Panel Presentation NCSLI 2013 Session 6B Easy Translation of TAR - - PowerPoint PPT Presentation

Information Technology Laboratory – National Institute of Standards and Technology

NCSLI 2013 Session 6B Easy Translation of TAR or TUR into Uncertainty

William Guthrie Statistical Engineering Division

Panel Presentation

Information Technology Laboratory – National Institute of Standards and Technology

Generic Uncertainty Expression

• Standard uncertainty for a measurement result

from a calibrated measurement device can be generally expressed as where

( ) ( ) ( )

2 2 n n

u y u D u C = +

( ) is the standard uncertainty of the calibration

n

u C

( ) is the standard uncertainty of the measurement

n

u D

1

Information Technology Laboratory – National Institute of Standards and Technology

Showing Next Level of Uncertainty

• Expanding this formula to show the next level of

uncertainty down the calibration chain gives

( ) ( ) ( ) ( ) ( ) ( )

2 2 2 2 2 1 1 n n n n n

u y u D u C u D u D u C

− −

= +   = + +  

2

Information Technology Laboratory – National Institute of Standards and Technology

Assumptions

• Next, assume

– TAR or TUR ≥4 at each level of calibration, – systems are in place that mitigate the effects of any

potential sources of uncertainty not accounted for in the TAR or TUR being used, and

– the numerator and denominator of the TAR or TUR are

approximately known multiples of the associated standard uncertainties

• Example based on ANSI/NCSL Z540.3 TUR

3

( ) ( ) ( ) ( ) ( ) ( )

540.3

Upper Device Spec.- Lower Device Spec. Upper 95% Cal. Unc.- Lower 95% Cal. Unc. 6 6 4 4

Z n n n k n n n

TUR u D u D u D r u C u C u C = ⋅ ≈ = ⋅ ≡ ⋅

use of 6/4 assumes approximate normality use of rk keeps results general

Information Technology Laboratory – National Institute of Standards and Technology

Relating Uncertainties at Different Levels

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2 1 1 2 2 2 2 1 1 2 2 2 2 1 1 2 2 2 1

4 4 16 16 16

n n k k n n n k n n n k n n n k n n

u D u D TUR r r u C u D u C r u D u D u C r u D u D u C r u D u D

− − − − − − −

≥ ⇒ = ≥   +     ⇒ ≥ +   ⇒ − − ≥ ⇒ − ≥

4

Information Technology Laboratory – National Institute of Standards and Technology

Put These Expressions Together …

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 1 1

16 16

n n n n n n k n n n k n n

r u D u y u D u D u C u D u D u C r u D u C D u D u

− − − − − −

  −      = + +     ≤ + + +   = +  +

5

Information Technology Laboratory – National Institute of Standards and Technology

… and Just Carry On …

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2 2 2 1 2 2 2 2 2 2 2

16 16

k n n n k n n n n

r u y u D u D u C r u D u D u D u C

− − −

≤ + +   = + + +  

( ) ( ) ( ) ( )

2 2 2 2 2 2 2 2

16 16

k k n n n n

r r u D u D u D u C −   = + + +    

( ) ( ) ( ) ( ) ( ) ( )

2 2 2 2 2 2 2 2 2 2 2 2

16 16

k n n k n n n n

r u D u r u u u D C D D D u

− − −

+ + + + −

                  ≤  

6

Information Technology Laboratory – National Institute of Standards and Technology

… Until You End Up Here!

Now we can just use for legacy systems with TAR or TUR ≥4. No further uncertainty analysis required! Similar results appear to hold for systems based on EOPR as well.

( ) ( ) ( ) ( )

2 1 2 2 2 2 2

16 16 16 if 1 16 16

i n k n i i k n i k n k

r u y u D r u D r u D r

− = ∞ =

  ≤       ≤     = < −

∑ ∑

( ) ( )

2

4 Device Accuracy 16 3

k

u y r = −

7