NOTES ON UNCERTAINTY ANALYSIS
FOR MEL LABS
by
Matt Young
For more detail, see also
http://www.mines.edu/Academic/courses/physics/phgn471/uncertainty.pdf
Copyright © 2002 by Matt Young. All rights reserved.
Matt Young’s Home Page
1. Definitions
Error = deviation from True Value
Uncertainty = estimate of probable error
2. Types of uncertainty
•
Measured or statistical uncertainty
•
•
for fluctuating or random variables
All other uncertainties
•
for variables whose statistics are not known
•
calculated from estimates
•
include but not limited to systematic errors
wpdocs/mel 2\uncert3.wpd. page 1
3. Uncertainty due to measured or statistical errors
•
Data are noisy, as due to voltage fluctuations
•
Temporarily ignore other sources of uncertainty
•
Measurand = quantity to be measured
•
Individual measured values are mi
•
Mean = µ, calculated from mean of data set
•
N = number of data points
•
True value = M = ???
•
µ is called an estimator of M
Calculate the standard deviation of the mean, or SDOM:
N
σm =
•
∑
i =1
(mi − µ ) 2
N ⋅ ( N − 1)
)m is the standard uncertainty due to random
errors alone
•
)m decreases in proportion to (N)
wpdocs/mel 2\uncert3.wpd. page 2
4. Significance of standard uncertainty )
•
M falls between µ – ) and µ + ) with 68 % probability
•
•
68 % confidence interval
M falls between µ – 2) and µ + 2) with 95 %
probability
•
•
95 % confidence interval
M falls between µ – 3) and µ + 3) with 99.7 %
probability
•
99.7 % confidence interval
Half the battle is learning the vocabulary!
wpdocs/mel 2\uncert3.wpd. page 3
Cheap and dirty way to estimate )
•
Calculate the maximum value of the dataset
minus the minimum value
•
Divide by 6·(N)
•
Result is a fair estimate of )
•
Use to check your calculation
¡Important note!
) is the 68 % confidence interval; the ’s
below are your best estimates of 99.7 % confidence
intervals. They are not directly comparable.
wpdocs/mel 2\uncert3.wpd. page 4
5. Estimated errors (all other errors)
•
Usually errors whose statistics are not known
P artial
 U n certain ty  


 =
 + ( G u essin g )
 A n alysis   D ifferen tiatio n 
Suppose m = m(a,b,c) & we calculate m by measuring
a, b, & c
Let a = extreme value of error of a
•
½ scale division, for example, or
•
mechanical tolerance in a part, for example
a is
•
an estimate or a guess
•
of the 99.7 % confidence interval
Estimate error m of m due to error a of a:
∂m
∆m a =
⋅ ∆a
∂a
•
•
Assumes a << a
a = half-width of 99.7 % confidence interval & is
•
always positive
•
not the standard uncertainty or standard
deviation
wpdocs/mel 2\uncert3.wpd. page 5
6. Relative error
Often, m = K·a (where K may be a function of b, c, ...)
•
Given a, calculate confidence interval of m:
∂
∆m a =
( K a ) ⋅ ∆a = K ⋅ ∆a
∂a
But K =m / a
∆m ∆a
=
so
m
a
Best estimator of m is its mean µ, so we write
∆m a ∆a
=
a
µ
Similarly, if
•
a is a random variable, and
•
we measure µa and )a
•
then
σm σa
=
µ
µa
where m means the measurand
wpdocs/mel 2\uncert3.wpd. page 6
7. Painless uncertainty analysis
•
When m is not a strong function (such as an
exponential) of a, b, c
•
•
Estimate a, b, c, ...
•
Calculate a/a, b/b, c/c, ...
•
Include only the largest in your analysis
We will use the ’s below to calculate standard
uncertainties
•
Note: If m = Kan, then, similarly
∆m a
∆a
=n⋅
a
µ
The relative error ma /µ of m due to a is proportional to
the relative error of a itself
wpdocs/mel 2\uncert3.wpd. page 7
8. Example 1 from MEL 2
Yield stress:
Y = p ⋅S / A
Y = yield stress
p = measured pressure at yield
S = area of piston
A = area of wooden block
•
Calculate component of uncertainty due to area of
block
•
A = a2, where a = 1.5 in
•
Guess:
1
∆a =
in ≅ 0 .0 3 in
32
∆A a
∆a
0 .0 3 in
=2
=2 ⋅
= 0 .0 4
1.5 in
A
a
•
Similarly,
S = π D 2 / 4 , D = 8 .7 5 in
G u ess: ∆D = 0 .0 1 5 in
∆S D / S = 2 ∆D / D = 0 .0 0 3
•
Ignore D in calculating Y
wpdocs/mel 2\uncert3.wpd. page 8
How to calculate )p from the calibration curve
105
Pressure, ksi
p
+ sigma
meas
100
p
meas
95
90
V meas
85
2.0
2.2
2.4
2.6
2.8
3.0
Voltage, V
Figure 1. Calibration curve.
•
Use the Excel spreadsheet to
•
Calculate line of best fit (solid line)
•
Find voltage Vmeas, as at yield stress, calculate
corresponding pressure pmeas
•
Use calculation of 68 % confidence interval
(dashed curves) to estimate the standard
deviation )
wpdocs/mel 2\uncert3.wpd. page 9
Now back to Example 1
Y = p ⋅ S / A = 2 6 0 0 k si , from above
By partial differentiation,
∆Y S = ∆S ⋅ p / A
∆Y A = ∆A ⋅ p S / A 2
•
S = 60 in2, S = 0.003·S, from above;
YS = 8 ksi; small as predicted
•
•
A = 0.04·A, from above
YA = 108 ksi
We do not have to estimate p, since we can
measure )p = 2 ksi from the calibration curve.
Thus,
u p =σp ⋅S / A ,
where up is the component of uncertainty of Y due to
the uncertainty )p of p, and
up = 50 ksi
wpdocs/mel 2\uncert3.wpd. page 10
9. Uncertainties at last!
(a) Type A or measured uncertainties
•
Standard uncertainty ur = the SDOM, that is,
•
ur = )
•
r stands for “random”
(b) Type B uncertainties, or all other uncertainties
•
Estimate 99.7 % confidence interval ma, mb,
mc, ..., as above
•
Assume uniform distribution of errors (not
Gaussian)
•
Standard deviation of uniform distribution
with half-width ma is ma/(3)
•
Define standard uncertainties as
ua = ma/(3), ub = mb/(3), ...
wpdocs/mel 2\uncert3.wpd. page 11
Back to Example 1 yet again
•
Type A (measured) uncertainty
up = 50 ksi
•
Type B (all other) uncertainties
YS = 8 ksi and
YA = 108 ksi, so
uS = (8 ksi)/(3) and
uA = (108 ksi)/(3), so
uS = 5 ksi and
uA = 62 ksi
wpdocs/mel 2\uncert3.wpd. page 12
10. Combined standard uncertainty
•
Uncertainties are added in quadrature (sum of
squares)
u c = u 12 + u 22 + u 32 +
•
uc is the combined standard uncertainty
•
Note that there may be more than 1 source of
random uncertainty
11. Expanded uncertainty
•
uc is multiplied by a coverage factor, usually 2
•
Express experimental results as
µ ± 2 uc
•
2 uc is the expanded uncertainty
•
The interval 2 uc defines the 95 % confidence
interval
•
The True Value is presumably within the
interval µ ± 2 uc, with 95 % probability
wpdocs/mel 2\uncert3.wpd. page 13
Example 1, one last time!
Table 1. Compilation of standard uncertainties
Source of
uncertainty
Type of
uncertainty
Standard
uncertainty, ksi
Measurement of
pressure
Measured (Type
A)
Area S of piston
Other (Type B)
5
Area A of block
Other (Type B)
62
Combined
standard
uncertainty
80
Expanded
uncertainty
160
50
Thus,
•
Y = 2600 ± 160 ksi
Note the use of round numbers for uncertainties and the
correct number of significant digits in the expression
for Y
wpdocs/mel 2\uncert3.wpd. page 14
12. Example 2. Uncertainty of Young’s modulus
•
Slope of curve of ) vs. Standard deviation of the slope of a line
y = A + Bx
•
Fitted equation is
•
Standard uncertainty )B of the slope B is
σ B2 = N σ y2 / ∆ , w h ere
1
2
σy =
N −2
N
∑( y
i =1
N
∆= N
∑x
i =1
i
− A − B xi ) 2 ,
N
2
i
−(
∑x )
i =1
2
i
John Taylor, Introduction to Error Analysis, University
Science Books, Mill Valley, California, latest edition.
•
Use slope uncertainty when interested in slope as
such (as when measuring Young’s modulus)
•
Use confidence interval of line when interested in
uncertainty of quantity on y-axis (such as
displacement or pressure)
wpdocs/mel 2\uncert3.wpd. page 15
Estimated (Type B) uncertainties
•
Formulas estimate uncertainty due to random errors
only, not calibration uncertainties of axes
E =σ / ε
∆E σ / E = ∆σ / σ (Here ) means stress not SDOM)
∆E ε / E = ∆ε / ε
•
Assume that electronics introduce negligible error
•
Write down a number for L, where L is any
length measurement. (Hint: what is the least
count of the dial indicator?)
•
•
Calculate for a representative value of Similarly, write down F, where F is a force
measurement, and calculate )
•
Calculate the appropriate E’s, divide by (3),
and combine in quadrature
Details are left as a proverbial exercise for the student
wpdocs/mel 2\uncert3.wpd. page 16
13. Example 3 from MEL 2
Use displacement sensor to measure E of steel specimen
•
Measured displacement = elongation of specimen +
elongation of shafts holding specimen, or
•
d m = d sp + d sh
•
Measure dm [Here d is displacement; L is original length]
•
Calculate
d sh = L sh × εsh , w h ere
εsh = F / ( A sh × E sh )
Assume Esh = 3 ×107 ± 3 ×106 psi (that is, ±10 %)
•
F = 6000 lb, Ash = 2 in2, Lsh = 10 in
•
Then dsh = 1 mil
•
Subtract systematic error, or bias, dsh from
[1 mil = 10-3 in]
measured value dm
wpdocs/mel 2\uncert3.wpd. page 17
Example 3, continued
•
Calculate confidence interval of correction, assuming
that Esp = 107 psi, F = 6000 lb, Lsp = 10 in
•
•
99.7 % confidence interval dm of bias:
•
Suppose εsp = 3 × 1 0 −3 (measured)
•
•
∆d sh = 0 .1 × d sh (Why?)
∆εsp = ∆d sh / L sp ... (Why?) ... = 10-5
•
∆E sp / E sp = ∆εsp / εsp ; ∆E sp =
1
×1 0 5
3
Standard uncertainty of E due to :
uε =
1
× 1 0 5 / 3 ≈ 1 9 0 00 p si
3
•
Around 0.2 %, for the made-up numbers I
have chosen
wpdocs/mel 2\uncert3.wpd. page 18