The purpose of the following procedure is to estimate the current measurement uncertainties that exist in testing.  The next step after analyzing current uncertainties in field measurements would be to reduce these uncertainties using alternative methods.  The problem at hand is not to determine what uncertainties the method invokes, but merely to understand the uncertainty in the calculations and measurement devices by following the EPA method, assuming it has no uncertainty in itself.  Essentially the uncertainty measurement involves no Bias created by a faulty method.

 

 

 

 

I.               Uncertainty Analysis Procedure

 

Uncertainty quantification is very judgmental and requires experience and understanding of the testing method

 

The quantity being determined is defined by Y, the parameters (inputs) used to determine Y are X1, X2, X3 É Xi.  These quantities are estimated by the terms y, x1, x2, x3 Éxi. 

 

The standard uncertainty of an input is u(xi), the combined standard uncertainty is uc(y), and the expanded uncertainty is the value which is reported.

 

The following procedure is merely a guide in how to obtain these quantities

 

K         Ð Coverage factor

n          Ð Number of recorded values

r           Ð Coefficient of level of Covariance

s          Ð Sample variance

u(xi)     Ð Standard Uncertainty

uc(y)    Ð Combined Uncertainty

Xi        Ð Input parameter of which the quantity is desired to be known

xi          Ð Input parameter that is actually measured

        Ð Mean value of measured parameter

Y         Ð Output result that is desired to calculate

y          Ð Output result that is actually calculated

 

  1. Write down equations that are used in calculations
  2. Determine uncertainty sources
    1. Variables being measured
    2. Contributions due to the testing process (some research might be necessary)

                                                     i.     Spatial variation

Uncertainty that results in too few measurements

            i.e. Calculations assume uniform flow in stack velocity

                                                      ii.     Human factors

Uncertainty that arises due to probe alignment errors

                                                        iii.     Precision

Uncertainty associated with reading the measurement

  1. List all instruments that are used in testing

Systematic uncertainties are instrument specific so instruments need only be listed once

  1. Determine systematic uncertainties
    1. Provided by manufacturer (x±a) Ð judgment  must be used based on assumed probability distribution

 

                                                     i.     For Rectangular distribution  

Used in situation where any value within the uncertainty can be assumed to have the same probability of occurring

            i.e. purity of substance

 

                                                      ii.     For Triangular distribution

Situation in which it is more probable that the measurement is closer to the mean

            i.e. Calibration of weighing scale

                                                        iii.     For Normal distribution

 

I>             Given as standard deviation (u =s), relative standard deviation (), or coefficient of variance CV% ()

II>           Uncertainty is given directly (±c)

I.               95% Confidence Interval (u=c/2)

II.             99.7% Confidence Interval (u=c/3)

    1. Determined experimentally from repeated observations, u=s (s is sample variance Ð defined in 8)

If test data is available from a certain procedure for which n>30 an uncertainty can be associated with this procedure through the variance (s)

    1. Determined theoretically

Estimating the uncertainty from well understood physical principles such as thermal expansion

    1. Listed in EPA methods as minimum accuracy
  1. Determine the sensitivity of parameters,

The sensitivity quantifies the effect the parameter has on the output

    1. Solve analytically
    2. Assuming y is linear in xi or u/x is small Ñ solve numerically by
    3. Solve experimentally (alter xi while observing the effect to y)

Calculated by

  1. Determine level of covariance, qik [u(xi,xk) = u(xi)u(xk)rik][ -1<qik<1]

Correlation between two input variables effect the combined standard uncertainty (The effect of one parameter might negate the effect of the other)

    1. Previous tests (covik )
    2. Analytical studies
    3. Experience

Educated Guess

  1. Calculate Combined Uncertainty

General calculation: 

  1. Determine overall uncertainty,
    1. sobs is the Random Uncertainty observed and calculated in two ways

                                                     i.     Normal distribution

I>             For instances when n > 30 (many samples)

II>          

                                                      ii.     StudentÕs t distribution

I>             Most likely case, n<30 (few samples)

II>           Approaches the normal distribution as n ˆ 30

III>        

  1. Expanded Uncertainty = K*u

K = Coverage factor that is a multiplier which is used to convert between uncertainty and reportable Confidence Interval.  It is dependent on degrees of freedom (n-1) and desired Confidence Interval

 

Must find the t-value for a two tailed Confidence Interval from software or tables.  The accompanied spreadsheet solves for this value

 

Glossary                                                                                   

 

Confidence Interval Ð Defines the amount of probability that the measurement falls between two values

                        For Example:  A 95% Confidence Interval categorizes the region of values (shaded) in which it is 95% probable that the value falls within.

                        Ref:  psych.rice.edu/online_stat/ chapter8/mean.html

 

Random Uncertainty Ð Uncertainty that is created during the test by the random dispersion of data

Systematic Uncertainties Ð Uncertainties that are based on the test setup or procedure and can be determined from previous tests.