow:
R ± 0.001 (Ω)
4.615 4.638 4.597 4.634 4.613 4.623 4.659 4.623
State the final result of the resistance with the
appropriate number of significant figures.
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Example 1: Solution
Error propagation
Mean:
Suppose the value of a quantity R(x,y,z,…) is
determined from the measured values of a number of
independent quantities x, y, z, … which are directly
measured.
1 8
R = å R i = 4.625 W
8 i= 1
Standard deviation:
sR =
Question: How to determine the error in R from the
errors associated with the measurements of x, y, z,
… respectively?
2
1 8
R i - R ) = 0.01868 W
(
å
8 - 1 i= 1
Standard error:
sR =
sR
= 0.006603 W
8
2
⎛ ∂R ⎞ 2 ⎛ ∂R ⎞ 2 ⎛ ∂R ⎞ 2
⎟ σ y +⎜
⎜
⎟ σx +⎜
⎟ σ z + ...
⎝ ∂x ⎠
⎝ ∂z ⎠
⎝ ∂y ⎠
2
σR =
Final result:
R ± s R = (4.625 ± 0.007)W
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Combining uncertainties
Combining uncertainties
Let A ± ΔA and B ± ΔB represent two measured quantities.
Let A ± ΔA and B ± ΔB represent two measured quantities.
The uncertainty in the product P = A × B is
The uncertainty in the sum S = A + B is
DS =
2
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2
2
(D A ) + (D B )
The uncertainty in the quotient Q = A / B is ALSO
The uncertainty in the difference D = A - B is ALSO
2
DD =
2
2
æD A ö
æD B ö÷
÷
D P = P çç
÷ + çç
÷
è A ÷
ø è B ÷
ø
2
æD A ö
æD B ö÷
÷
D Q = Q çç
+ çç
÷
÷
÷
è A ø è B ÷
ø
2
(D A ) + (D B )
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Example 2
Example 2: Solution
Suppose the area A of a rectangular plate is to be
determined. Several independent measurements of the
length L and width W of the plate were obtained:
Mean:
L=
1 8
å Li = 24.246 cm
8 i= 1
L ± 0.01 (cm)
Standard deviation:
24.26 24.23 24.22 24.25 24.28 24.26 24.24 24.23
sL =
W ± 0.01 (cm)
2
1 8
Li - L ) = 0.01996 cm
(
å
8 - 1 i= 1
Standard error:
50.36 50.35 50.38 50.41 50.36 50.39 50.37 50.32
sL =
Estimate the area of the plate and its uncertainty.
sL
= 0.007055 cm
8
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Example 2: Solution
Example 2: Solution
Mean:
Best estimated for the length and breath:
8
W =
1
å W i = 50.368 cm
8 i= 1
L ± s L = (24.246 ± 0.007)cm
W ± s W = (50.37 ± 0.01)cm
Standard deviation:
8
sW =
1
å (W i - W
8 - 1 i= 1
Best estimated for the area:
2
)
A = L ´ W = 1221.222 cm 2
= 0.02712 cm
Standard error:
2
s
æs ö
÷ + çæ
ç W
s A = A çç L ÷
è L ø èW
Standard error:
sW =
sW
= 0.009590 cm
8
2
ö÷
= 0.425 cm 2
÷
ø
Final result:
A ± s A = (1221.2 ± 0.4)cm 2
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Which is the BEST line?
Least squares fit
The best straight line to fit a set of measured data
points (x1, y1), (x2, y2), …, (xn, yn) is
y best = m best x + c best
Assumption:
• The uncertainty in our measurements of x is
negligible but not in y
• The uncertainty in our measurements of y is the
same
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Least squares fit: Formulas
D = n å x i2 -
(å
2
xi )
mbest
σm
best
= n
Example 3
A student measures the pressure P of a gas at five
different temperatures T by keeping the volume constant.
His data are should in the following table:
Slope:
1
= ( n∑ xi yi − ∑ xi ∑ yi )
Δ
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σ2
Δ
P ± 5 (mm of mercury)
y-intercept:
cbest =
1
xi2 ∑ yi − ∑ xi ∑ xi yi )
(
∑
Δ
σc
best
=
σ2
Δ
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∑x
2
i
85
95
105
T = A + BP
T ± 1 (°C)
Standard deviation for y:
σy =
75
50.36 50.35 50.38 50.41 50.36
1
2
( yi − cbest − mbest xi )
∑
n−2
Estimate the value of absolute zero and its error with
appropriate number of significant figures.
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Example 3: Solution
Example 4
slope, mbest = 3.71 °C/(mm of mercury)
y-intercept, cbest = −263.35 °C
σy = 6.6808 °C
σm = 0.2113 °C/(mm of mercury)
σc = 18.2045 °C
A student wants to measure the acceleration of gravity g
by measuring the period T and the length ℓ of a simple
pendulum.
ℓ ± 0.1 (cm)
57.3
The best fit of a straight line,
T = − 263.35 + 3.71 P
61.1
73.2
83.7
95.0
T ± 0.001 (s)
T = 2p
l
g
1.521 1.567 1.718 1.835 1.952
Final result:
Estimate the acceleration of gravity g and its uncertainty
with appropriate number of significant figures.
A ± s A = (- 263 ± 8)°C
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Example 4: Solution
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Example 4: Solution
Acceleration of gravity:
slope, mbest = 0.03987 s2/cm
y-intercept, cbest = 0.0268 s2
σm = 0.0002067 s2/cm
σc = 0.01558 s2
g=
4p 2
= 990.1748 cm/ s2
m
Standard deviation of g:
sg = g
The best fit of a straight line,
T2 = 0.03987 ℓ + 0.0268
sm
= 5.13274 cm/ s2
m
Standard error of g:
sg =
sg
n
= 2.2954 cm/ s2
Final result:
g ± s g = (990 ± 2)cm/ s2
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