Reporting Measured Values

When scientists make a measurement, they do not simply write down a number — they must report that number in a way that honestly communicates how reliable and precise the measurement is. A result such as 'the length is about 3 metres' is far less useful than '3.47 ± 0.02 m', because the second version tells you both the best estimate and the range of uncertainty. Reporting measured values correctly is one of the most fundamental skills in experimental physics. The precision of a reported value is conveyed through significant figures (also called significant digits). Every digit that is known with confidence, plus one estimated digit, should be retained; all others are meaningless noise. Alongside significant figures, every physical quantity must be accompanied by its correct SI unit, because a number without a unit is scientifically meaningless — '9.81' could be anything, whereas '9.81 m s⁻²' is unambiguously the acceleration due to gravity. Uncertainty is the formal way of expressing the limits of a measurement. It can arise from the resolution of an instrument (for example, the smallest division on a ruler), from random scatter in repeated readings, or from systematic errors in the experimental setup. By quoting uncertainty explicitly — as an absolute uncertainty, a fractional uncertainty, or a percentage uncertainty — a scientist allows others to judge whether a result is good enough for its intended purpose and to combine it correctly with other measurements.

Key takeaways
  • Identify Significant Figures
  • Determine Absolute Uncertainty
  • Calculate Fractional and Percentage Uncertainty
  • Propagate Uncertainties Through Calculations
  • Write the Final Result in Standard Form

Identify Significant Figures

Count every non-zero digit, every zero between non-zero digits, and every trailing zero that appears after a decimal point. For example, 0.00470 has three significant figures (4, 7, and the trailing 0). Leading zeros are never significant — they are just placeholders. When you make a measurement, the number of significant figures you report should match the precision of the instrument: a balance reading to 0.01 g gives results to 3 s.f. if the reading is, say, 2.35 g.

Determine Absolute Uncertainty

The absolute uncertainty Δx\Delta x is expressed in the same units as the measurement xx. For a single reading with an analogue instrument, the absolute uncertainty is typically half the smallest scale division. For a digital instrument, it is ±1 in the last displayed digit. For a set of repeated measurements, the uncertainty is estimated from the range: Δx=xmax⁡−xmin⁡2\Delta x = \frac{x_{\max} - x_{\min}}{2}, or more rigorously from the standard deviation of the mean.

Calculate Fractional and Percentage Uncertainty

Dividing the absolute uncertainty by the measured value gives the fractional (relative) uncertainty: Δxx\frac{\Delta x}{x}. Multiplying by 100% gives percentage uncertainty: Δxx×100%\frac{\Delta x}{x} \times 100\%. These dimensionless quantities are essential when combining uncertainties in calculations. For example, if a length is 12.4 ± 0.2 cm, the percentage uncertainty is 0.212.4×100%≈1.6%\frac{0.2}{12.4} \times 100\% \approx 1.6\%.

Propagate Uncertainties Through Calculations

When a derived quantity depends on several measured quantities, uncertainties must be propagated. For addition or subtraction, add absolute uncertainties: if z=x+yz = x + y, then Δz=Δx+Δy\Delta z = \Delta x + \Delta y. For multiplication, division, or powers, add fractional uncertainties: if z=xm⋅ynz = x^m \cdot y^n, then Δzz=∣m∣Δxx+∣n∣Δyy\frac{\Delta z}{z} = |m|\frac{\Delta x}{x} + |n|\frac{\Delta y}{y}. This ensures that the uncertainty in the final answer is never understated.

Write the Final Result in Standard Form

A correctly reported result has the form: x±Δxx \pm \Delta x (unit). The uncertainty should be rounded to one or two significant figures, and the measured value should be rounded to the same decimal place as the uncertainty. For example, if a calculation gives 9.81274 m s⁻² with an uncertainty of 0.034 m s⁻², round the uncertainty to 0.03 m s⁻² and report: g=9.81±0.03g = 9.81 \pm 0.03 m s⁻². Never quote a result to more decimal places than its uncertainty allows.

The Key Equation

$$\frac{\Delta z}{z} = |m|\frac{\Delta x}{x} + |n|\frac{\Delta y}{y}
- **\Delta z** — Absolute uncertainty in z (same as z): The uncertainty in the derived (calculated) quantity z. - **z** — Derived quantity (varies): The quantity calculated from measured values, e.g. area, volume, speed. - **\Delta x,\, \Delta y** — Absolute uncertainties in measured quantities (same as x and y respectively): The uncertainties in the directly measured inputs to the calculation. - **m,\, n** — Exponents (powers) (dimensionless): The powers to which x and y are raised in the formula for z, e.g. $z = x^m y^n$. ## Worked Example **Problem.** A student measures the diameter of a sphere using vernier calipers and records d = 4.52 \pm 0.02 cm. Calculate the volume of the sphere and express it with its absolute uncertainty and correct significant figures. (Volume of a sphere: $V = \frac{4}{3}\pi r^3$, where $r = d/2$.)

V = 48.4 \pm 0.6 \text{ cm}^3

∗∗Answer.∗∗Thevolumeofthesphereis48.4±0.6cm3(percentageuncertainty≈1.3**Answer.** The volume of the sphere is 48.4 \pm 0.6 cm³ (percentage uncertainty \approx 1.3%).
Figure
Diagram
Anatomy of a Reported Measurement9.81 ± 0.03 m s⁻²Complete reported valueBest estimate9.813 sig. figs.Absolute uncertainty± 0.03same decimal placeSI Unitm s⁻²always requiredPercentage Uncertainty(0.03 / 9.81) × 100% ≈ 0.31%
Origins · 1 minLord Kelvin (William Thomson) and how this was discovered
LK
Lord Kelvin (William Thomson)
1824–1907 · British (Irish-born)

Lord Kelvin was one of the foremost advocates for rigorous quantitative measurement in physics. He established the absolute thermodynamic temperature scale (the kelvin, symbol K, is named in his honour) and famously insisted that true scientific understanding requires the ability to express a phenomenon in numbers. He made major contributions to thermodynamics, electromagnetism, and the laying of the transatlantic telegraph cable, all of which depended on precise, well-reported measurements. Kelvin once estimated the age of the Earth at 20–400 million years using cooling calculations — a boldly quantitative claim that was later revised when radioactive heating was discovered. He showed that even a carefully reasoned measurement is only as good as the completeness of the physical model behind it.

  • Significant Figures — Significant figures are the direct tool used to encode precision when writing a measured value; understanding them is prerequisite to correct reporting.
  • Types of Experimental Error — Random and systematic errors are the sources of measurement uncertainty; knowing their origin helps you estimate and minimise the uncertainty you must report.
  • SI Units and Base Quantities — Every reported value must carry a correct SI unit; familiarity with the SI system is essential for meaningful reporting.
  • Uncertainty Propagation in Calculations — Once individual measurements are reported with uncertainty, propagation rules determine the uncertainty in any derived quantity calculated from them.