Precision, Accuracy and Uncertainty

Spec mapping: OCR H556 Module 2.2 — Making measurements and analysing data. This lesson defines precision, accuracy and resolution, anchors the (value ± uncertainty) grammar that every numerical answer in the H556 papers must satisfy, and establishes the absolute / percentage uncertainty forms used throughout Module 2.2 and the Practical Endorsement. (Refer to the official OCR H556 specification document for exact wording.)

Having distinguished random from systematic errors in the previous lesson, we now put numbers on the idea of uncertainty. When a physicist writes

$l = (1.250 \pm 0.005)\,\text{m}$

they are making a precise, mathematically meaningful statement: "I believe the true length of this rod lies somewhere in the interval $1.245\,\text{m}$ to $1.255\,\text{m}$ , and my best single estimate of it is $1.250\,\text{m}$ ". This lesson teaches you how to derive, write and interpret such statements.

We cover four things: the distinction between precision and accuracy (already met qualitatively, now made numerical); resolution and how it bounds the uncertainty of a single reading; absolute vs percentage uncertainty; and the number of significant figures appropriate for a quoted uncertainty.

Key definition. The uncertainty of a measurement is a quantitative statement of the interval within which the true value is believed to lie. The value ± uncertainty pair is the irreducible grammar of every quantitative claim in physics — neither half on its own is enough.

Precision vs Accuracy — the Numerical View

The precision of a measurement (or a set of measurements) is the closeness of agreement between repeated readings under unchanged conditions: it is about spread. The accuracy of a measurement is the closeness of agreement between a (mean) measured value and the true (accepted) value: it is about bias.

These two attributes are genuinely independent. A balance with a sticky pan but a perfectly calibrated zero may give tightly clustered readings that all sit 2 g above the true mass — precise but inaccurate. A pair of cheap rulers used by a careful student may give widely scattered readings that average to the true length — accurate but imprecise.

The canonical dartboard 2×2 picture cements the four cases.

The key numerical translations are:

Precise $\Longleftrightarrow$ small spread of repeated readings; quantified by the standard deviation $\sigma$ (or by half-range at A-Level).
Accurate $\Longleftrightarrow$ small distance from the true value to the mean of repeated readings; quantified by $|\bar{x} - x_{\text{true}}|$ .

In the language of Lesson 5: precision is the absence of significant random error; accuracy is the absence of significant systematic error. The two are decoupled because they have different physical origins — random errors come from noise and resolution limits, while systematic errors come from calibration, zero, parallax and method.

Resolution and the Uncertainty of a Single Reading

The resolution of an instrument is the smallest change in the measured quantity that the instrument can reliably register. It is fundamentally a property of the instrument, not the operator. Resolution sets the minimum uncertainty of a single reading; the actual uncertainty is normally at least the resolution and sometimes larger (if operator factors dominate).

Convention for the uncertainty of a single reading

Analogue scale (ruler, thermometer, voltmeter dial): uncertainty is half the smallest division.
- $30\,\text{cm}$ ruler with $1\,\text{mm}$ divisions $\rightarrow \pm 0.5\,\text{mm}$ on a single reading.
- Alcohol thermometer with $1\,^{\circ}\text{C}$ divisions $\rightarrow \pm 0.5\,^{\circ}\text{C}$ .
Digital instrument (multimeter, balance, timer display): uncertainty is the last displayed digit (some textbooks halve it; OCR accepts either provided you are consistent).
- Digital multimeter reading $2.35\,\text{V} \rightarrow \pm 0.01\,\text{V}$ .
- Digital balance reading $48.73\,\text{g} \rightarrow \pm 0.01\,\text{g}$ .
Stopwatch: operator reaction time dominates over instrument resolution. Even if the display reads to $0.01\,\text{s}$ , the realistic uncertainty per start/stop is $\pm 0.1$ to $\pm 0.2\,\text{s}$ .
Vernier instruments (calliper, micrometer): uncertainty is half the vernier resolution. A vernier callipers with a $0.05\,\text{mm}$ resolution has a single-reading uncertainty of $\pm 0.025\,\text{mm}$ — round to $\pm 0.03\,\text{mm}$ if you prefer one significant figure.

A digital-balance reading and an analogue-thermometer reading are governed by different conventions even when their numerical resolutions look similar; this catches candidates in OCR exams who default to "half the smallest division" everywhere.

Worked Example 1 — quoting single readings

State the measurement and its absolute uncertainty for each.

(a) A $30\,\text{cm}$ ruler reading $148\,\text{mm}$ for the length of a metal rod. (b) A digital voltmeter showing $3.27\,\text{V}$ . (c) A human-operated stopwatch showing $14.58\,\text{s}$ . (d) A vernier callipers (resolution $0.05\,\text{mm}$ ) reading $12.35\,\text{mm}$ . (e) A micrometer (resolution $0.01\,\text{mm}$ ) reading $0.34\,\text{mm}$ .

Solution.

(a) $(148 \pm 0.5)\,\text{mm}$ — half the smallest division on an analogue ruler. (b) $(3.27 \pm 0.01)\,\text{V}$ — last digit of a digital display. (c) $(14.58 \pm 0.2)\,\text{s}$ — reaction time dominates, not the display resolution. (d) $(12.35 \pm 0.025)\,\text{mm}$ , conventionally rounded to $(12.35 \pm 0.03)\,\text{mm}$ . (e) $(0.34 \pm 0.005)\,\text{mm}$ or $(0.34 \pm 0.01)\,\text{mm}$ — one or two readings of $0.01\,\text{mm}$ , depending on convention.

PAG 1 ("measurement of physical quantities") tests exactly these conventions — the markers want to see you choose the right rule for the right instrument, and to write the uncertainty alongside the value, not separately.

Uncertainty from Repeated Measurements

When you take $N$ repeated readings under unchanged conditions, the best estimate of the true value is the mean

$\bar{x} = \frac{1}{N}\sum_{i=1}^{N} x_i.$

The uncertainty in the mean can be estimated by the half-range method:

$\Delta \bar{x} \approx \frac{x_{\max} - x_{\min}}{2}.$

This is conservative (it tends to overstate uncertainty compared to the rigorous statistical $\sigma/\sqrt{N}$ standard-error formula used at undergraduate level), but it is easy to compute by hand and matches OCR mark-scheme expectations. At undergraduate physics you will replace it with the standard error of the mean

$\text{SE} = \frac{\sigma}{\sqrt{N}}$

where $\sigma$ is the sample standard deviation. OCR mark schemes accept half-range up to A-Level; you should know the standard-error form exists but you will not be examined on it.

Worked Example 2 — mean and half-range

A student times $10$ oscillations of a pendulum six times and records (in seconds):

$14.5,\ 14.8,\ 14.3,\ 14.7,\ 14.6,\ 14.4.$

Calculate the mean time for 10 oscillations and its absolute uncertainty.

Solution.

$\bar{t} = \frac{14.5 + 14.8 + 14.3 + 14.7 + 14.6 + 14.4}{6} = \frac{87.3}{6} = 14.55\,\text{s}.$

Range = $14.8 - 14.3 = 0.5\,\text{s}$ , so half-range $\Delta \bar{t} \approx 0.25\,\text{s}$ .

Quoted result: $(14.55 \pm 0.25)\,\text{s}$ , or to OCR's significant-figure convention $(14.5 \pm 0.3)\,\text{s}$ .

The period of a single oscillation is $T = 14.55 / 10 = 1.455\,\text{s}$ , with uncertainty $0.25/10 = 0.025\,\text{s}$ , so $T = (1.46 \pm 0.03)\,\text{s}$ .

Dividing by $N$ when timing oscillations is why OCR practical methods always recommend measuring a "bunch" of cycles, fringes or periods rather than one at a time. The percentage uncertainty in the per-cycle answer drops by exactly the factor of $N$ that you divided by.

Absolute vs Percentage Uncertainty

Uncertainty is reported in two equivalent forms:

Absolute uncertainty $\Delta x$ — same units as the quantity. Example: $l = (1.250 \pm 0.005)\,\text{m}$ .
Percentage uncertainty $\dfrac{\Delta x}{x} \times 100\%$ — dimensionless. Example: $l = 1.250\,\text{m}\,\pm\,0.4\%$ .

Both forms carry the same physical information. Which one you use depends on the context.

Context	Preferred form
Stating a single measurement	Absolute
Comparing different measurements	Percentage
Propagating through $\times$ or $\div$ (next lesson)	Percentage
Propagating through $+$ or $-$ (next lesson)	Absolute
Quoting a final result	Absolute, with the value rounded to the matching decimal place

The conversion rules are

$\frac{\Delta x}{x}\times 100\% = \text{percentage uncertainty}, \qquad \Delta x = \frac{\text{percentage uncertainty}}{100} \times x.$

Worked Example 3 — converting between forms

(a) $l = (45.0 \pm 0.5)\,\text{cm}$ . Find the percentage uncertainty.

$\frac{0.5}{45.0}\times 100 = 1.11\% \approx 1.1\%.$

(b) $t = 3.2\,\text{s}$ with a percentage uncertainty of $2\%$ . Find the absolute uncertainty.

$\Delta t = \frac{2}{100} \times 3.2 = 0.064\,\text{s} \approx 0.06\,\text{s}.$

Quote $t = (3.20 \pm 0.06)\,\text{s}$ .

Why Percentage Uncertainty Matters

The reason percentage uncertainty is so useful is that two measurements with the same absolute uncertainty can have wildly different reliability. A length of $1000 \pm 1\,\text{mm}$ is extraordinarily precise ( $0.1\%$ uncertainty). A length of $2 \pm 1\,\text{mm}$ is barely meaningful ( $50\%$ uncertainty). The percentage form captures this at a glance, and is the form in which uncertainties combine for multiplication and division (next lesson).

Worked Example 4 — identifying the biggest percentage uncertainty

In a Young modulus practical a student measures:

Diameter of wire $d = (0.34 \pm 0.01)\,\text{mm}$ (micrometer).
Length of wire $L = (1.500 \pm 0.005)\,\text{m}$ (metre ruler).
Mass added $m = (2.000 \pm 0.001)\,\text{kg}$ (digital balance).
Extension $x = (1.8 \pm 0.1)\,\text{mm}$ (vernier scale on Searle apparatus).

Identify the largest percentage uncertainty.

Solution.

Quantity	Value	$\Delta$	$\Delta/x \times 100\%$
$d$	$0.34\,\text{mm}$	$0.01\,\text{mm}$	$2.94\%$
$L$	$1.500\,\text{m}$	$0.005\,\text{m}$	$0.33\%$
$m$	$2.000\,\text{kg}$	$0.001\,\text{kg}$	$0.05\%$
$x$	$1.8\,\text{mm}$	$0.1\,\text{mm}$	$5.56\%$

Extension $x$ has the largest raw percentage uncertainty at $\approx 5.6\%$ . But — and this is the move that distinguishes the top band — the diameter $d$ is squared in the cross-sectional area $A = \pi d^{2}/4$ , so its effective contribution to the final percentage uncertainty on $E$ is

$2 \times 2.94\% = 5.88\%,$

slightly larger than $x$ . We will see exactly this propagation in the next lesson. For now: when a quantity is squared (cubed, square-rooted) in the final formula, weight its percentage uncertainty by the power before comparing.

Significant Figures in Uncertainties

OCR's rule (and that of most physics courses):

Uncertainties are quoted to 1 significant figure (occasionally 2 if justified).
The measured value is rounded so that its last decimal place matches that of the uncertainty.

Raw	Rounded correctly
$1.23456 \pm 0.0234\,\text{m}$	$1.23 \pm 0.02\,\text{m}$
$9.8712 \pm 0.43\,\text{m s}^{-2}$	$9.9 \pm 0.4\,\text{m s}^{-2}$
$273.152 \pm 5.7\,\text{K}$	$273 \pm 6\,\text{K}$
$0.00472 \pm 0.00018\,\text{A}$	$0.0047 \pm 0.0002\,\text{A}$

Precision, Accuracy and Uncertainty

Precision, Accuracy and Uncertainty

Precision vs Accuracy — the Numerical View

Resolution and the Uncertainty of a Single Reading

Convention for the uncertainty of a single reading

Worked Example 1 — quoting single readings

Uncertainty from Repeated Measurements

Worked Example 2 — mean and half-range

Absolute vs Percentage Uncertainty

Worked Example 3 — converting between forms

Why Percentage Uncertainty Matters

Worked Example 4 — identifying the biggest percentage uncertainty

Significant Figures in Uncertainties

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