Measurement, SI Units, and the Theory of Errors

Contents

Measurement
The International System of Units
The theory of errors
A worked example: timing a pendulum

Measurement

A measurement is a physical interaction between a system and an apparatus whose outcome is a number (or a set of numbers) together with a unit and a statement of uncertainty. All three components are mandatory: a number without a unit does not identify a quantity, and a number without an uncertainty does not identify how strongly it constrains a theory [JCGM:2008] [Taylor:1997]. The quantity a measurement intends to determine is called the measurand.

Definition 2.1 (Physical quantity).

A physical quantity is a property of a system that can be compared quantitatively with a reference property of the same kind, the unit. The result of the comparison is the numerical value, so that every quantity $Q$ decomposes as $Q = \{Q\}\,[Q]$ with $\{Q\}$ a real number and $[Q]$ the unit.

The International System of Units

Axiom 2.2 (SI).

All physical quantities in this treatise are expressed in the International System of Units (SI), in every branch of physics.

Since the 2019 revision, the SI is defined not by artefacts but by fixing the exact numerical values of seven defining constants — the caesium hyperfine frequency $\Delta\nu_{\mathrm{Cs}}$, the speed of light $c$, the Planck constant $h$, the elementary charge $e$, the Boltzmann constant $k$, the Avogadro constant $N_A$, and the luminous efficacy $K_{\mathrm{cd}}$ — from which the seven base units (second, metre, kilogram, ampere, kelvin, mole, candela) are derived [BIPM:2019]. The complete tables, with the CODATA 2018 values of the remaining measured constants, are collected in Appendix B [Tiesinga:2021].

Two practical consequences of Axiom 2.2 run through the whole treatise. First, equations keep their constants: $c$, $G$, $\hbar$ and $k$ are never set to one, so every formula can be evaluated against an experiment without unit bookkeeping. Where a natural-unit shortcut is genuinely illuminating it appears inside a flagged remark with the conversion stated. Second, every reported number carries its unit through the typography itself, e.g. $g_0 = 9.80665\,\mathrm{m}/\mathrm{s}^{2}$ (exact, by convention [BIPM:2019]).

Dimensional analysis

Every physical quantity has a dimension, a formal product of powers of the base dimensions $\mathsf{T}, \mathsf{L}, \mathsf{M}, \mathsf{I}, \Theta, \mathsf{N}, \mathsf{J}$. An equation between quantities is admissible only if both sides carry the same dimension; this trivial requirement is unreasonably powerful, because it constrains the possible form of an answer before any dynamics is solved.

Theorem 2.3 (Buckingham $\pi$ theorem).

Let a physical relation connect $n$ quantities $q_1,\dots,q_n$ whose dimensions involve $r$ independent base dimensions. Then the relation can be rewritten as a relation among $n-r$ independent dimensionless combinations $\pi_1,\dots,\pi_{n-r}$ of the $q_i$. Rests on Axiom 2.2.

Derivation. Derives Theorem 2.3. Write the dimension of each $q_i$ as an integer (or rational) exponent vector in the $r$-dimensional space of base dimensions. The map sending a monomial $q_1^{a_1}\cdots q_n^{a_n}$ to its dimension is linear in the exponents $a_i$, with matrix $M$ of shape $r\times n$ and rank $r' \le r$. Dimensionless monomials are precisely the kernel of $M$, which by the rank–nullity theorem (Linear Algebra and Representation Theory) has dimension $n - r'$. Choosing a basis $\pi_1,\dots,\pi_{n-r'}$ of the kernel, any dimensionally consistent relation among the $q_i$, being invariant under independent rescalings of the base units, can depend on the $q_i$ only through kernel monomials, hence is a relation among the $\pi_j$. With the customary assumption that the $r$ base dimensions actually occur independently, $r' = r$.

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Example 2.4.

For a pendulum of length $\ell$, mass $m$, in gravity $g$, with period $T$: the four quantities involve $\mathsf{T},\mathsf{L},\mathsf{M}$, so there is $4-3 = 1$ dimensionless group, $\pi = T\sqrt{g/\ell}$. Dimensional analysis alone forces $T = C\sqrt{\ell/g}$ with $C$ a pure number — mass cannot appear. Dynamics is needed only for $C = 2\pi$ (Oscillations and Mechanical Waves).

The theory of errors

No measurement returns the measurand exactly. The deviations are of two kinds, treated by two different disciplines.

Definition 2.5 (Random and systematic errors).

A random error varies unpredictably between repetitions of the measurement under nominally identical conditions; it is treated statistically, and shrinks under repetition. A systematic error biases every repetition the same way (a miscalibrated scale, a clock that runs slow); it is immune to repetition and must be hunted by varying the apparatus and the method. The corresponding quantitative statements are called type A and type B uncertainty evaluations [JCGM:2008].

Repeated measurements

Let $x_1,\dots,x_N$ be independent repetitions of a measurement of a quantity whose true value is $\mu$, each modelled as a random variable with mean $\mu$ and variance $\sigma^2$ (the probabilistic machinery is developed in Probability and Statistics).

Definition 2.6 (Sample mean and variance).

\begin{equation}\tag{2.1} \bar x = \frac{1}{N}\sum_{i=1}^{N} x_i\ec \qquad s^2 = \frac{1}{N-1}\sum_{i=1}^{N}\left(x_i - \bar x\right)^2\ep \end{equation}

Theorem 2.7 (The mean improves as $1/\sqrt{N}$).

$\bar x$ is an unbiased estimator of $\mu$, and its standard deviation is $\sigma/\sqrt{N}$. Rests on Definition 2.6.

Derivation. Derives Theorem 2.7. Linearity of the expectation gives $\avg{\bar x} = \frac1N\sum_i \avg{x_i} = \mu$. For the variance, since the $x_i$ are independent, variances add:

\begin{equation}\tag{2.2} \operatorname{var}\bar x = \frac{1}{N^2}\sum_{i=1}^{N}\operatorname{var} x_i = \frac{\sigma^2}{N}\ec \end{equation}

whence the standard deviation of the mean is $\sigma/\sqrt N$. An analogous computation shows $\avg{s^2} = \sigma^2$, which is the reason for the $N-1$ (Bessel) denominator.

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The standard uncertainty quoted for a type-A evaluation is therefore $u = s/\sqrt{N}$. The ubiquity of the Gaussian distribution for random errors is explained by the central limit theorem, proved as Theorem 11.46: a sum of many small independent disturbances tends to a Gaussian whatever the individual disturbances look like. The form that applies here is the Lindeberg–Feller one (Theorem 11.48), since the disturbances acting on a measurement — a thermal drift, a vibration, a quantisation step, a reading error — are independent but certainly not identically distributed.

Two hypotheses of that theorem are worth keeping in view, because a measurement can violate either. The contributions must have finite variance: a heavy-tailed error source need not average down at all (Remark 11.47). And none may dominate — Theorem 11.48 implies that every individual source contributes a vanishing fraction of the total variance, so a single dominant source defeats its hypothesis. The converse does not hold, and the theorem is not a licence to argue backwards: failing the condition does not by itself make the total non-Gaussian, and satisfying every variance share is not enough on its own.

The rate of approach is $O(M^{-1/2})$ in the number $M$ of contributing sources, by the Berry–Esseen bound Equation (11.40) — which additionally assumes a finite third absolute moment. That bound is uniform in the deviation, so it constrains the absolute error of the Gaussian approximation equally everywhere; in the tails, where the probability being approximated is itself tiny, the relative error is correspondingly larger. Since a coverage factor $k=3$ is a claim about a tail probability, it deserves more caution than the same approximation applied near the centre.

Propagation of uncertainty

Derived quantities inherit uncertainty from their inputs.

Theorem 2.8 (First-order propagation).

Let $f(x_1,\dots,x_n)$ be differentiable, and let the $x_i$ be independent measurements with standard uncertainties $u_i$ small enough that $f$ is approximately linear over their spread. Then the standard uncertainty of $y = f(x_1,\dots,x_n)$ is

\begin{equation}\tag{2.3} u_y^2 = \sum_{i=1}^{n}\left(\pdv{f}{x_i}\right)^{\!2} u_i^2\ep \end{equation}

Derivation. Derives Theorem 2.8. Expand $f$ to first order about the means, $y - \avg{y} \approx \sum_i \pdv{f}{x_i}(x_i - \avg{x_i})$. Squaring and taking expectations, the cross terms $\avg{(x_i-\avg{x_i})(x_j-\avg{x_j})}$ vanish for $i\ne j$ by independence, leaving exactly Equation (2.3). For correlated inputs the covariances reappear as $2\sum_{i<j}\pdv{f}{x_i}\pdv{f}{x_j}\operatorname{cov}(x_i,x_j)$ [JCGM:2008].

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Corollary 2.9 (Quotients and powers).

For $y = x_1^{a_1}\cdots x_n^{a_n}$, relative uncertainties combine as $\left(u_y/y\right)^2 = \sum_i a_i^2 \left(u_i/x_i\right)^2$. Rests on Theorem 2.8.

Derivation. Derives Corollary 2.9. Apply Theorem 2.8 to $\ln y = \sum_i a_i \ln x_i$, for which $\pp \ln y/\pp \ln x_i = a_i$.

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Combining and fitting data

Theorem 2.10 (Weighted mean).

Given independent measurements $x_i$ of the same quantity with uncertainties $u_i$, the linear combination of minimal variance is

\begin{equation}\tag{2.4} \hat x = \frac{\sum_i x_i/u_i^2}{\sum_i 1/u_i^2}\ec \qquad u_{\hat x}^2 = \left(\sum_i \frac{1}{u_i^2}\right)^{-1}\ep \end{equation}

Rests on Theorem 2.7.

Derivation. Derives Theorem 2.10. Seek $\hat x = \sum_i w_i x_i$ with $\sum_i w_i = 1$ (unbiasedness) and $\operatorname{var}\hat x = \sum_i w_i^2 u_i^2$ minimal. With a Lagrange multiplier $\lambda$ (Lagrangian Mechanics uses the same device for constraints), $\pp_{w_i}\!\left[\sum_j w_j^2 u_j^2 - \lambda\left(\textstyle\sum_j w_j - 1\right)\right] = 0$ gives $w_i = \lambda/(2u_i^2)$; the constraint fixes $\lambda$, yielding Equation (2.4), and substitution gives the stated variance.

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Theorem 2.11 (Least-squares line).

Given points $(x_i, y_i)$, $i = 1,\dots,N$, with equal uncertainties on the $y_i$ and negligible uncertainty on the $x_i$, the line $y = a + b x$ minimizing $\chi^2 = \sum_i (y_i - a - b x_i)^2$ has

\begin{equation}\tag{2.5} b = \frac{\overline{xy} - \bar x\,\bar y}{\overline{x^2} - \bar x^2}\ec \qquad a = \bar y - b\,\bar x\ec \end{equation}

where overbars denote sample means. Rests on Definition 2.6.

Derivation. Derives Theorem 2.11. $\pp_a\chi^2 = -2\sum_i(y_i - a - bx_i) = 0$ gives $\bar y = a + b \bar x$; $\pp_b \chi^2 = -2\sum_i x_i (y_i - a - b x_i) = 0$ gives $\overline{xy} = a\bar x + b\overline{x^2}$. Solving the two linear equations yields Equation (2.5). The second-derivative matrix is positive definite (its determinant is proportional to the variance of the $x_i$), so the extremum is the minimum.

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Reporting

A result is reported as $y = (\hat y \pm u)\,[\,\text{unit}\,]$, the uncertainty with at most two significant figures and the value rounded to match. Where higher confidence is wanted, an expanded uncertainty $U = k\,u$ is quoted with the coverage factor $k$ stated ($k = 2$ covers about $95\,\mathrm{\%}$ for a Gaussian) [JCGM:2008]. When an experiment box in this treatise quotes asymmetric bounds (as the gravitational-wave observations of Experiment: Gravitational Waves do), they are the bounds of the quoted posterior interval of the discovery analysis.

A worked example: timing a pendulum

Suppose a pendulum of length $\ell = 0.9500 \pm 0.0005\,\mathrm{m}$ swings with small amplitude, and $50$ periods are timed five times with a stopwatch reading to $0.01\,\mathrm{s}$: $T_{50} =$ $97.81\,\mathrm{s}$, $97.95\,\mathrm{s}$, $97.73\,\mathrm{s}$, $98.02\,\mathrm{s}$, $97.89\,\mathrm{s}$. The sample mean is $\bar T_{50} = 97.88\,\mathrm{s}$ with $s = 0.11\,\mathrm{s}$, so $u_{\bar T_{50}} = s/\sqrt 5 = 0.05\,\mathrm{s}$, and the period is $T = 1.9576 \pm 0.0010\,\mathrm{s}$. With $T = 2\pi\sqrt{\ell/g}$ (derived in Oscillations and Mechanical Waves), $g = 4\pi^2 \ell/T^2$; Corollary 2.9 gives

\begin{equation}\tag{2.6} \frac{u_g}{g} = \sqrt{\left(\frac{u_\ell}{\ell}\right)^{\!2} + \left(2\,\frac{u_T}{T}\right)^{\!2}} \approx 1.2\times 10^{-3}\ec \end{equation}

so $g = 9.79 \pm 0.01\,\mathrm{m}/\mathrm{s}^{2}$ — consistent with the local value and eight orders of magnitude less precise than the ballistic gravimetry of Section 20.4, which is the same physics with the error budget engineered down [Niebauer:1995] [Taylor:1997].