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A femtometre is $10^{-15}$ metre, written $\mathrm{fm}$. It is a convenient unit of measurement for the dimensions of an atomic nucleus, rather than the much larger dimensions of a chemical atom. For example, a length of $5\mathrm{fm}$ is $5\times 10^{-15}\mathrm{m}$.

Scattering probes such small lengths through momentum transfer rather than a miniature mechanical ruler. The relation between wavelength and momentum explains why increasingly short-distance structure requires increasingly large momentum transfer.

Measurement uncertainty describes the spread of values reasonably attributable to a specified measurand, using the available information. It differs from measurement error, which is a difference from a reference value: the uncertainty can be quantified even when that difference is unknown. A useful result therefore states what was measured, the unit of measurement, the value, and how its uncertainty was evaluated.^[ref]

For a wire's electrical resistance obtained from $R=V/I$, uncertainty in the voltage and electric current propagates through that measurement model. Shared measurement calibration effects can make the inputs correlated, so adding percentages or treating every contribution as independent can be wrong. The measurement uncertainty budget records those assumptions. Report a standard uncertainty when a standard-deviation scale is intended, or an expanded uncertainty with its stated coverage factor; a bare plus-or-minus number does not specify which was used.

The metre, symbol $\mathrm{m}$, is the SI base unit of measurement for length. It is defined by fixing the speed of light in vacuum to $299792458\mathrm{m}{\mathrm{s}}^{-1}$, with the second defined independently. Thus light in vacuum travels one metre in $1/299792458$ of a second.

The exact defining constant does not make every measured length exact. Realizing a length standard can involve optical interferometry, frequency references, alignment and corrections for propagation through air; their imperfections contribute to measurement uncertainty. The definition specifies the reference, whereas an experiment supplies an imperfect realization.^[ref]

A micrometre, symbol $\mu \mathrm{m}$, is a unit of measurement for length equal to one millionth of a metre:The SI prefix micro supplies the exact factor $10^{-6}$.^[ref]

For an invented measurement example, a metal foil of thickness $25\mu \mathrm{m}$ has thickness $0.025\mathrm{mm}$. Forty such layers have combined thickness $1000\mu \mathrm{m}=1\mathrm{mm}$ if they lie flat without gaps. The calculation makes a small length tangible without assuming a particular measuring instrument.

Unit conversion does not remove measurement uncertainty. A reported thickness of $(25\pm 1)\mu \mathrm{m}$ becomes $(0.025\pm 0.001)\mathrm{mm}$: both the value and its uncertainty receive the same exact scale factor.

A physical quantity is a property of a phenomenon, body or substance whose magnitude can be expressed by a number and a reference, usually a unit of measurement. For example, the length $L=2\mathrm{m}$ and the same length $L=200\mathrm{cm}$ have different numerical values but represent the same physical quantity. The numerical value alone is therefore incomplete.

A physical quantity must also be specified physically. A gas's pressure at a stated position and temperature, the mass of a specimen, and a detector's electric current are different measurands even if an instrument happens to display the same number for them. Dimensional analysis constrains which quantities can be added or compared, while a system of units supplies a consistent convention for expressing them.^[ref]

A sensitivity coefficient describes how an estimated output of a measurement model changes when one input changes slightly. For $y=f(x_1,\dots ,x_n)$, the first-order coefficients, evaluated at the input estimates, areA sensitivity coefficient can have a sign and a unit of measurement: for an output in ohms and an input in volts, its unit is ohms per volt. The GUM also describes estimating sensitivity coefficients experimentally by varying inputs.^[ref]

In a measurement uncertainty budget, propagation of uncertainty uses these coefficients to transform input variances and covariances:Here $u_i$ denotes an input standard uncertainty. The derivative's sign disappears from its independent variance term but still matters in a covariance term.^[ref]

For an original electrical resistance example, $R=V/I$, $V=10\mathrm{V}$ and $I=2\mathrm{A}$ give $R=5\Omega$ andIndependent input standard uncertainties of0.1volt and0.02ampere each contribute0.05ohm on the output scale, giving $u_R=\sqrt{0.05^2+0.05^2}\Omega \approx 0.0707\Omega$. A larger sensitivity coefficient alone does not identify the dominant uncertainty: the input standard uncertainty and any correlations also matter.

The derivative is a local approximation. If the input spread spans a strongly nonlinear part of the measurement model, first-order propagation of uncertainty can be inadequate even when every derivative has been computed correctly.

An SI prefix supplies an exact decimal factor when attached to a unit of measurement. For example, milli means $10^{-3}$ and micro means $10^{-6}$; a millimetre and a micrometre are correspondingly scaled metres. The prefix symbol and unit symbol form one unit that can itself be raised to a power.^[ref]

For an original conversion example,The area conversion squares the factor as well as the unit. Applying just one factor of $10^{-3}$ to that area would give $0.004{\mathrm{m}}^2$, a value 1000 times too large. Scientific notation makes the power explicit.

Prefix symbols are case-sensitive and cannot be stacked. For mass, prefixes attach to gram, because the base-unit name kilogram already contains a prefix. SI prefixes denote powers of ten, even in a computing context; they do not silently change into powers of two.^[ref] An exact unit conversion scales any stated measurement uncertainty with the value; it does not make an uncertain measurement exact.

A standard uncertainty expresses a component of measurement uncertainty as an estimated standard deviation. It has the same unit of measurement as the measurand: $u(L)=0.2\mathrm{mm}$ describes a spread in a length, rather than a percentage or a known measurement error. A Type A evaluation of uncertainty uses statistical analysis of observations; a Type B evaluation of uncertainty uses other information, such as a reference certificate or instrument specification. These classify the evaluation method, not whether a physical effect is random or systematic.^[ref]

For $n$ independent repeated readings with common variance, the sample mean has estimated standard uncertainty $s/\sqrt{n}$, where $s$ is their sample standard deviation. This is the standard error of the mean; the scatter $s$ instead describes individual readings. Repetition reduces this contribution, but a shared measurement calibration contribution remains. A quoted tolerance $\pm a$ supplies a different example: assigning a continuous uniform distribution within those bounds gives $u=a/\sqrt{3}$. That assignment is a model of the available information, not a consequence of repeating measurements.^[ref]

A unit of measurement is a chosen reference for comparing magnitudes of a physical quantity. A value written as a number times a measurement unit changes its numerical part when the measurement unit changes, while representing the same physical quantity.