Download ME 120 Experimental Methods

ME 120 Experimental Methods
Uncertainty Analysis
BJ Furman
12SEP05
BJ Furman
SJSU MAE
Goal of Uncertainty Analysis
‹ Come
value
up with a probable bound on a measured
xtrue = xmeas ± ux
at C% confidence or n:1 odds
‹ The
uncertainty, ux, will be calculated from the
precision and bias errors, px and bx, associated
with the measurement and measurement
system.
™
The total uncertainty, ux is:
ux=(b2x +p2x)1/2
BJ Furman
SJSU MAE
1
Classes of Experiments
‹ Two
™
™
classes of experiments:
Single-sample (i.e., a single measurement of the
measurand)
Repeat-sample (several measurements of the
measurand under identical conditions)
(note: “Sample” is used by some to refer to the
set of data taken during repeated
measurements of a measurand under fixed
operating conditions. So a single-sample
experiment would be a sample with only one
measurement.)
BJ Furman
SJSU MAE
Statistical Measurement Theory
‹
Precision errors are distributed, that is they follow a
distribution that characterizes the probability that an
error of a particular size will occur.
™
Ex. Gaussian or normal distribution
• Most physical properties that are continuous
or regular in time and space.
™
Other distributions are possible:
• Log-normal (failure or durability projections)
• Weibull (fatigue tests)
• Poisson (events randomly occuring in time)
• Binomial (reliability testing)
‹
The measurement sample (a limited set) is drawn
from a population (the group of all measurements)
BJ Furman
SJSU MAE
2
The Normal Distribution
error data is often normally
distributed
‹ Precision
™
The probability density function (PDF) for a normal
distribution is:
⎡ ( x − μ )2 ⎤
1
f ( x) =
exp ⎢ −
2σ 2 ⎥⎦
σ 2π
⎣
x = value of measurement
μ = mean of population
σ= standard deviation of population (σ2 is called the variance)
(note that the mean and standard deviation of the population are usually
unknown, so we will use estimates of them from a sample: ( x , S x )
BJ Furman
SJSU MAE
The Normal Distribution, cont.
‹ Standard
z=
Normal distribution:
x−μ
σ
⇒
f ( z) =
2
1
e− z / 2
σ 2π
Source: Beckwith, T. G., Marangoni, R. D., Lienhard, J. H., Mechanical Measurements, Addison-Wesley, Reading, MA, 1995
BJ Furman
SJSU MAE
3
Uncertainty Bounds from Population Statistics
‹
If the population statistics (μ and σ) are known, and
assuming only precision errors, then with c%
confidence, a single measurement is bounded by:
μ – zc/2 σ < x < μ + zc/2 σ
or alternatively,
we expect the measurand to lie in the range of:
μ ± zc/2 σ with c% confidence
where zc/2 is half the area under the standard normal
distribution corresponding to c% probability
BJ Furman
SJSU MAE
Uncertainty Bounds from Population Statistics, cont.
‹
Ex. μ=200 °C σ=3 °C for a particular chemical process
from long term measurements. What range do we
expect to see the temperature be in 95% of the time?
™ Population or sample statistics?
™ What type of distribution?
•
•
•
•
Plot the data on a histogram, and look at the shape
Plot the data on normal probability paper
Use a ‘goodness-of-fit’ test, like χ2
Use a statistical computer program like Dataplot
(http://www.itl.nist.gov/div898/software/dataplot/homepage.htm)
™
Assuming normally distributed data:
• Lookup zc/2 , corresponding to 95% probability from the standard normal curve
table
• Range will be:
{
200 ±1.96*(3 °C) = 200 ±5.88
BJ Furman
(should we round??)
SJSU MAE
4
Sample Statistical Parameters
‹ Population
statistical parameters are usually
unknown, so we need to rely on estimates
n
∑x
i
™
Sample mean (for n measurements): x =
™
Sample standard deviation (n measurements):
n
Sx =
∑(x − x)
i =1
i
n −1
n
2
=
i =1
n
( ∑ xi ) − nx 2
2
i =1
BJ Furman
n −1
SJSU MAE
Sample Statistical Parameters, cont.
‹ Suppose
™
™
m replicates of n samples
What do you expect to see with x1 , x2 , x3 K , xm ?
The sample means have a distribution!
• The distribution of sample means tends toward normal as
n increases! (the Central Limit Theorem)
• There will be a standard deviation associated with the
distribution of sample means (also called the estimated standard
error):
Sx =
Sx
n
What happens when the sample size increases?
BJ Furman
SJSU MAE
5
Uncertainty Bounds from Sample Statistics
‹ We
expect the true mean value of the
measurand to lie in the range of:
xtrue = x ± tα / 2,ν S x
(at C% confidence)
where
xtrue is the true mean
x is the sample mean
tα / 2,ν is Student's t distribution with ν degrees of freedom
and α is the level of significance (and C=1-α )
ν= n-1
(note as n→∞, tα/2,ν approaches Zα/2)
Who was Student?
BJ Furman
SJSU MAE
Example
Sample No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Value
0.98
1.07
0.86
1.16
0.96
0.68
1.34
1.04
1.21
0.86
1.02
1.26
1.08
1.02
0.94
1.11
0.99
0.78
1.06
0.96
C%=
95%
0.025
α/2 =
n=
20 Number of samples
19 Degrees of freedom
ν=
xbar=
1.02 Sample average
Sx= 0.157677 Sample standard deviation
Sxbar= 0.035258 Estimated standard error
t-stat=
2.093
t-stat*Sxbar=
0.07
true mean in:
1.02 +-0.07
95%
Confidence
From Figliola and Beasly, p. 125
BJ Furman
SJSU MAE
6
Comparison of Sample Means
‹ The
t distribution is also useful in determining
whether or not the means of two populations
are significantly different
x1 − x2
™
Calculate the the t-statistic: t =
™
See if it falls in the range of ± tα/2,ν ,
( S12 / n1 ) + ( S 22 / n2 )
2
⎡⎣( S12 / n1 ) + ( S 22 / n2 ) ⎤⎦
Where ν =
( S12 / n1 ) 2 ( S 22 / n2 ) 2
−
n1 − 1
n2 − 1
(rounded down)
If it does, than the population means are not significantly different at the stated level
of confidence.
BJ Furman
SJSU MAE
Confidence Interval on Standard Deviation
‹
The distribution describing the variation in
standard deviations of samples is Chi-squared,
so a bound on s can be calculated as:
( N − 1) s 2
χ (2α / 2, N −1)
≤σ ≤
( N − 1) s 2
χ (12 −α / 2, N −1)
C%=(1-α)
From: http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm
BJ Furman
SJSU MAE
7
Design-Stage Uncertainty Analysis
‹ What
data?
™
uncertainty arises before we start taking
Uncertainty from ability of measurement instrument
to resolve
• Ures= ±1/2 resolution
™
(at C% Confidence)
Uncertainty from instrument errors
• Uins=
±
k
∑e
i =1
2
i
(C% confidence)
BJ Furman
SJSU MAE
Instrument Error Calculation
‹
Ex: LCKD “button” load cell from Omega
(http://www.omega.com/Pressure/pdf/LCKD.pdf)
SPECIFICATIONS:
Excitation: 5 Vdc, 7 Vdc max
Output: 2 mV/V nominal
5-Point Calibration:
0%, 50%, 100%, 50%, 0%
Linearity: ±0.25% FSO
Hysteresis: ±0.25% FSO
Repeatability: ±0.10% FSO
U ins = ± (lin.) 2 + (hys.) 2 + (repeat.) 2 + ( z.b.) 2 + (th. span) 2 + (th. zer.) 2
Zero Balance: ±2% FSO
Operating Temp Range (stated accuracy applies):
-NA–54 to 107°C (–65 to 225°F)
Compensated Temp Range:
U ins = ± (0.25%) 2 + (0.25%) 2 + (0.10%) 2 + ((16 deg F )(0.01%)) 2 + ((16 deg F )(0.005%)) 2
16 to 71°C (60 to 160°F)
Thermal Effects (applies outside compensated range):
Span: ±0.01% of FSO/°F
Zero: ±0.005% of FSO/°F
ins
Safe Overload: 150% of Capacity
Ultimate Overload: 300% of Capacity
Weight: < 0.5 oz (< 14 g)
Suppose 0-100 lb FS range used at 80 °C,
and that zero balance can be adjusted out:
U
= ± (0.0625) + (0.0625) + (0.01) + (0.0256) + (0.0064)
U ins = ± 0.167 = ±0.41% FSO
Note: this is the instrument (device) uncertainty, which is one part of the overall
uncertainty for any particular measurement using this device. What else will contribute
to the total uncertainty, and how will you calculate the total uncertainty?
BJ Furman
SJSU MAE
8
Error Propagation
‹
“When experimental data are used to compute a final
result, the uncertainty of the data must be propagated
to determine the uncertainty in the result.” (BM&L, p.
116)
2
U f ( x1 , x2K, xn )
Ex:
I=
2
⎛ ∂f
⎞
⎛ ∂f ⎞ ⎛ ∂f
⎞
= ⎜
u1 ⎟ + ⎜
u2 ⎟ + K ⎜
un ⎟
⎝ ∂x1 ⎠ ⎝ ∂x2 ⎠
⎝ ∂xn ⎠
1
wh 3
12
2
where w and h are measured values
BJ Furman
SJSU MAE
References
‹
Beckwith, T. G., Marangoni, R. D., Lienhard, J. H., Mechanical
Measurements, Addison-Wesley, Reading, MA, 1995.
‹
Figliola, R. S., Beasley, D. E., Theory and Design for Mechanical
Measurements, 3rd ed., J. Wiley & Sons, New York, 2000.
NIST/SEMATECH e-Handbook of Statistical Methods,
http://www.itl.nist.gov/div898/handbook/, September, 2002.
Lane, D. M., HyperStat Online,
http://davidmlane.com/hyperstat/index.html, August 2002.
McClelland, G. H., Seeing Statistics java applets,
http://psych.colorado.edu/~mcclella/java/zcalc.html, 1999.
Weisstein, E., Eric Weisstein’s World of Mathematics,
http://mathworld.wolfram.com/topics/ProbabilityandStatistics.html, 2002.
School_of_Mathematics_and_Statistics
University_of_St_Andrews,_Scotland, web page on William Sealey Gosset,
‹
‹
‹
‹
‹
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gosset.html , 1997.
BJ Furman
SJSU MAE
9
References, cont.
‹
‹
Dallal, G. E., “What Student Did,”
http://www.tufts.edu/~gdallal/student2.htm, October 2000.
Omega Engineering, Inc., http://www.omega.com/, September 2002.
BJ Furman
SJSU MAE
Putting it all together
BJ Furman
SJSU MAE
10