University Physics I · Mathematical & Physical Foundations · 1.3

Significant Figures & Uncertainty

A reading tells you where a value probably sits, and how widely “probably” spreads. Learn to find that spread, carry it through a calculation, and stop quoting digits at the point it stops supporting them.

Open the lab ↓Practice & worked examples ↓

Build the model

Connect the measurement to the mechanism.

A measurement is a claim about an interval, not a point. Every instrument has a smallest step it can resolve, and repeating a reading shows how far the result moves when nothing about the object has changed. Random effects scatter readings either side of the true value and average away as 1/√N; systematic effects shift every reading the same way and survive any number of repeats.

So a result is finished only when it is written as a best estimate, an uncertainty in the same unit, and a stated coverage: x = x̄ ± u. Derived quantities inherit those intervals through the partial derivatives of the formula — absolute uncertainties combine for sums, relative ones for products and powers. Significant figures are only the shorthand: once u is known, round u first, then quote the value to the same decimal place.

The rest of what the calculator prints is decoration.

Simple definition: The uncertainty of a measurement is a number in the same unit as the value, saying how far the true value could reasonably sit from it — by convention one standard deviation of the spread, unless a coverage factor is stated.
Example: L = 50.00 ± 0.05 mm puts the length within 0.05 mm of 50.00 mm — a relative uncertainty of 0.05/50.00 = 0.0010, or 0.10%. At k = 1 that interval holds the value about two times in three.

Best estimate and scatterx̄ = (1/N) Σ xᵢ · s = √[ Σ(xᵢ − x̄)² / (N − 1) ]

s is the spread of one reading. More data pins s down; it does not make s smaller.

s carries the unit of x; N − 1 because x̄ came from the same N readings

Uncertainty of the meanu(x̄) = s / √N

Only random scatter obeys this. A systematic offset sits inside x̄ and survives every repeat.

Same unit as x. Four times the readings halve u; nine times cut it to a third.

One reading, from resolutionu = a/√3

Use it when one reading is all you have and the value is equally likely anywhere inside the step. On an analogue scale, interpolate first, then judge a.

a is the half-width of the interval. Digital step d → a = d/2, so u ≈ 0.29d.

General propagationu_f² = (∂f/∂x)² uₓ² + (∂f/∂y)² u_y² + …

The partial derivative says how hard f leans on that input. Correlated inputs need extra cross terms.

Independent inputs only. Every term carries the units of f squared.

Sums and differencesf = ax ± by → u_f = √[(a uₓ)² + (b u_y)²]

A difference of two close values keeps u but loses relative precision — the small-difference trap.

Absolute uncertainties combine. The two terms must already share a unit.

Products, quotients, powersf = k xᵐ yⁿ → u_f/|f| = √[(m uₓ/|x|)² + (n u_y/|y|)²]

A quantity entering squared contributes twice its relative uncertainty, so measure that one best.

Relative uncertainties combine; each exponent multiplies its own term.

Resolution is the smallest step, not a guarantee of truth

Every instrument reports in steps. A metre rule marked in millimetres, a caliper stepping in 0.01 mm, a timer ticking in 0.001 s: each can only place a value inside one step. When a single reading is all you have, that step defines an interval of half-width a, and if the value is equally likely anywhere inside it the standard uncertainty is a/√3 — about 0.29 of a step. Resolution is a floor on how well one reading can be known. It says nothing about whether the instrument is telling the truth: a caliper displaying 0.01 mm steps while sitting 0.15 mm off zero is precise and wrong.

Repeat the measurement and watch it move

Time the same pendulum ten times and you get ten different numbers. The best estimate is the mean x̄. The spread is the sample standard deviation s, with N − 1 in the denominator because x̄ was taken from the same data and one degree of freedom is already spent. s answers a question about a single reading — how far does one timing wander? — and it does not shrink as data accumulates; it only becomes better determined. What shrinks is the uncertainty of the mean, u(x̄) = s/√N. Four times the readings halve it, nine times reduce it to a third. That √N is also why brute repetition stops paying: going from 100 timings to 400 buys one more factor of two.

Accuracy and precision fail in different directions

Precision is agreement among your own readings; accuracy is agreement with the true value. They are independent, and a measurement can fail at either. Random effects — reaction time, draughts, a last digit flickering — push readings both ways, so they appear as scatter and average away as 1/√N. Systematic effects — a stretched tape, an uncorrected zero, a clock running 0.5% fast, a scale read at an angle — shift every reading by nearly the same amount. Repetition cannot find them, because they sit inside x̄ and barely touch s. They are exposed only by changing something: calibrate against a standard, measure the quantity a second way, reverse the instrument end for end, or check the zero before and after the run.

Write the result as an interval, and say what it covers

A finished result is x = x̄ ± u, with u in the same unit as x̄ — and the ± is ambiguous until you say what it covers. The standard uncertainty is the k = 1 case: for an approximately normal distribution it brackets the true value about 68% of the time. Multiply by a coverage factor, usually k = 2, for an expanded uncertainty covering roughly 95%, and state which one you quoted. The relative uncertainty u/|x| is the form that travels through multiplication, and it is what shows which input actually limits the result. When one input contributes several times the relative uncertainty of another, the smaller term is already free, and a better instrument for it buys nothing.

Derived quantities inherit the intervals

Density comes from a mass and two lengths; a spring constant from a force and an extension. If f depends on independent inputs x, y, …, its uncertainty follows from the partial derivatives: u_f² = (∂f/∂x)²uₓ² + (∂f/∂y)²u_y² + … . Each partial derivative measures how hard f leans on that input, and the squares mean the terms add in quadrature rather than linearly, because independent errors are as likely to cancel as to reinforce. Two shortcuts follow, and they cover most of this course. For sums and differences, absolute uncertainties combine in quadrature. For products, quotients and powers, relative uncertainties do, with each exponent multiplying its own term — which is why the quantity that enters squared usually deserves the best instrument.

Round the uncertainty first, then the value

Take a cylinder: m = 44.75 ± 0.02 g, d = 12.70 ± 0.02 mm, L = 50.00 ± 0.05 mm. With V = πd²L/4 the calculator gives ρ = m/V = 7065.2 kg m⁻³. The relative terms are 2(0.02/12.70) = 0.315% for the diameter, 0.05/50.00 = 0.100% for the length and 0.02/44.75 = 0.045% for the mass; in quadrature they give 0.333%, so u_ρ = 0.00333 × 7065.2 ≈ 24 kg m⁻³. The diameter, entering squared, carries almost all of it; a better balance would spend on the caliper and nothing on the scale. Now report. Round u to one significant figure, or two when one would discard too much — here 24 kg m⁻³ — then round the value to that same decimal place: ρ = 7065 ± 24 kg m⁻³, or (7.065 ± 0.024) × 10³ kg m⁻³.

Change one variable at a time

Make the relationship visible.

Interactive model

Scatter of one reading s80 ms

Systematic uncertainty b10 ms

Readings averaged N25

Drag N from 25 to 100 and the dashed random curve halves from 16 to 8 ms while the solid one only eases from 19 to 13 ms, because it can never sink below the floor b — then raise b to 40 ms, where the break-even N falls to 4 and averaging past four readings buys almost nothing.

RANDOM PART s/√N16.0 ms

COMBINED u18.9 ms

EXPANDED U (k = 2)37.7 ms

BREAK-EVEN N64 readings

Live interpretationRANDOM PART s/√N: 16.0 ms. COMBINED u: 18.9 ms. EXPANDED U (k = 2): 37.7 ms. BREAK-EVEN N: 64 readings

Catch the common trap

Explain before calculating.

A pendulum's period is timed 25 times with a stopwatch that runs 0.5% fast. The readings scatter with s = 0.08 s. What changes if 100 timings are taken?

Choose an answer to test the model.

Practice & worked examples

Reason from the model, then test the result.

EasyA digital timer resolves 0.001 s and reads 1.842 s for one swing of a pendulum. That single reading is all you have. Quote the result with its standard uncertainty.

One reading, so the interval comes from the display step: d = 0.001 s, and the true value can sit anywhere within half a step of what is shown — a half-width a = d/2 = 0.0005 s.
Nothing favours the middle of that step, so treat the value as equally likely anywhere inside it: u = a/√3 = 0.0005/1.732 = 0.000289 s.
Round u to one significant figure, u = 0.0003 s, then round the reading to that same decimal place — which means writing the trailing zero the display never showed.

Answert = 1.8420 ± 0.0003 s (k = 1). That is only the resolution floor: start-stop reaction time adds a random spread some two hundred times larger, which repeats can measure and this single reading cannot.

MediumTiming 10 swings of that pendulum five times gives 18.62, 18.71, 18.55, 18.68 and 18.59 s. Find the mean, the spread of one reading, the uncertainty of the mean, and quote the period of one swing.

Mean of the 10-swing times: x̄ = 93.15/5 = 18.630 s.
Deviations from the mean are −0.01, +0.08, −0.08, +0.05 and −0.04 s; their squares sum to 0.0170 s².
Divide by N − 1 = 4, not by 5, because x̄ came from these same five readings: s = √(0.0170/4) = √0.00425 = 0.0652 s. That is the spread of one timing.
The mean is better determined than any single reading: u(x̄) = s/√N = 0.0652/√5 = 0.0292 s.
Dividing by 10 swings divides value and uncertainty alike: T = 1.8630 s with u_T = 0.00292 s. Round u first, to 0.003 s, then match the value to that decimal place.

AnswerT = 1.863 ± 0.003 s (k = 1), a relative uncertainty of 0.16%. Timing ten swings rather than one bought that: the reaction-time error is paid once per run, not once per swing.

HardA longer pendulum, timed by photogate, gives L = 0.9420 ± 0.0010 m and T = 1.9470 ± 0.0025 s. Use g = 4π²L/T² to find g with its uncertainty, and say which measurement is worth improving.

g = 4π²L/T² is a product of powers, so relative uncertainties combine — and T enters to the power −2, so its term is doubled.
Best estimate: g = 39.4784 × 0.9420 / 1.9470² = 37.1887 / 3.79081 = 9.8102 m s⁻².
Relative terms: u_L/L = 0.0010/0.9420 = 0.106%, and 2u_T/T = 2 × 0.0025/1.9470 = 0.257%.
In quadrature: u_g/g = √(0.106² + 0.257²) = 0.278%, so u_g = 0.00278 × 9.8102 = 0.027 m s⁻². Round u to two significant figures here, since one would jump it to 0.03.
The timing term is 2.4 times the length term, so it carries 85% of the variance. Halving u_T would pull u_g down to 0.016 m s⁻²; measuring L perfectly would only reach 0.025 m s⁻².

Answerg = 9.810 ± 0.027 m s⁻² (k = 1), a relative uncertainty of 0.28%. Buy a better clock, not a better ruler.