MER331 – Lab 1
An Introduction to
Uncertainty Analysis
Error Sources
The measurement process consists of three distinct steps: calibration, data acquisition, and data
reduction
–calibration error
–data acquisition error
–data reduction error
Uncertainty Analysis
Error - difference between true value and
measured value.
Two general categories of error (exclude
gross blunders)
Fixed error – remove by calibration
Random error – quantify by uncertainty
analysis
The term uncertainty is used to refer to “a
possible value that an error may have”
Stages of Uncertainty Analysis
1. Design Stage Uncertainty Analysis
•
Uncertainty analysis can be used to assist in the
selection of equipment and procedures based on
their relative performance and cost.
2. Advanced-Stage and Single Measurement
Uncertainty Analysis
•
Consider procedural and test control errors that
affect the measurement
Uncertainty Analysis
Example: Each group has a piece of paper and 2 measurement tools (labeled 1, 2). You must use one measurement
tool to measure the height (H) and the other to measure width
(W). Does it matter which tool you use to measure H?
A = H*W (H is the longer side)
How do we estimate the uncertainty in our measurement of A?
Step 1: Figure out the uncertainty in H and W.
Step 2: Figure out how the uncertainty in H and W affects the
uncertainty in A
Step 1 - Estimating Uncertainty Interval
If you have a statistically
significant sample use: ±2s
for a 95% confidence interval.
The general practice in
engineering is 95%
confidence of 20 to 1
odds.
Step 1 - Estimating Uncertainty Interval
What do you do if you DON’T
have a statistically significant
sample?
Use ± ½ smallest scale
division as an estimate.
( this is called a 0th order
analysis)
Step 1 - Estimating Uncertainty Interval
What do we use for Rulers 1 and 2?
Some nomenclature:
H = height, W = Width
dH = uncertainty in H in units of H
uH = percent uncertainty = dH/H
dW = uncertainty in W in units of W
uW = percent uncertainty = dW/W
Step 2: How does the uncertainty in H and W affect
uncertainty in A?
The calculated result, R is assumed to be calculated from a
set of measurements: R = f(X1, X2, X3,…, XN)
In our case we would write this as: R = A, X1 = H, X2 = W
or A = f(H,W) where f = H*W
The effect of the uncertainty in a single measurement (i.e.
one of the X’s) on the calculated result, R, if only that one
X were in error is:
R
dRxi 
dX i
X i
Partial Derivative of R wrt Xi - A Sensitivity Coefficient
Step 2: How does the uncertainty in H and W affect
uncertainty in A?
In our case:
A
dAH 
dH and
H
A
dAW 
dW
W
Now calculate the partial derivatives:
A  H *W
A
 W and
H
A
H
W
Step 2: How does the uncertainty in H and W affect
uncertainty in A?
When several independent variables (X’s) are used
in calculating the Result, R, the individual terms
are combined by a root-sum-square method
(Method due to Kline and McClintock (1953))
 N  R

dR   
dX i 
i 1  X i

2
1/ 2



Step 2: How does the uncertainty in H and W affect
uncertainty in A?
In our case we write this as:
1/ 2
R
 A
2
2
dA  ( dH ) (
dW ) 
W
 H

Now Calculate your uncertainties in A.
Uncertainty as a Percentage
In most situations the overall uncertainty in a given result is
dominated by only a few of its terms. Ignore terms that are
smaller than the largest term by a factor of 3 or more.
It is difficult (impossible) to compare errors with different units
associated with them (e.g. how big is a 2 gram error
compared to a 2 second error?)
To solve this we “nondimensionalize” the errors:
u xi 
dX i
Xi
uR, xi
dRx
X i R


u xi
R
R X i
i
So, as a percentage…
1/ 2
 N  X R


uR 
   i
u xi  
R  i 1  R X i
 

dR
2
1/ 2
 H A
2 W R
2
u A  (
u H ) (
uW ) 
A W
 A H

Uncertainty Analysis
There are three sources of uncertainty in
experimental measurements:
1.
Interpolation uncertainty (zeroth order)
2.
Unsteadiness (1st order)
3.
Instrument Calibration (Nth order)
Zero-Order Uncertainty
At zero-order uncertainty, all variables and
parameters that affect the outcome of the
measurement, including time, are assumed to be
fixed except for the physical act of observation
itself.
Any data scatter is the results of instrument
resolution alone uo.
Higher-Order Uncertainty
Higher order uncertainty estimates consider the
controllability of the test operating conditions.
For a first order estimate we might make a series of
measurements over time and calculate the variation
in that measurement. The first order uncertainty of
that measurand is then:
u1 = ± 2s* at (95%)
* Note: Assuming we make enough measurements
Nth-Order Uncertainty
As the final estimate, instrument calibration characteristics
are entered into the scheme through the instrument
uncertainty, uc. A practical estimate of the Nth order
uncertainty uN is:
N 1 2
2
u N  (uc )   ui
i 1
(95%)
Uncertainty analyses at the Nth order allow for the direct
comparison between results of similar tests obtained using
different instruments or at different test facilities.
Report Results at uN level
Nth order Uncertainty
N 1 2
2
u N  (uc )   ui
i 1
First-order Uncertainty
u1 > u 0
Zero- order Uncertainty
u0 = ±1/2 resolution
Your Homework for Lab next week
Next week in lab we will be measuring viscosity
and density. You are to develop a tool* to
calculate the uncertainty in your measurement
of both.
* tool can be in Matlab or Excel
Density Measurement
You will measure density by measuring the volume
and mass of the fluid.
o The volume, V, will be measured in a graduated
cylinder.
o The mass, m, will be measured on a balance as
the difference between the full mf and empty mass
me of the graduated cylinder.
Density will be calculated as:
rfluid = (mf-me)/V
Density Measurement
Calculate the uncertainty in r:
o
Estimate uncertainty interval for each measured
quantity, (mf, me and V)
o Analyze the propagation of uncertainty into results
calculated from experimental data.
o Check your results using me = 100 ± 1g, mf = 150
± 1g, and V = 94 ± 0.5 mL (I will ask for your
answer in lab next week).
Viscosity Measurement
You will estimate the viscosity of a fluid using a falling
ball viscometer. The viscosity, m, (in units of cp) is
calculated as the product of the calibration constant,
K, the time, t, and the difference between the falling
ball and fluid density:
m = Kt(rball-rfluid)
o
o
You will measure time using a stopwatch.
The values of K and rball will be given to you.
Viscosity measurement
.
Calculate the uncertainty in m:
o Estimate uncertainty interval for each
measured quantity (t, , rfluid).
o Analyze the propagation of uncertainty into
results calculated from experimental data.
o Test your calculations for K = 5 ± 1 cp-mL/gs, t = 30 ± .01s, rball= 5.2 ± 2 g/mL, and rfluid=
1 ± 1g/mL