2.
Measurement Process Characterization
2.3. Calibration 2.3.3. Calibration designs 2.3.3.2. General solutions to calibration designs


Requirements 
Solutions for all designs that are cataloged in this Handbook are
included with the designs. Solutions for other designs can be
computed from the instructions below given some familiarity with
matrices. The matrix manipulations that are required for the
calculations are:


Notation 


Convention for showing the measurement sequence 
The convention for showing the measurement sequence is illustrated with
the three measurements that make up a 1,1,1
design for 1 reference standard, 1 check standard, and 1 test item.
Nominal values are underlined in the first line .
1 1 1 Y(1) = +  Y(2) = +  Y(3) = +  

Matrix algebra for solving a design 
The (mxn) design matrix X is
constructed by replacing the pluses (+), minues () and blanks with
the entries 1, 1, and 0 respectively.
The (mxm) matrix of normal equations, X'X, is formed and augmented by the restraint vector to form an (m+1)x(m+1) matrix, A: 

Inverse of design matrix 
The A matrix is inverted and shown in the
form:


Estimates of values of individual artifacts 
The leastsquares estimates for the values of the individual artifacts
are contained in the (mx1) matrix,
B, where
where Q is the upper left element of the A^{1} matrix shown above. The structure of the individual estimates is contained in the QX' matrix; i.e. the estimate for the ith item can be computed from XQ and Y by


Clarify with an example 
We will clarify the above discussion with an example from the mass
calibration process at NIST. In this example, two NIST kilograms are
compared with a customer's unknown kilogram.
The design matrix, X, is The measurements obtained, i.e., the Y matrix, are The value of the reference standard, R^{*}, is 0.82329. Then
This yields the following leastsquares coefficient estimates: 

Standard deviations of estimates 
The standard deviation for the
ith item is:
where
The process standard deviation, which is a measure of the overall precision of the (NIST) mass calibrarion process,
is the residual standard deviation from the design, and s_{days} is the standard deviation for days, which can only be estimated from check standard measurements. 

Example 
We continue the example started above. Since n = 3 and
m = 3, the formula reduces to:
We obtain the following computations
