---
title: Measurement and Dimensions
module: Foundations
moduleNumber: 0
lessonNumber: 1
order: 1
summary: >
  Every physical quantity is a number attached to a unit, and that pairing is what
  lets you check an equation before computing anything, since terms that add together
  must carry the same dimensions. We build the SI base units and the notion of
  dimension, then use dimensional analysis to test a proposed relation and form
  scaling groups — a method that fixes a formula's shape but never its numerical
  constants. The lesson also sets how precisely a result may be stated, through
  significant figures, propagated uncertainty, and order-of-magnitude checks that
  catch errors a raw calculator answer hides.
topics: [Foundations]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 1 — Measurement and Vectors; §§1-2–1-5"
---

## Quantities, units, and dimensions

A physical measurement is a numerical value with a unit. The unit identifies the
quantity being measured and makes arithmetic physically interpretable. Mechanics
uses length, mass, and time as frequent base quantities, with SI units metre,
kilogram, and second. Derived units retain their base-unit structure. Force, for
example, has unit

$$
1\ \mathrm N=1\ \mathrm{kg\,m\,s^{-2}}.
$$

Dimension records physical type independently of a chosen unit. Length has
dimension $[L]$, time has $[T]$, mass has $[M]$, and force has
$[MLT^{-2}]$. A valid additive equation has the same dimensions in every term.

**Conversion factors.**

Conversion factors are ratios of equal quantities. They are arranged so unwanted
units cancel algebraically:

$$
72\ \mathrm{km\,h^{-1}}
\left(\frac{1000\ \mathrm m}{1\ \mathrm{km}}\right)
\left(\frac{1\ \mathrm h}{3600\ \mathrm s}\right)
=20\ \mathrm{m\,s^{-1}}.
$$

The written cancellation checks the direction of each conversion. Compound units
must be converted through every power of each base unit; a cubic centimetre differs
from a cubic metre by a factor of $10^6$, not $10^2$.

**Dimensional analysis.**

Dimensions test a proposed relation before numerical substitution. In

$$
x=x_0+v_0t+\frac12at^2,
$$

every term has dimension $[L]$. Dimensional agreement is necessary but not
sufficient: it cannot determine the dimensionless factor $1/2$ or distinguish a
sum from a difference. It can reject an equation that adds a velocity to an
acceleration or reports a force with energy units.

> **Worked example (Pendulum period from dimensions).** Suppose the period $T$ of a
> simple pendulum depends on its length $L$, the free-fall acceleration $g$, and the
> bob mass $m$: write $T=C\,L^a g^b m^c$ with $C$ dimensionless. Matching dimensions,
>
> $$
> [T]=[L]^a\,[LT^{-2}]^b\,[M]^c=[L]^{a+b}\,[T]^{-2b}\,[M]^c.
> $$
>
> Equating exponents gives $c=0$, so mass drops out; $-2b=1$, so $b=-\tfrac12$; and
> $a+b=0$, so $a=\tfrac12$. Hence
>
> $$
> T=C\sqrt{\frac{L}{g}}.
> $$
>
> Dimensions fix the form but not $C$. The small-angle solution supplies $C=2\pi$,
> a value dimensional analysis alone cannot recover.

## Measurement resolution and instruments

Reported digits imply measurement resolution. A quantity written
$2.31\ \mathrm m$ resolves length at approximately $0.01\ \mathrm m$ under the
stated instrument convention. Random uncertainty is reduced through repeated
measurement; systematic calibration error is not removed by repetition. A result
should state its uncertainty with the same unit as the measured quantity.

**SI definitions and derived quantities.**

The SI system fixes base units through reproducible physical standards. The second
is defined through a specified atomic transition of caesium; the metre is defined
through the distance light travels in a stated fraction of a second; the kilogram is
defined through a fixed numerical value of Planck's constant. Daily mechanics work
does not require reproducing these standards, but the definitions explain why SI
units are not arbitrary artifacts of a particular object or laboratory.

Derived quantities are formed by multiplication and division of base quantities.
Velocity has dimensions $[LT^{-1}]$, acceleration has $[LT^{-2}]$, momentum has
$[MLT^{-1}]$, and energy has $[ML^2T^{-2}]$. The joule may be written

$$
1\ \mathrm J=1\ \mathrm{N\,m}=1\ \mathrm{kg\,m^2\,s^{-2}}.
$$

Writing units through the calculation makes incompatible dimensions visible before a
final numerical answer can conceal them. A result for a stopping distance must reduce to metres; a result for a
period must reduce to seconds. Units never determine the physical model, but they
test whether the algebra is compatible with the proposed quantity.

**Conversion chains and scale factors.**

Unit conversion is multiplication by one, expressed in a form that cancels the
existing unit. Several factors may be chained when a quantity includes multiple
base dimensions. Density illustrates a dimensional trap because both mass and volume
units change:

$$
1.00\ \mathrm{g\,cm^{-3}}
\left(\frac{1\ \mathrm{kg}}{1000\ \mathrm g}\right)
\left(\frac{100\ \mathrm{cm}}{1\ \mathrm m}\right)^3
=1.00\times10^3\ \mathrm{kg\,m^{-3}}.
$$

The cube applies to the length conversion because the denominator contains cubic
centimetres. Treating $\mathrm{cm^3}$ as though it converted with a factor of 100
would give an error of four orders of magnitude. Prefixes are scale factors, not
new dimensions: kilo means $10^3$, milli means $10^{-3}$, and micro means
$10^{-6}$.

> **Worked example (Unit conversion, pressure).** Convert a tire pressure of
> $32\ \mathrm{psi}$ to pascals using $1\ \mathrm{psi}=6894.76\ \mathrm{Pa}$:
>
> $$
> 32\ \mathrm{psi}\left(\frac{6894.76\ \mathrm{Pa}}{1\ \mathrm{psi}}\right)
> =2.2\times10^5\ \mathrm{Pa}.
> $$
>
> The two significant figures of $32\ \mathrm{psi}$ limit the result to two; the
> conversion factor is a defined ratio and adds no precision. The answer carries
> pressure units, not newtons or square metres alone.

## Uncertainty and numerical reporting

Significant figures describe the precision supported by a measurement. In addition
and subtraction, the final decimal place is limited by the least precise addend. In
multiplication and division, relative uncertainty more directly expresses the error scale. For a
quantity $q=ab/c$ with independent small uncertainties,

$$
\frac{\delta q}{q}\approx
\frac{\delta a}{a}+\frac{\delta b}{b}+\frac{\delta c}{c}.
$$

A power $q=a^n$ obeys

$$
\frac{\delta q}{q}\approx |n|\frac{\delta a}{a}.
$$

The first-order estimates apply when uncertainties are small and
independent. Correlated measurements require their covariance; systematic errors
must be modeled separately rather than treated as random scatter.

> **Worked example (Propagated uncertainty in an area).** A rectangular plate
> measures $L=(20.0\pm0.1)\ \mathrm{cm}$ and $W=(10.0\pm0.1)\ \mathrm{cm}$. Its
> area is $A=LW=200\ \mathrm{cm^2}$, with fractional uncertainty
>
> $$
> \frac{\delta A}{A}\approx\frac{0.1}{20.0}+\frac{0.1}{10.0}=0.015.
> $$
>
> So $\delta A\approx0.015\times200\ \mathrm{cm^2}=3\ \mathrm{cm^2}$, giving
> $A=(200\pm3)\ \mathrm{cm^2}$. Extra digits in the area would claim precision the
> measurements do not support.

**Order of magnitude and limiting checks.**

An order-of-magnitude estimate retains powers of ten and the governing variables
while omitting numerical factors not resolved by the available information. A fall
time from height $h$ has scale $t\sim\sqrt{h/g}$; doubling height changes this time
by a square-root factor rather than a factor of two. Such estimates identify whether
a detailed numerical result has a plausible scale.

> **Worked example (Free-fall time).** A body released from rest falls freely
> through a height $h$. Ignoring air resistance and taking $g$ constant,
> $h=\tfrac12 g t^2$ gives $t=\sqrt{2h/g}$. With $h=45\ \mathrm m$ and
> $g=9.81\ \mathrm{m\,s^{-2}}$,
>
> $$
> t=\sqrt{\frac{2(45\ \mathrm m)}{9.81\ \mathrm{m\,s^{-2}}}}=3.0\ \mathrm s.
> $$
>
> Quadrupling $h$ only doubles $t$: the square-root scaling $t\sim\sqrt{h/g}$ made
> explicit.

Dimensional checks should be paired with limiting checks. An expression for a range
should vanish when launch speed vanishes. A formula for force should vanish when the
interaction coefficient vanishes. An equation may be dimensionally correct and
still physically wrong, so dimensions, limits, signs, and experimental uncertainty
provide independent checks.

## Calibration and systematic error

An instrument does not report a quantity directly. It reports a response that is
converted to a quantity by a measurement model. A spring scale gives an extension,
a thermistor gives a resistance, and a photogate gives a time interval. Calibration
establishes the connection between that response and a known reference. For a
simple linear model, a reading $R$ may be converted through $x=aR+b$, where $a$
is a scale factor and $b$ is a zero offset. Both parameters have uncertainty and
both can drift with temperature, loading history, or electronic aging.

Calibration must be performed over the range in which the instrument will be used.
A balance adjusted near 100 g can still respond nonlinearly near its maximum load,
so a straight-line fit that looks adequate in a narrow central region can miss
curvature at the ends.

$$
% caption: Calibration points define a response line whose intercept fixes the instrument's zero offset.
\begin{tikzpicture}[scale=0.9, >=Latex]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (5.6,0) node[right] {reference value};
  \draw[->, black] (0,0) -- (0,3.4) node[above] {instrument response};
  \draw[acc, thick] (0,0.55) -- (5.1,3.0);
  \draw[dashed, black] (0,0.55) -- (0.6,0.55);
  \fill[acc] (0,0.55) circle (1.7pt);
  \node[left, black] at (0,0.55) {zero};
  \foreach \x/\y in {0.75/0.95,1.55/1.28,2.35/1.66,3.15/2.02,3.95/2.44,4.6/2.74}{
    \draw[fill=white, draw=black, thick] (\x,\y) circle (1.5pt);
  }
  \node[acc, anchor=south east] at (5.05,2.9) {calibration line};
  \node[black, anchor=north west] at (2.7,1.55) {readings};
\end{tikzpicture}
$$

**Random variation and systematic shift.**

Repeated readings quantify random variation. Under stable conditions, many small
independent disturbances often produce a roughly symmetric distribution about a
mean. The standard deviation describes the spread of individual readings; the
standard error of the mean describes uncertainty in an estimated mean and decreases
as $1/\sqrt{N}$ when the readings are independent. Taking more readings therefore
helps with random scatter, but it does not repair a shared bias.

A systematic shift moves the whole distribution. A ruler with a worn zero edge,
a stopwatch with a fixed start delay, or a sensor calibrated at the wrong
temperature can give highly repeatable but inaccurate values. Comparing two
methods, measuring a known reference, changing the operator, or reversing an
apparatus can expose such shifts. Agreement among repeated readings is evidence of
precision, not by itself evidence of accuracy.

Uncertainty statements should say what they represent. A range such as
$x=(12.4\pm0.2)\ \mathrm{cm}$ may be an instrument-resolution estimate, a
standard deviation, or a coverage interval; these have different meanings. The
method used to obtain the interval belongs with the result. When measurements are
correlated—for example, many lengths read from the same miscalibrated rule—their
common calibration uncertainty should not be divided by $\sqrt{N}$.

$$
% caption: Random scatter widens each distribution about its own mean; a systematic shift displaces the whole distribution from the true value.
\begin{tikzpicture}[scale=0.9, >=Latex]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[right] {measured value};
  \draw[->, black] (0,0) -- (0,3.1) node[above] {count};
  \draw[acc, thick, domain=0.4:4.0, samples=90]
    plot (\x,{2.3*exp(-((\x-2.2)/0.62)^2)});
  \draw[black, thick, dashed, domain=2.0:5.6, samples=90]
    plot (\x,{2.3*exp(-((\x-3.8)/0.62)^2)});
  \draw[dotted, black] (2.2,0) -- (2.2,2.5);
  \draw[dotted, black] (3.8,0) -- (3.8,2.5);
  \draw[<->, black] (2.2,1.1) -- (3.8,1.1);
  \node[black, anchor=south] at (3.0,1.12) {shift};
  \node[acc, anchor=south] at (2.2,2.5) {true};
  \node[black, anchor=south] at (3.8,2.5) {biased};
\end{tikzpicture}
$$

## Dimensional models and scaling

The Buckingham-pi method organizes dimensional analysis when several variables
are present. Write each variable as powers of the base dimensions and place those
exponents in a matrix. For drag force $F$, density $\rho$, speed $v$, and a body
size $D$, the rows for $M$, $L$, and $T$ yield the dimensionless combination
$F/(\rho v^2D^2)$. Dimensional reasoning then permits
$F=C\rho v^2D^2$, where the coefficient $C$ cannot be determined by dimensions.
Its value can depend on shape, surface condition, and flow regime.

The matrix identifies dimensionless groups and reduces the number of independent
variables. A physical model must still establish that the chosen list contains every
relevant effect. Viscosity, for
example, introduces a further dimensionless ratio and changes the drag law when
the flow regime changes. A successful scale model must preserve the relevant
dimensionless ratios. Geometric copying alone leaves those governing ratios
uncontrolled.

Scale analysis expresses the same dependence more compactly. If a result depends on
$v^2$, a 10% speed change causes about a 20% change in that contribution. If a
quantity varies as $1/r^2$, doubling distance reduces it by a factor of four.
Proportional changes expose scale before numerical calculation and are often more informative than
the final numerical value. They also flag whether an approximation can be
neglected: a term smaller by $10^{-6}$ is usually irrelevant to a percent-level
experiment but not necessarily to a part-per-million measurement.

A dimensional matrix can also expose a missing parameter: if a fitted dimensionless
coefficient shifts whenever the surrounding fluid changes, it is absorbing omitted
physics, and adding fluid density and viscosity produces new groups that turn the
unexplained variation into a testable prediction. A dimensional consequence and an
experimentally established coefficient are separate claims — the group follows from
dimensions, its value and validity range from data — so a coefficient measured in
one narrow experiment carries its conditions rather than the label "universal."

| Audit item | Compact representation | Evidence required |
| --- | --- | --- |
| Drag group | $\Pi_D=F/(\rho v^2D^2)$ | variables and their units |
| Local speed response | $\Delta F/F\approx2\Delta v/v$ at fixed $\rho$ and $D$ | perturbation or fitted slope |
| Validity domain | stated ranges of speed, size, and fluid | residuals inside and outside the range |

Conversions and measurement models require the same record. Every fitted coefficient
must carry dimensions or be explicitly dimensionless; every reference quantity must
be stated; and every extrapolation should be marked as such. A
calibration obtained over 0--10 N establishes the response over that interval.
An extension to 100 N requires a separate test. Scale arguments set the likely
size of a failure; comparison with standards establishes whether it occurs.

## Instrument response and calibration curves

Between the physical interaction and the reported number lies an instrument model
that must state the input quantity, the observed response, the operating range, and
the corrections applied. Treating a display as the quantity itself hides assumptions
that can dominate an uncertainty budget.

Offset, sensitivity, nonlinearity, and loading are distinct effects. An offset
adds a constant error. Sensitivity changes the scale factor. Nonlinearity changes
the relation between response and input across the range. Loading occurs when the
instrument alters the system it measures: a heavy probe can change a vibrating
system, and a voltmeter with finite input resistance can change a circuit. A zero
check detects only one point of a response curve; it cannot establish the slope
or curvature away from zero.

Corrections should be applied through an explicit equation. If an indicated length
is $L_i$ and a calibrated zero error is $z$, the corrected length might be
$L=L_i-z$. The sign follows from the definition of $z$; the written model fixes
the convention. When a correction is comparable to its own
uncertainty, reporting extra digits in the corrected value creates a false sense
of control. The correction and its uncertainty should be retained separately
until the final calculation.

**Calibration curves, residuals, and validity range.**

Calibration data test the response at several known inputs, so they are stronger
than a single zero adjustment. A fitted straight line gives a sensitivity and an
offset; the residuals — observed minus predicted response — show whether the line
is adequate. Scatter about zero is consistent with the model, whereas a smooth bend
or a run of same-sign residuals reveals missed structure. Calibration points should
bracket the intended use: extrapolation beyond the largest reference is unreliable
even where interpolation is excellent, and a polynomial that lowers residuals inside
the range can behave implausibly outside it. The calibration equation is evidence
for a specified interval, not a law of the device.

Increasing and decreasing tests identify hysteresis. When the response at a given
input depends on the earlier loading path — through friction, thermal lag, or
mechanical relaxation — a single curve is incomplete, and the path dependence must
enter the model or the uncertainty.

$$
% caption: Residuals are the vertical gaps between measured points and the fitted line; alternating, unpatterned scatter supports a straight response.
\begin{tikzpicture}[scale=0.9, >=Latex]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (5.9,0) node[right] {reference load};
  \draw[->, black] (0,0) -- (0,3.4) node[above] {sensor output};
  \draw[acc, thick] (0.3,0.4) -- (5.3,2.9);
  \draw[dotted, black] (0.7,0.75) -- (0.7,0.60);
  \draw[dotted, black] (1.4,0.80) -- (1.4,0.95);
  \draw[dotted, black] (2.1,1.48) -- (2.1,1.30);
  \draw[dotted, black] (2.8,1.47) -- (2.8,1.65);
  \draw[dotted, black] (3.5,2.20) -- (3.5,2.00);
  \draw[dotted, black] (4.9,2.88) -- (4.9,2.70);
  \draw[acc, very thick] (4.2,2.18) -- (4.2,2.35);
  \foreach \x/\y in {0.7/0.75,1.4/0.80,2.1/1.48,2.8/1.47,3.5/2.20,4.2/2.18,4.9/2.88}{
    \draw[fill=white, draw=black, thick] (\x,\y) circle (1.4pt);
  }
  \node[acc, anchor=south east] at (5.1,2.7) {linear model};
  \node[acc, anchor=west] at (4.33,2.24) {residual};
\end{tikzpicture}
$$

**Propagation with correlation.**

Adding relative uncertainties assumes small independent errors and gives a
conservative first estimate. The general first-order rule for a result $q(x,y)$
carries a covariance term:

$$
\sigma_q^2\approx
\left(\frac{\partial q}{\partial x}\sigma_x\right)^2+
\left(\frac{\partial q}{\partial y}\sigma_y\right)^2+
2\frac{\partial q}{\partial x}\frac{\partial q}{\partial y}\cov(x,y).
$$

Positive correlation raises the uncertainty of a sum and lowers that of a
difference; negative correlation reverses this. A shared calibration offset is the
familiar case: two masses weighed on the same balance carry a common scale error
that cancels in their difference, so treating the readings as independent overstates
that difference's uncertainty. Correlation is recognized from procedure — a shared
timebase, thermometer, or fitted curve — and rounding intermediate values before
combining them destroys the information needed to detect it.

$$
% caption: The tilt of a joint-uncertainty ellipse encodes correlation: up-right for positive, down-right for negative.
\begin{tikzpicture}[scale=0.9, >=Latex]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.4) -- (6.0,0.4) node[right] {$x$};
  \draw[->, black] (0.5,0) -- (0.5,3.4) node[above] {$y$};
  \draw[acc, thick, rotate around={30:(2.1,1.8)}] (2.1,1.8) ellipse (1.4 and 0.5);
  \fill[acc] (2.1,1.8) circle (1.6pt);
  \node[acc, anchor=south] at (2.1,3.0) {positive};
  \draw[black, thick, rotate around={-30:(4.7,1.8)}] (4.7,1.8) ellipse (1.4 and 0.5);
  \fill[black] (4.7,1.8) circle (1.6pt);
  \node[black, anchor=south] at (4.7,3.0) {negative};
\end{tikzpicture}
$$

## Reporting and statistical inference

A reproducible result includes the information needed to interpret the reported
number. State the measured quantity, unit, uncertainty, method used for the
uncertainty, and conditions that materially affect the result.
For example, a length reported as $(24.31\pm0.05)\ \mathrm{mm}$ should identify
whether $0.05\ \mathrm{mm}$ is a resolution estimate, a standard uncertainty, or
an interval with a specified coverage. The uncertainty is usually rounded to one
or two significant digits; the central value is rounded to the same decimal place.

Experimental design begins before measurements are collected. Identify the desired
quantity, the model linking it to observations, the dominant uncertainty sources,
and a check that can detect a systematic error. Repetition should be allocated
where it lowers an important random component. Calibration effort should be
allocated where a common bias would dominate every reading. A measurement with
very fine resolution may still be a poor design if it is sensitive to an
uncontrolled temperature, alignment, or timing condition.

Independent cross-checks use different failure modes. A period measured with a
photogate and a distance measured from video frames can test a model without sharing
the same timing mechanism. Plan a blank measurement, reversal, or known reference
with the experiment. Record these checks even when they agree with expectation;
they support the stated limits.

**Significant figures, rounding, and guard digits.**

Significant figures communicate the resolution supported by an uncertainty analysis.
They prevent a displayed number from claiming resolution that the measurement and
model cannot support. The important place in an addition or subtraction is set by
the least certain decimal place.
For multiplication and division, a relative uncertainty calculation is generally
more informative than a mechanical count of digits. The final reported digits
should follow from the uncertainty, not from the number of digits carried by a
calculator.

Keep guard digits through intermediate steps. Early rounding can shift a small
difference or distort a regression slope, especially when several corrections are
combined. Suppose a calibration calculation gives $2.3467\ \mathrm{N}$ with an
uncertainty of $0.031\ \mathrm{N}$. Rounding the uncertainty to
$0.03\ \mathrm{N}$ and the value to the same hundredth gives
$(2.35\pm0.03)\ \mathrm{N}$. Writing $2.3467\ \mathrm{N}$ beside that
uncertainty would imply that the last two digits are experimentally meaningful.

Exact counts and defined conversion factors are different from measured values.
There are exactly 12 items in a dozen, and a metre-to-millimetre conversion is an
exact scale relation. Their digits do not limit the result. A measured length,
however, carries finite resolution even when a digital display shows many places.
The last displayed digit is often uncertain; it is not a guaranteed exact digit.

Rounding must preserve sign and scale. A small negative result rounded to a stated
place remains negative unless it is demonstrably zero within the reporting rule.
Scientific notation makes scale visible: $3.2\times10^{-6}$ and
$0.0000032$ represent the same magnitude, but the former is less likely to hide
the position of the significant digits.

**Distributions, intervals, and coverage.**

An uncertainty interval is meaningful only with its method and coverage. For a
roughly normal distribution, a standard deviation describes the typical spread of
individual observations. About 68% of such observations lie within one standard
deviation of the mean, while about 95% lie within two standard deviations. These
percentages are properties of the distribution model, not a universal guarantee
for every finite data set.

The uncertainty of a mean is smaller than the spread of individual readings: for
$N$ independent readings of standard deviation $s$, the standard error is
$s/\sqrt{N}$, and a confidence interval built from it says how the method performs
over repeated samples, not that a fixed constant varies between experiments. A
histogram supports this comparison only when its bin width is stated, since changing
the width can make the same data look smooth, skewed, or multimodal; for small sets
a dot plot preserves more, and a long tail or two clusters calls for physical
investigation before averaging. Coverage from random scatter alone can be narrow
while the true value lies outside it because a calibration shift was omitted, so
combining random and systematic components needs a stated model, not a bare percent
sign.

$$
% caption: For a normal model about 68% of observations fall within one standard deviation of the mean and about 95% within two.
\begin{tikzpicture}[scale=0.9, >=Latex]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.4) -- (6.4,0.4) node[right] {value};
  \draw[->, black] (0.4,0.2) -- (0.4,3.2) node[above] {density};
  \fill[acc!12] (2.6,0.4) -- plot[domain=2.6:4.0, samples=50]
    (\x,{0.4+2.3*exp(-((\x-3.3)/0.72)^2)}) -- (4.0,0.4) -- cycle;
  \draw[acc, thick, domain=1.1:5.5, samples=100]
    plot (\x,{0.4+2.3*exp(-((\x-3.3)/0.72)^2)});
  \draw[dotted, black] (3.3,0.4) -- (3.3,2.8);
  \draw[<->, black] (2.6,1.05) -- (4.0,1.05);
  \node[black, fill=white, inner sep=1pt] at (3.3,1.05) {68\%};
  \draw[<->, black] (1.88,0.64) -- (4.72,0.64);
  \node[black, fill=white, inner sep=1pt] at (3.3,0.64) {95\%};
\end{tikzpicture}
$$

**Fitted parameters and covariance.**

Fitting a line $y=a+bx$ yields an intercept $a$ and slope $b$ whose uncertainties
are generally coupled: a larger slope compensated by a smaller intercept fits the
same data nearly as well, and the off-diagonal covariance is positive or negative
with the tilt of the allowed region. Reporting $a$ and $b$ separately is then
insufficient, since a prediction $y_*=a+bx_*$ has a variance that includes the
covariance term. Measurements spread broadly in $x$ constrain the slope more than
clustered ones, and placing the fitted origin near the data centre reduces the
coupling.

A residual plot remains essential after a fit. A straight line through curved data
can give a small average residual yet an obvious pattern against $x$, marking a
missing term; random error alone leaves unstructured scatter.

## Validation, traceability, and reproducibility

Dimensional agreement is a necessary condition for a model, but validation must
go further. A proposed expression should be checked against limiting behaviour,
symmetry, known reference cases, and data over the range where it will be used.
The drag scale $F\sim\rho v^2D^2$ has the correct dimensions for an inertial
fluid-drag regime, yet it cannot describe every fluid flow. At sufficiently low
speed or small size, viscosity introduces another scale and the functional form
changes.

Dimensionless plots test scaling models across systems. If a model predicts that
$F/(\rho v^2D^2)$ is constant under specified conditions, data from several body
sizes and speeds should collapse onto one band when plotted in those variables.
Failure to collapse may indicate an omitted variable, a calibration problem, or a
change of physical regime. A graph is more informative when axes express the
model's dimensionless prediction than when each raw variable is plotted alone.

Validation has a scale boundary: agreement at 1 m/s does not establish a model at
100 m/s if a neglected compressibility effect grows with speed, so a claim states
its tested range, the ratios held fixed, and the residual size relative to the
measurement uncertainty. Using the same data to fit parameters and to declare
success hides overfitting; a separate data set, an independent quantity, or a
withheld range is a stronger test, and the evidence should say whether it tests the
fitted parameters, the functional form, or the stated range — one residual summary
cannot establish all three.

| Test | Quantity held out or varied | Interpretation |
| --- | --- | --- |
| Reference case | independently known value | calibration and units |
| Dimensionless collapse | size or speed at fixed governing ratios | functional form and omitted variables |
| Range extension | condition outside fit interval | validity-domain limit |
| Residual pattern | $r_i=y_i-y_{\rm model,i}$ | bias, regime change, or noise model |

**Quantitative checks on a proposed model.**

For each observation, compare the residual $r_i=y_i-y_{\rm model,i}$ with that
observation's expected uncertainty. Residuals much larger than expected point to a
wrong model, an underestimated uncertainty, or an apparatus failure; residuals
uniformly much smaller are also suspect, from overstated uncertainties or from
tuning parameters on the same data being judged.

When uncertainties are independent and approximately normal, the reduced
chi-squared statistic is a compact check. For $N$ observations and $p$ fitted
parameters,

$$
\chi^2_\nu=\frac{1}{N-p}\sum_i\frac{r_i^2}{\sigma_i^2}.
$$

A value near one is compatible with the assumed statistical model but does not prove
the physical one; a large value signals disagreement beyond the assigned
uncertainties; a value well below one suggests conservative uncertainties or
correlated points. It cannot diagnose a model on its own, since a systematic curve
can leave a moderate mean squared residual while remaining physically decisive.

Predictions should be tested where they were not fitted — a small-angle pendulum
model at a new length, a different bob mass, or a deliberately larger angle chosen
to stress a neglected term. A model may pass a goodness-of-fit check yet omit a
mechanism too small to resolve, so the honest conclusion is bounded: the data are
consistent with the model at the tested precision and range, not universally.

$$
% caption: Data from several scales collapse onto one band in dimensionless variables; a separated trend signals missing physics or a regime change.
\begin{tikzpicture}[scale=0.9, >=Latex]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.4) -- (6.2,0.4) node[right] {dimensionless input};
  \draw[->, black] (0.4,0.2) -- (0.4,3.3) node[above] {dimensionless response};
  \draw[dashed, black] (0.7,2.1) -- (5.8,2.1);
  \foreach \x/\y in {0.9/2.05,1.5/2.16,2.1/2.06,2.7/2.15,3.3/2.03,3.9/2.12,4.5/2.07,5.1/2.14,5.6/2.05}{
    \draw[fill=acc!15, draw=acc, thick] (\x,\y) circle (1.5pt);
  }
  \foreach \x/\y in {0.9/0.85,1.6/1.0,2.3/1.25,3.0/1.55,3.8/1.95,4.6/2.5}{
    \draw[fill=white, draw=black, thick] (\x,\y) circle (1.5pt);
  }
  \node[acc, anchor=south] at (2.6,2.35) {one band};
  \node[black, anchor=north west] at (2.1,1.05) {second regime};
\end{tikzpicture}
$$

**SI traceability and local standards.**

The SI fixes units through numerical values of physical constants — the second by
the caesium frequency, the metre by the speed of light, the kilogram by Planck's
constant. A laboratory works through a shorter chain: a certified reference, a local
calibration, and an instrument used under stated conditions. Traceability means this
chain is documented with its uncertainties, not that every reading matches a national
standard. The record names the reference, its certificate, the date and method of
comparison, and the uncertainty transferred. A balance checked against one reference
mass is not thereby calibrated for every load or temperature; several references
across the range test the scale factor, and a repeated zero check tests the offset.

Unit definition and instrument calibration are separate. Converting
$37.2\ \mathrm{cm}$ to $0.372\ \mathrm m$ uses an exact factor and changes neither
the length nor its relative uncertainty, whereas the reading from the rule stays
uncertain, and traceability concerns that reading. A reference artifact — gauge
block, mass set, or timing source — has its own valid conditions and finite
calibration interval; using it as though it were perfect transfers its unknown
deviations into every result.

**Uncertainty budgets and dominant contributions.**

An uncertainty budget lists each input, its assigned uncertainty, the route by
which it reaches the result, and the combined total, separating random, calibration,
resolution, environmental, and model-assumption components. A long list of small
terms is not thorough if it omits the one effect that shifts every result the same
way. In $q=f(x_1,x_2,\ldots)$ each input contributes about
$|\partial q/\partial x_i|\,u(x_i)$; independent terms combine in quadrature and
correlated ones through their covariances. The sensitivity coefficient matters as
much as the input uncertainty, so a precisely measured variable can dominate if the
model amplifies it, while a coarse one is harmless if the result barely depends on it.

Effort should target the dominant contribution: reducing a term already far below
the total changes little, while reducing the largest term helps most. If temperature
drift dominates a length measurement, repeating readings at the same drift does not
help — the temperature must be measured or stabilized. Budgets should also carry a
known model discrepancy, such as a small-angle or uniform-density approximation, as
a separate bound rather than folding it into a random standard deviation.

$$
% caption: An uncertainty budget ranks each contribution after its sensitivity is applied, so effort targets the dominant term rather than the largest raw input.
\begin{tikzpicture}[scale=0.9, >=Latex]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (2.0,0.3) -- (2.0,3.4);
  \draw[->, black] (2.0,0.3) -- (6.4,0.3) node[right] {relative contribution};
  \fill[acc!15] (2.0,2.75) rectangle (5.9,3.1);
  \draw[acc, thick] (2.0,2.75) rectangle (5.9,3.1);
  \node[left, acc] at (1.9,2.92) {temp};
  \fill[black] (2.0,2.05) rectangle (4.3,2.4);
  \draw[black, thick] (2.0,2.05) rectangle (4.3,2.4);
  \node[left, black] at (1.9,2.22) {zero};
  \fill[black] (2.0,1.35) rectangle (3.4,1.7);
  \draw[black, thick] (2.0,1.35) rectangle (3.4,1.7);
  \node[left, black] at (1.9,1.52) {random};
  \fill[black] (2.0,0.65) rectangle (2.8,1.0);
  \draw[black, thick] (2.0,0.65) rectangle (2.8,1.0);
  \node[left, black] at (1.9,0.82) {model};
\end{tikzpicture}
$$

**Dimensional similarity and scale models.**

Matching dimensionless ratios, not shape, is what makes a scale model predictive.
In fluid flow the Reynolds number compares inertial and viscous effects; in a
free-surface flow the Froude number compares inertial and gravitational effects. A
model that copies geometry but changes these numbers reproduces an appealing picture
while missing the governing force balance. When not every group can be held equal
between model and full scale, the report states which mismatch remains and estimates
its effect. Similarity also bounds what an apparatus can test: a pendulum's period
compares across sizes at fixed amplitude-to-length ratio and common gravity, but air
resistance upsets the comparison for very light bobs, and a bridge model that does
not preserve its stress ratio shows geometric deflection yet misses material failure.

**Data records, reporting, and reproducibility.**

Reproducibility begins with raw observations: preserve the original readings, units,
instrument settings, timestamps, exclusions, and any correction applied afterward,
with each derived column linked to the equation that produced it. Separating
observation from interpretation lets the interpretation be updated for a better model
without losing the original data. Measured and inferred quantities belong in separate
rows, so a raw reading can be re-evaluated without reconstructing the apparatus
history.

| Layer | Retained item | Audit question |
| --- | --- | --- |
| Observation | raw reading, unit, time, and instrument state | can the original value be reconstructed? |
| Correction | calibration or baseline rule | what sign and range were applied? |
| Derived quantity | stated equation and unrounded inputs | which assumptions enter the result? |
| Interpretation | fitted parameter or model test | does it remain valid if the model changes? |

Keep reporting conventions consistent: one unit per column stated in the heading,
error bars labelled as standard deviations, standard errors, or a coverage interval,
and axes labelled with quantity and unit. Note the conditions held fixed, the
rejected data and the reason, and any departure from the planned procedure; these
define the conditions under which the stated uncertainty applies.

**Review before release.**

A separate review reconstructs the chain from the final number back to the
observations, checking that units are consistent, calibration dates cover the
measurement, corrections carry the stated sign, uncertainty components are not
double-counted, and plotted values match the table — a correct formula applied to
the wrong column yields a plausible but invalid result. The review also confirms
that the reported precision matches the evidence: rounding, axis ranges, and the
stated uncertainty should agree, and any sensitivity to a reasonable change of
assumption belongs in the report. Archiving the raw data, calibration records, and
calculation method lets another worker reproduce the result or find why nominally
similar experiments disagree.

[^tipler15]: Tipler and Mosca, _Physics for Scientists and Engineers_, 6th ed., §§1-2–1-5.