This analysis demonstrates the analysis of uncertainty for values predicted from a calibration curve where there is uncertainty in both the predictors and the predicted values. The example is taken from a study conducted by Michele Donnelly of the Fire Science Division of the National Institute of Standards and Technology. It relates to the volume fraction of nitrogen required to extinguish flame on the surface of a plastic cylinder. Measurements of flux (X) are made on the surface of the object, and corresponding rotameter readings (Y) are converted to flowrate (Y') from a calibration curve for the rotameter. Flowrate is then converted to nitrogen volume fraction. The steps in the analysis are as follows. 1. Create a calibration curve for the rotatmeter. 2. Compute predicted values of nitrogen flowrate from rotameter readings on twenty-one cylinders in the study. 3. Compute the standard deviations of the predicted values of flowrate. 4. Convert the flowrates to nitrogen volume fractions (%). 5. Calculate the uncertainties due to the calibration curve for the nitrogen volume fractions. 6. Calulate the uncertainties due to flux measurements for the nitrogen volume fractions. 7. Combine the last two components into a total uncertainty. 1. A calibration curve is created for the rotameter. This calibration curve is used to convert future rotameter readings to flowrate. Data for estimating the coefficients of the calibration curve consist of ten repetitions at eight ball settings. X Y 40 7.871 40 7.877 40 7.863 40 7.877 40 7.871 40 7.877 40 7.863 40 7.874 40 7.871 40 7.865 50 9.608 50 9.591 50 9.600 50 9.582 50 9.582 50 9.600 50 9.587 50 9.587 50 9.608 50 9.600 70 12.94 70 12.93 70 12.94 70 12.93 70 12.94 70 12.94 70 12.91 70 12.92 70 12.95 70 12.94 90 15.94 90 15.84 90 15.91 90 16.01 90 15.98 90 15.89 90 15.82 90 15.89 90 15.88 90 15.90 30 5.981 30 5.980 30 5.983 30 5.985 30 5.985 30 5.981 30 5.981 30 5.978 30 5.981 30 5.977 80 14.40 80 14.39 80 14.40 80 14.41 80 14.42 80 14.39 80 14.43 80 14.37 80 14.44 80 14.37 100 17.23 100 17.17 100 17.23 100 17.17 100 17.15 100 17.19 100 17.21 100 17.24 100 17.23 100 17.22 20 4.012 20 4.014 20 4.015 20 4.011 20 4.013 20 4.011 20 4.020 20 4.014 20 4.009 20 4.011 The Omnitab output from a quadratic fit to the data where two of the measurements have been excluded as outliers follows. ESTIMATES FROM LEAST SQUARES FIT TERM COEFFICIENT S.D. OF COEFF. RATIO ACCURACY* ------------------------------------------------------------------- 0 -.14444077 .017538628 -8.24 5.32 1 .21711582 .00067569158 321.32 6.69 2 -.00043623472 .0000055532573 -78.55 6.10 ------------------------------------------------------------------- RESIDUAL STANDARD DEVIATION = .028181005 BASED ON DEGREES OF FREEDOM 78 - 3 = 75 Plots for assessing the adequacy of the quadratic model are shown below. The F statistic and the standardized residuals and normal probability plot indicate that there is some variation that is not accounted for by the quadratic model. These may be caused by inaccuracies in the ball readings which we cannot identify. In general, the fit is satisfactory. 2. Calibrated values of nitrogen flowrate (LPM) are computed as a function of flux from the calibration curve: The standard deviations for the predicted values of flow from the quadratic calibration curve are combined with the residual standard deviation of the fit, , to estimate the uncertainty for flowrate. 3. The flowrate is converted to nitrogen volume fraction (%), Z, where and C = 10 LPM for air is considered to be a constant. 4. The uncertainty of the nitrogen value fraction due to the uncertainty of the calibration curve is calculated by propagation of error to be: 5. The uncertainty of the nitgroen volume fraction due to flux is calculated. The measured flux is considered to have an expanded uncertainty of at most ± 3%. If we assume that 3% represents the outer limits of a normal distribution, this corresponds to a standard deviation for flux of The uncertainty in nitrogen volume fraction due to the uncertainty in the flux measurement is calculated by propagation of error to be: 6. The standard deviation for nitrogen volume is the root-sum-square of the uncertainties due to the calibration curve and error in the measurement of flux as follows: For the samples in the study, the measured values of flux, nitrogen volume fractions and associated uncertainties are shown in the table below. The portion of the uncertainty due to the calibration curve is shown in the 3rd column. The portion of the uncertainty due to flux is shown in the 4th column. The total standard uncertainty is shown in the fifth column. Table of measured values of flux ( ) and nitrogen volume fraction (%) and associated components of uncertainty Flux Cal N2 Vol SD N2 Vol Due to flux Total SD X Z 9.3 34.23508 0.06576 0.20946 0.30488 10.0 37.40776 0.06259 0.22584 0.30575 9.6 35.06355 0.06494 0.21647 0.30672 19.3 46.13260 0.05387 0.45153 0.48476 4.3 24.34408 0.07566 0.09497 0.29604 5.0 29.68942 0.07031 0.11074 0.27073 4.6 21.93403 0.07807 0.10172 0.31322 5.0 32.50115 0.06750 0.11074 0.25605 5.4 30.65654 0.06934 0.11979 0.26944 19.3 45.60766 0.05439 0.45153 0.48527 17.0 44.51841 0.05548 0.39431 0.43377 21.4 47.63400 0.05237 0.50458 0.53324 21.4 46.64503 0.05335 0.50458 0.53408 30.0 45.06976 0.05493 0.72987 0.75156 2.1 21.93403 0.07807 0.04598 0.29979 1.8 21.93403 0.07807 0.03936 0.29884 12.9 38.86245 0.06114 0.29460 0.35600 12.5 36.64890 0.06335 0.28503 0.35379 24.3 47.63400 0.05237 0.57911 0.60424 25.7 46.13260 0.05387 0.61561 0.64039 27.1 47.63400 0.05237 0.65246 0.67486