Substitution measurement method

A substitution measurement method compares an unknown with a reference by placing them successively in the same relevant position of a measuring system. The comparison seeks a match or measures a difference while retaining as much of the system response as possible. The VIM lists substitution among measurement methods; the detailed sequence belongs to the measurement procedure.^[ref]

For a physical application, NIST's double substitution weighing procedure compares a standard and an unknown of the same nominal mass twice, using a weighing instrument as a comparator. It also determines response with a known sensitivity mass and includes measurement-assurance checks.^[ref] The measurement standard supplies the reference value; the comparator need not establish the whole mass independently from its own zero.

An original simplified drift model shows why sequence matters. Let a comparator indication at equally spaced time index $t$ bewith fixed response coefficient $a$, additive offset $b$ and linear instrumental drift $d$. Observe standard S at $t=0$, unknown X at $t=1,2$, and S again at $t=3$. The mean X indication and mean S indication have the same mean time $3/2$, givingBoth the constant offset and the stipulated linear drift cancel. For $m_S=m_0=100\mathrm{g}$, $a=500{\mathrm{g}}^{-1}$, $b=10$ and $d=2$, an unknown $m_X=100.004\mathrm{g}$ gives indications $10,14,16,16$. Their paired means differ by two, recovering $0.004\mathrm{g}$.

This teaching sequence isolates one cancellation mechanism; it is not the full NIST procedure. Nonlinear drift, unequal timing, changing response or different positioning can defeat it. NIST's procedure separately addresses buoyancy and uncertainty contributions.^[ref] A close match does not remove the reference's measurement uncertainty or establish that the substitution left every relevant condition unchanged.