Only those who experiment make errors, and who experiments, errs! a manifesto for the calculation of the propagation of uncertainties

Gonçalves, Luís Moreira

doi:10.21577/0100-4042.20170460

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Assuntos Gerais • Quím. Nova 43 (2) • Feb 2020 • https://doi.org/10.21577/0100-4042.20170460 copy

Only those who experiment make errors, and who experiments, errs! a manifesto for the calculation of the propagation of uncertainties

Authorship SCIMAGO INSTITUTIONS RANKINGS

Making experimental measurements, whether we are at peace in the face of such a reality or not, invariably involves experimental uncertainty. Unfortunately, presenting uncertainty alongside the results is not as widespread as it should be. The author attempts here to simplify, improve understanding, and appeal to a healthy living with uncertainty. One of the main tools scientists should know that there is available is the uncertainty propagation equation, which can be applied to all equations involving experimental results.

Keywords:
central limit theorem; experimental errors; normal distribution; propagation of uncertainty

$Δ y = \sqrt{\sum_{i} {(\frac{δ y}{δ x_{i}})}^{2} {(Δ x_{i})}^{2}}$
$y = k a$	$Δ y = k Δ a$
$y = a + b$	$Δ y = \sqrt{{(Δ a)}^{2} + {(Δ b)}^{2}}$
$y = a - b$	$Δ y = \sqrt{{(Δ a)}^{2} + {(Δ b)}^{2}}$
$y = a b$	$Δ y = y \sqrt{{(\frac{Δ a}{a})}^{2} + {(\frac{Δ b}{b})}^{2}}$
$y = \frac{a}{b}$	$Δ y = y \sqrt{{(\frac{Δ a}{a})}^{2} + {(\frac{Δ b}{b})}^{2}}$
$y = k^{a}$	$Δ y = y \ln (a) Δ a$
$y = \log_{k} a$	$Δ y = \frac{Δ a}{\ln (k) a}$

$Δ y = \sqrt{\sum_{i} {(\frac{δ f}{δ x_{i}})}^{2} {(Δ x_{i})}^{2}}$
$Abs = ε b [X] (assu \min do ε e b como cons \tan tes)$	$Δ Abs = ε b Δ [X]$
$n_{total} = n_{1} + n_{2}$	$Δ n_{total} = \sqrt{{(Δ n_{1})}^{2} + {(Δ n_{2})}^{2}}$
$m_{x} = m_{final} - m_{inicial}$	$Δ m_{x} = \sqrt{{(Δ m_{final})}^{2} + {(Δ m_{inicial})}^{2}}$
$V = RI$	$Δ V = V \sqrt{{(\frac{Δ R}{R})}^{2} + {(\frac{Δ I}{I})}^{2}}$
$[A] = \frac{n_{A}}{V}$	$Δ [HA] = [A] \sqrt{{(\frac{Δ n_{A}}{n_{A}})}^{2} + {(\frac{Δ V}{V})}^{2}}$
$T = 10^{- Abs}$	$Δ T ≅ 2, 303 T Δ Abs$
$pH = \log [H^{+}]$	$Δ pH ≅ 0, 434 \frac{Δ H^{+}}{H^{+}}$

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[1] *e-mail: lmgoncalves@iq.usp.br