# Uncertainty calculation Met Cal vs GUM

An particular way to calculate of uncertainty is available in several guides of expression of uncertainty.
All of them explain like  find the type A uncertainties based  on a sequence of measurements at a particular test point.
Most of them used the formula:

S1 = (SDEV /  Square root(N) ) where:
1. N is the number of measurements
2. SDEV is the standard deviation of the measurements

But in the document this formula have.a new parameter , the F factor
https://community.flukecal.com/sites/community.flukecal.com/files/private/uncertainty_in_version_7_metcal.pdf

So, the new formula in this document is
S1 = (SDEV /  Square root(N) ) * F
F is based on the half the values of Student’s T distribution and the number of
degrees of freedom.
How you know, at the end ,you need to report the expanded uncertainty on based in Student's T again.
So, I think that is like expand the uncertainty twice.

I don't understand very well, what is the reason for this.
Could someone explain the reason?

Date Marcus Flack

The expanded uncertainty is the standard uncertainy multiplied by the coverage factor.  The S1 paramater is an uncertainty component of U2 based on the standard deviation of the UUT measurements.  The value of F is used to properly weight the S1 component relative to its number of samples assuming that the measurements follow a normal distribution pattern.  Does this answer your question?

Note that the S1 value cannot be determined with only 1 measurement.  NMEAS = 3 or more should be used to get a meaningful value for S1.  You can override S1 and or U2 by using the VSET FSC. Patrick Butler

Unless overridden or disabled, the value of F is determined per Table G.2 of Annex G of the document ANSI/NCSL Z540-2-1997. The values of F used by MET/CAL are exactly half the values shown in the 95.45% column of Table G.2.
Note that MET/CAL uses the simplifying assumption that the number of degrees of freedom is one less than the number of measurements (NMEAS). If this assumption is not acceptable, it may be possible for the metrologist / procedure writer to directly calculate F and override MET/CAL's built-in determination of F. William (Bill) Spath

Additionally per the VSET help file

USE_ST

The USE_ST parameter is used to enable or disable the use of the Student's T distribution to determine the factor F used in the measurement uncertainty calculation. Legal values for USE_ST are "Yes" and "No". The basic measurement uncertainty calculation is: Standard Uncertainty = RSS(U1, U2, U3, ..., U10) where

U2 = RSS(S1, S2) and where

S1 = (SDEV / (NMEAS ^ 0.5)) * F

Recall that F is a factor based on the Student's T distribution and the number of degrees of freedom. (In MET/CAL's uncertainty calculation the simplifying assumption is made that the number of degrees of freedom is one less than number of measurements.) When USE_ST is set to "Yes", the factor F is determined, unless overridden or disabled, per Table G.2 of Annex G of the document ANSI/NCSL Z540-2-1997. The values of F used by MET/CAL are exactly half the values shown in the 95.45% column of Table G.2.

When USE_ST is set to "No", the factor F is set to 1.0, so the calculation of S1 becomes simply:

S1 = (SDEV / (NMEAS ^ 0.5))

By default (as distributed) MET/CAL V6.0 sets USE_ST to "No". In other words, the factor F is set to 1 and it is presumed that the Coverage Factor (COV_FAC), typically set to 2, and used to determine the Expanded Uncertainty based on the Standard Uncertainty, is sufficient to incorporate the confidence in the standard deviation of the measured values as a function of the number of measurements. When NMEAS is 10 or more, F is close to 1 in any case, and so this presumption would appear to be justified. For small NMEAS values, on the other hand, the Student's

T-based F value can be significant (for example, F is 6.985 when NMEAS is 2), and the decision as to whether it's appropriate to set F to 1 unconditionally or not has to be based on the judgement of the metrologist. Based on comments from various European and American sources, it was determined that the best approach for MET/CAL is to provide the option and allow each site to decide how to implement this aspect of the uncertainty calculation.

There are three ways to set USE_ST:

1. Database

As shipped, USE_ST is set to "No" in the V6.0 database. That is, the use of the Student's T-based factor F in the uncertainty calculation is, by default, not enabled. To change the database specification choose "Uncertainty..." in the "Configure" menu of the Metrology Database Customization and Configuration application. A dialog will appear which allows the USE_ST parameter specified. This value will apply to all workstations at your site, unless overridden at the workstation level by an initialization file specification or at the procedure level by a VSET specification.

2. Initialization File

USE_ST may be specified in the "[startup]" section of the MET/CAL initialization file. An initialization file specification overrides the database specification for all measurement uncertainty calculations on the workstation.

3. VSET Statement

USE_ST may be set at the procedure level in a VSET statement.

For example:

VSET use_st = yes

A VSET specification of USE_ST overrides any initialization file and database specification, and remains in effect for the duration of procedure execution until overridden or reset in a subsequent VSET statement.

To reset a VSET specification of USE_ST specify:

VSET use_st = *

NOTE: USE_ST is supported in MET/CAL V7 for compatibility with MET/CAL V6. USE_ST is non-preferred in V7. It is recommended that USE_ST be set to "No" when using MET/CAL V7.