= Substitution measurement method
A \b[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>.<International vocabulary of metrology by JCGM>[VIM3entry2.5, note, substitution measurement method]{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.<Weighing by Double Substitution by National Institute of Standards and Technology>[SOP4–2019, p1, Purpose; pp6–8, weighing sequences and observations]{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 $t$ be
$$
y(t)=a(m-m_0)+b+dt,
$$
with 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 $3/2$, giving
$$
\bar y_X-\bar y_S=a(m_X-m_S).
$$
Both 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.<Weighing by Double Substitution by National Institute of Standards and Technology>[SOP4–2019, §3, calculations; uncertainty section, air-buoyancy and density contributions]{ref} A close match does not remove the reference's <measurement uncertainty> or establish that the substitution left every relevant condition unchanged.
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