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