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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