Accuracy, Precision & Uncertainty

Open any CODATA table and this module's vocabulary is the entire plot. The gravitational constant $G$ is the cautionary tale: individual experiments report uncertainties of a few parts in $10^5$ — magnificent precision — yet laboratories disagree by more than ten times their quoted bars. Tight clusters, mutually off-centre: unfound systematic errors are hiding, and metrologists have spent two decades hunting them with independent methods — torsion balances versus atom interferometers — for exactly the reason your syllabus teaches: unrelated designs rarely share a bias. The same discipline decides discoveries. When Fermilab's muon $g\!-\!2$ experiment reported a discrepancy with theory, the whole argument was a fight between error budgets — every known systematic listed and defended before anyone said "new physics". A GPS fix, a blood assay, a diagnostic test's false-negative rate: each is a $\bar{x} \pm \Delta x$ sentence in different clothes. The habit you are building — how wrong could this be, and in which way? — is the professional reflex.

Picture four dartboards. The bullseye is the true value; each dart is one measurement. The first player lands a tight cluster dead centre: accurate and precise. The second lands a cluster just as tight, but high and right: precise, not accurate — something in the technique pushed every dart the same way. The third's darts spread widely but centre on the bullseye: accurate on average, not precise. The fourth sprays darts off-centre: neither.

Now translate darts into physics. What spreads a cluster? Random errors: unpredictable, both-ways fluctuations — your stopwatch thumb firing early this time and late the next, draughts nudging the balance pan, the judgement call when a pointer sits between markings. Their defining signature: they average toward zero, because overshoots and undershoots cancel across many readings. What shifts a cluster off-centre? Systematic errors: flaws that push every reading the same way. A metre rule with a worn zero end. A voltmeter reading 0.2 V with nothing connected. A slow stopwatch. Parallax from always sitting slightly to the left. Student B's pendulum length, short by one bob-radius in every trial. The killer property: repetition is blind to them — every dart inherits the same shift, so the average inherits it too. You cannot average your way out of a systematic error; you can only find it and remove it, by calibrating, subtracting a measured zero error, or redesigning the measurement. One more instrument word belongs in the picture: resolution — the smallest change an instrument can display (1 mm for a metre rule, one last-digit unit for anything digital). Resolution caps how fine your readings can be; it promises nothing about how true they are. A 0.01 mm digital caliper that was dropped last week resolves beautifully and lies consistently.

A professional measurement is a claim with a confidence band attached:

where: $\bar{x}$ — the best estimate (for repeated readings, their mean); $\Delta x$ — the absolute uncertainty, in the same units as $x$. Read it as a promise: "the true value very likely lies between $\bar{x} - \Delta x$ and $\bar{x} + \Delta x$." So $L = 252.0 \pm 0.4\ \mathrm{mm}$ claims the truth sits between 251.6 and 252.4 mm. The uncertainty is not an apology — it is the most useful part of the sentence, telling the reader exactly how hard they may lean on the value. The band is also how physicists decide whether two results agree: they are consistent when their bands overlap. Measure $g$ as $9.79 \pm 0.03$ while your friend gets $9.85 \pm 0.02$: your band tops out at $9.82$, theirs starts at $9.83$ — no overlap, so at least one experiment carries an unaccounted error. This overlap logic is exactly how exams phrase "are these results consistent?" — and how collaborations decide whether a discrepancy is a discovery or a mistake. Where does $\Delta x$ come from for a single reading? From the instrument, by convention: an analogue pointer can honestly be judged to about half the smallest division; on a digital display everything past the last shown digit is invisible, so the uncertainty is one unit of that digit:

where: the smallest division is the instrument's resolution (its least count in Indian-board language). Board conventions differ: CBSE, JEE and NEET usually treat the full least count as the reading error (metre rule → $\pm 1$ mm); CIE 9702 accepts half a division per scale judgement — and a rule-measured length involves two judgements (both ends), totalling a full millimetre anyway. Do not fight your board's convention; state it and use it. Repetition upgrades a single reading into a data set: the best estimate is the mean, and a serviceable exam-level uncertainty is half the spread:

where: $x_{\max} - x_{\min}$ — the range of the readings. Read it physically: the mean lets the two-way random wobbles fight and partially cancel, while the half-range honestly reports how big they were. More readings pin down the centre of the scatter more reliably — why every practical mark scheme rewards "repeat and average". (University work replaces the half-range with proper statistics and the mean's uncertainty shrinks roughly like $1/\sqrt{n}$; your board wants the idea, not the formula: more repeats, tighter mean.) But watch what the mean does not do — this deserves a derivation, because it is the single most examinable idea in the module.

The dartboard, formalised: the cluster's spread ($\pm 0.4$) is your precision; its centre's displacement from truth ($2.0$) is your inaccuracy — and repetition only ever fixes the first.

Random errors in the wild: fluctuating last digits on a multimeter, scatter in repeated stopwatch times, the judgement wobble between rulings, environmental noise. Treatment: repeat and average; use resolution finer than the effect you chase; time twenty oscillations, not one, so the timing wobble is shared across twenty periods. Systematic errors in the wild: zero error (reads non-zero at zero input — the vernier and screw-gauge sub-species get their own module later this chapter); calibration error (a stretched tape, a slow stopwatch); parallax (a fixed slanted view, misaligning pointer and scale the same way every time); technique offsets (pendulum length to the top of the bob; counting from "one" instead of "zero", quietly turning 20 oscillations into 19). Treatment — the word that earns marks — calibrate: check against a standard, measure and subtract the zero error, view scales perpendicularly (good ammeters put a mirror behind the needle for this), or redesign so the offset cancels. One subtlety examiners love: human reaction time with a stopwatch. The trial-to-trial scatter of your reaction is random; a consistent anticipate-the-start, lag-the-stop bias is systematic. The full-mark answer names both parts — and the fix is a fiducial marker at the centre of the swing, where the bob moves fastest and the crossing is sharp.

The third row is the exam's favourite: a smaller uncertainty does not certify a better experiment. Student B lives in that row.

Board level — timing a pendulum and reporting the period (CIE/ICSE practical pattern)

A student times 20 complete oscillations of a pendulum three times with a digital stopwatch (resolution 0.01 s): 30.2 s, 30.6 s, 30.4 s. Human reaction time is about 0.2 s. Report the period with its uncertainty, classify the errors present, and state one improvement.

Step 1. Best estimate of the 20-oscillation time: mean $= (30.2 + 30.6 + 30.4)/3 = 91.2/3 = 30.4\ \mathrm{s}$.

Step 2. Uncertainty of that time from the spread, equation (3): $\Delta t = (30.6 - 30.2)/2 = 0.2\ \mathrm{s}$ — matching the quoted reaction time and dwarfing the stopwatch's 0.01 s resolution. The human, not the instrument, sets the uncertainty here; writing that sentence is typically worth a mark.

Step 3. Divide by 20 — both the value and the uncertainty: $T = 30.4/20 = 1.52\ \mathrm{s}$ and $\Delta T = 0.2/20 = 0.01\ \mathrm{s}$. Report $\boxed{T = 1.52 \pm 0.01\ \mathrm{s}}$. This is why we time 20 oscillations: the reaction-time wobble is paid once per timing, then shared across 20 periods.

Step 4. Classify: trial-to-trial scatter (reaction jitter, ±both ways) is random. Candidate systematics: counting from "one" instead of "zero" (turns 20 swings into 19, shifting every trial), or timing from an extreme where the bob lingers.

Step 5. Improvement: a fiducial marker at the lowest point of the swing, where the bob moves fastest; count "zero" as it first crosses.

Competitive level — four data sets, one diagnosis (JEE-Main pattern)

The accepted value is $g = 9.81\ \mathrm{m/s^2}$. Four students each measure $g$ four times: *A: 9.80, 9.82, 9.81, 9.79 · B: 9.95, 9.44, 10.12, 9.68 · C: 9.42, 9.44, 9.43, 9.41 · D: 9.20, 10.30, 9.60, 10.00. Which data set is precise but not accurate, and what type of error dominates it?*

Step 1. Triage precision first — precision is just the spread, no truth needed. Half-ranges: A: $(9.82-9.79)/2 = 0.015$; B: $(10.12-9.44)/2 = 0.34$; C: $(9.44-9.41)/2 = 0.015$; D: $(10.30-9.20)/2 = 0.55$. Tight clusters: A and C.

Step 2. Now accuracy — compare means with 9.81. A: mean $= 39.22/4 = 9.805$, essentially on target. C: mean $= 37.70/4 = 9.425$, about $0.385$ low — roughly 26 half-ranges away from truth. A is accurate and precise; C is precise and badly inaccurate.

Step 3. Answer: $\boxed{\text{data set C — dominated by a systematic error}}$. A tight cluster far from the accepted value is the fingerprint of a one-way shift, not random scatter. For completeness: B is roughly accurate on average ($39.19/4 \approx 9.80$) but imprecise; D is neither. Timing note: 60–90 seconds. Spread first (eyeball max minus min), means second — and only for the tight clusters. The precise-but-inaccurate answer only ever hides in a tight cluster sitting off the accepted value.

Swapping the definitions under pressure. The single-mark classic. Anchor them asymmetrically: accuracy needs the true value in its sentence ("close to the true/accepted value"); precision must not mention truth at all ("repeated readings close to each other"). A definition of precision containing "true value" scores zero.

Calling zero error random. A zero error is the textbook systematic error — it shifts every reading identically and must be subtracted, not averaged over. If a question says an ammeter reads 0.2 A with the circuit open, correct every reading by −0.2 A; do not fold it into the uncertainty.

"Take more readings" as the fix for everything. Repetition answers random error only. If the question plants a stretched tape or a slow stopwatch, the expected fix is calibration, technique or instrument change — "repeat and average" earns nothing against a systematic fault. Match the medicine to the disease.

Parallax named but not explained. Boards want the mechanism, not the word: the line of sight is not perpendicular to the scale, so the pointer appears displaced against the markings. Add the cure — view squarely, use the mirror strip behind the needle — and the mark is safe.

Quoting uncertainty with false precision. Quote $\Delta x$ to 1 significant figure (2 when the leading digit is 1) and cut the value at the same decimal place — $9.8 \pm 0.3$, never $9.8134 \pm 0.3$. The full reporting grammar lives in the significant-figures module; examiners treat a mismatched pair as an error.

Repeatability vs reproducibility (CIE/IB). Same experimenter, same apparatus → repeatability. Different team, different method, same answer → reproducibility — tested directly at AS level, and the strongest weapon against hidden systematics, because unrelated setups rarely share a bias.

Using the wrong $g$ convention. Questions state their constant: CBSE and NEET commonly command $g = 9.8\ \mathrm{m/s^2}$ (sometimes 10), JEE-Main states it per paper, CIE and IB use 9.81. Using your memorised favourite instead of the paper's stated value is a self-inflicted systematic error — exactly the irony the examiner intends you to appreciate.

Texts. NCERT Physics Part 1, Ch. 1 (Units and Measurement) — the CBSE/NEET framing of errors and least counts · CIE 9702 syllabus, AS Practical Skills appendix — the accuracy/precision/repeatability definitions mark schemes quote · IB Physics guide (2025), Tool 3 "Processing uncertainties" · J. R. Taylor, An Introduction to Error Analysis, Ch. 1–4 — the classic treatment one level up.

Provenance. All conventions as per NCERT Ch. 1, CIE 9702 practical guidance and the IB 2025 guide. A verifier should recompute: worn-rule set $252.4+251.7+252.0+252.3+251.6 = 1260.0 \Rightarrow \bar{x} = 252.0$, deviations sum to 0, half-range $0.4$, offset $2.0 = 5\times$ half-range; pendulum $91.2/3 = 30.4$, $(30.6-30.2)/2 = 0.2$, $30.4/20 = 1.52$, $0.2/20 = 0.01$; data sets $39.22/4 = 9.805$, $37.70/4 = 9.425$, $39.19/4 = 9.7975$, half-ranges $0.015, 0.34, 0.015, 0.55$; overlap check $9.79+0.03 = 9.82 < 9.85-0.02 = 9.83$.

Prerequisites. Physical Quantities & Measurement. Leads to. Absolute, Relative & Percentage Errors (turning this module's vocabulary into computable numbers) · Significant Figures, Rounding & Reporting (the grammar for writing values and uncertainties down).