Significant Figures in Physics Lab Reports: A Complete Working Guide

A physics lab report touches sig figs at every stage: reading an instrument (which sets your starting uncertainty), calculating a result (which propagates that uncertainty through the math), and comparing your answer to an accepted value (which quantifies how well the experiment actually worked). Get the instrument-reading convention right, know which physical constants in your formula are exact and which are themselves measured, and carry the uncertainty through correctly — and the rest of a lab report’s sig fig requirements follow automatically.

This is the second subject-specific pillar on this site, after Significant Figures in Chemistry. Where that page’s standout fact was about atomic mass, this one’s is about physical constants: since 2019, several of the constants a physics formula relies on are exactly defined, with zero uncertainty — while others, used just as often, are genuinely experimental values still being refined today. Knowing which is which changes how you treat them in a calculation. Everything below builds directly on reading instrument uncertainty and error propagation covered earlier in this site — this page is where those ideas meet an actual lab bench.

<!– BLOCK: B02 – Inline Mini-Calculator –> <!– DEV NOTE: Embed the Error Propagation Calculator here, pre-loaded with a physics-formula preset. Shortcode: [sfc_error_propagation_calculator preset=”pendulum_g” show_both_methods=”true”]. Reuses the same tool built for Pillar 6, with a physics-specific starting example instead of a generic one. –>

[Live Error Propagation Calculator embeds here] — Pre-loaded with the pendulum example below; change the measured length, period, and their uncertainties to see how the calculated value of g and its uncertainty change.

Three Places Sig Figs Enter a Physics Lab Report

Reading uncertainty: where it actually comes from. For an analog scale (ruler, protractor, analog dial), the conventional reading uncertainty is half the smallest marked division — you can reliably estimate a position to about that precision by eye, but not much finer. For a digital display, the convention is ±1 in the last displayed digit, unless the manufacturer states a larger accuracy specification (many digital instruments are less accurate than their display resolution suggests, and the stated spec should win when it’s larger). For manually triggered timing (a stopwatch started and stopped by hand), the dominant uncertainty is almost never the stopwatch’s own resolution — it’s human reaction time, typically 0.1–0.3 seconds, which is usually far larger than the 0.01 s the display shows.

Which constants in your formula are exact — and which aren’t. Since the 2019 redefinition of the SI, seven physical constants have been fixed by international agreement to have exact values with zero uncertainty: the speed of light, the Planck constant, the elementary charge, the Boltzmann constant, the Avogadro constant, the caesium-133 hyperfine transition frequency, and the luminous efficacy of a defined visible radiation. Treat these exactly the way you’d treat an exact number — they never limit a calculation’s precision. Most other constants a physics formula uses are not on this list, including the gravitational constant, G, which remains a genuinely experimental value, redetermined periodically, and is in fact one of the least precisely known constants in physics. Mathematical constants like π are a separate case entirely: they’re not physical measurements at all, so they carry no uncertainty and can be used to as many digits as a calculation needs.

Propagating uncertainty through a physics formula. Most physics-lab formulas are combinations of multiplication, division, and powers — our error propagation guide covers the general rules; the one that trips people up most in a physics context specifically is the power rule, because so many common formulas square or take a square root of a measured quantity (period, distance, velocity). Squaring a measured value doubles its relative uncertainty’s contribution to the result — a fact easy to miss if you’re only thinking in sig-fig-counting terms rather than actual propagated uncertainty. See Example 3.

Worked Examples

Example 1 — Reading uncertainty, two ways

Analog: a ruler marked in millimeters is used to measure a block’s length, read as 45 mm. Conventional uncertainty: half the smallest division = ± 0.5 mm → report as 45.0 ± 0.5 mm.

Digital: a digital caliper displays 12.47 mm. Absent a stated manufacturer accuracy spec, conventional uncertainty is ± 1 in the last digit = ± 0.01 mm → report as 12.47 ± 0.01 mm.

Neither convention claims the instrument is accurate — only that this is the precision it can be read to. An uncalibrated digital caliper can still read 12.47 ± 0.01 mm and be systematically wrong, exactly as covered in accuracy vs. precision.

Example 2 — Exact constants vs. a genuinely measured one

Exact, by definition (2019 SI redefinition):

• Speed of light: c = 299,792,458 m/s, exactly
• Planck constant: h = 6.62607015 × 10⁻³⁴ J·s, exactly
• Elementary charge: e = 1.602176634 × 10⁻¹⁹ C, exactly
• Avogadro constant: N_A = 6.02214076 × 10²³ /mol, exactly

Not exact — a real, still-improving experimental value:

• Gravitational constant: G ≈ 6.674 × 10⁻¹¹ N·m²/kg², with a relative uncertainty around 2 × 10⁻⁵ — roughly ten thousand times less precisely known than c, h, or e, which have zero uncertainty by definition.

If a calculation uses c, h, e, or N_A, treat them as exact — infinite significant figures, never the limiting factor. If it uses G, treat it as a measured quantity with its own real (comparatively large) uncertainty that has to be propagated like any other input.

Example 3 — A full calculation: finding g from a pendulum

Measured: length L = 0.950 ± 0.002 m, period T = 1.955 ± 0.004 s (averaged over multiple swings). Formula: g = 4π²L/T².

Raw calculation, guard digits kept: g = 4π² × 0.950 / (1.955)² = 39.4784 × 0.950 / 3.822025 ≈ 9.813 m/s²

Relative uncertainties: u(L)/L = 0.002/0.950 ≈ 0.211%. u(T)/T = 0.004/1.955 ≈ 0.205% — but T is squared in the formula, so its contribution to g’s relative uncertainty doubles: 2 × 0.205% = 0.409%.

Quadrature (GUM) method: √(0.211%² + 0.409%²) ≈ 0.460% → absolute uncertainty = 9.813 × 0.00460 ≈ 0.045

Answer: g = 9.813 ± 0.045 m/s² (quadrature). The simple worst-case method (0.211% + 0.409% = 0.620%) gives a wider ± 0.061 m/s² instead — both defensible, per our error propagation guide, depending on which one your course expects.

Example 4 — Comparing to the accepted value

Measured: g = 9.813 m/s² (Example 3). Accepted standard value: g_n = 9.80665 m/s² exactly — a value fixed by international agreement in 1901, distinct from the actual local gravitational acceleration, which genuinely varies by location and altitude.

Percent error = |9.813 − 9.80665| / 9.80665 × 100% = 0.00635 / 9.80665 × 100%

Answer: 0.065% percent error — small enough to indicate a well-executed experiment, and a result that sits comfortably within the ± 0.045 m/s² uncertainty band calculated in Example 3, which is exactly what a successful comparison to an accepted value should show.

Example 5 — Why raw data shouldn’t be rounded early

A student records five trial times for the pendulum: 1.95 s, 1.96 s, 1.95 s, 1.96 s, 1.96 s — already rounded to 2 decimal places at the moment of recording, because “that’s what the final answer needed anyway.” The stopwatch actually displayed 3 decimal places each time. Averaging the pre-rounded values and averaging the full-precision values can land on different last digits — the same double-rounding effect covered generally elsewhere on this site, here costing real data rather than just an arithmetic exercise. Record raw data to the full precision the instrument actually shows; round only the final calculated result.

Where This Still Trips People Up

• A digital reading isn’t automatically more accurate, just more precise-looking. ± 1 in the last digit describes readability, not whether the instrument was ever calibrated — see accuracy vs. precision.
• π is not in the same category as G. π is a mathematical constant with no physical measurement behind it at all — treat it as exact to as many digits as needed. G is a physical constant that genuinely had to be measured, and still carries real uncertainty.
• Stopwatch resolution overstates real timing precision. A stopwatch showing hundredths of a second doesn’t mean your timing is good to 0.01 s — human reaction time dominates, and reporting timing uncertainty as “± the stopwatch’s resolution” understates the true uncertainty substantially.
• Squaring doesn’t leave sig figs unchanged just because it “looks like” ordinary multiplication. Treating T² as simple multiplication (matching T’s own sig fig count) happens to give a similar classroom-level answer, but it hides that the relative uncertainty genuinely doubles — a distinction that matters the moment you’re doing real propagation instead of digit-counting. See Example 3.
• Not every constant with a lot of digits is exact. G is often written to 4–5 sig figs on a formula sheet, which can look just as “fixed” as c or h — the digit count gives no visual clue that one is exact and the other isn’t.

Instrument Uncertainty by Type

The general pattern: the “obvious” number (the smallest marked division, the display’s last digit) is a reasonable starting estimate, but the actual limiting factor is sometimes somewhere else entirely — human reaction time being the clearest example.

Where the Constants Data Comes From

The exact/measured split in Example 2 comes directly from the BIPM’s own account of the SI’s seven defining constants, adopted at the 26th General Conference on Weights and Measures in 2018 and in force since 20 May 2019. The gravitational constant, by contrast, has no such fixed definition — its currently accepted value comes from CODATA’s periodic recommended-values adjustment, which is revised as new experiments improve on it, precisely because it remains a measured quantity rather than a defined one.

Common Mistakes

1. Rounding raw data at the moment of recording instead of keeping full instrument precision until the final calculation — see Example 5.
2. Reporting timing uncertainty as the stopwatch’s display resolution, ignoring the much larger human reaction time contribution.
3. Treating every constant in a formula as equally exact, when only the seven SI defining constants actually carry zero uncertainty — see Example 2.
4. Applying the multiplication sig-fig rule to a squared term without recognizing that relative uncertainty, not just sig fig count, doubles under squaring.
5. Calculating percent error against the measured value instead of the accepted value in the denominator — it’s always accepted value on the bottom.
6. Assuming a digital instrument’s readout is accurate just because it’s precise — see the accuracy/precision gotcha above.

Practice Problems

Concept: Reading uncertainty from instruments

Q1. A ruler marked in millimeters is used to measure a length. What’s the conventional reading uncertainty? A) ± 1 mm B) ± 0.5 mm C) ± 0.1 mm D) ± 2 mm Answer: B) ± 0.5 mm — half the smallest division.

Q2. A digital scale displays 24.5 g with no stated accuracy specification. What’s the conventional reading uncertainty? A) ± 0.5 g B) ± 5 g C) ± 0.1 g D) ± 1 g Answer: C) ± 0.1 g — ± 1 in the last displayed digit.

Concept: Exact vs. measured constants

Q3. Since the 2019 SI redefinition, which of these has exactly zero uncertainty? A) Gravitational constant, G B) Planck constant, h C) Local gravitational acceleration, g D) None of these Answer: B.

Q4. Why does the gravitational constant G remain experimentally measured while the Planck constant h does not? A) G is less scientifically important B) G was not fixed by the 2019 redefinition and continues to be determined by experiment, unlike h, e, k, and N_A C) G changes measurably over time D) G has no physical units Answer: B.

Concept: Power propagation

Q5. In g = 4π²L/T², if T has a 0.2% relative uncertainty, what’s its contribution to g’s relative uncertainty before combining with L’s? A) 0.1% B) 0.2% C) 0.4% D) 0.04% Answer: C) 0.4% — squaring doubles the relative uncertainty contribution.

Q6. If T’s relative uncertainty dominates the combined uncertainty in g, which measurement most needs improving to tighten the final result? A) L, always B) T, since squaring amplifies its relative uncertainty C) Neither matters D) The constant 4π² Answer: B.

Concept: Percent error against an accepted value

Q7. A pendulum experiment measures g = 9.81 m/s², against the accepted value 9.80665 m/s². What’s the percent error, to 2 significant figures? A) 0.034% B) 0.34% C) 3.4% D) 0.0034% Answer: A) 0.034%.

Q8. Percent error is always calculated relative to which value? A) The measured value B) The accepted/true value C) The average of both D) Neither — it’s a plain difference with no denominator Answer: B.

Concept: Lab report data-handling conventions

Q9. When recording raw data in a lab notebook, the correct practice is to: A) Round immediately to the number of sig figs expected in the final answer B) Record the full precision the instrument actually shows, and round only the final calculated result C) Always record exactly 3 sig figs regardless of the instrument D) Record only whole numbers Answer: B.

Q10. Why shouldn’t intermediate values be rounded throughout a multi-step lab calculation? A) It’s fine to round at every step B) Rounding at each step compounds error and can shift the final answer’s last digit — the same double-rounding problem covered generally elsewhere on this site C) Only final answers need any precision D) Intermediate values don’t need units Answer: B.

Exact vs. Measured, Side by Side

Exact — defined, zero uncertainty:

• c = 299,792,458 m/s
• h = 6.62607015 × 10⁻³⁴ J·s
• e = 1.602176634 × 10⁻¹⁹ C
• N_A = 6.02214076 × 10²³ /mol

Measured — real, ongoing uncertainty:

• G = 6.674 × 10⁻¹¹ N·m²/kg² ± ~0.00015 × 10⁻¹¹ (relative uncertainty ≈ 2 × 10⁻⁵, still being refined by CODATA)

Quick Reference

Continue Learning

Related fundamentals:

• Accuracy, Precision, and Uncertainty
• Error Propagation: How Uncertainty Moves Through a Calculation
• Exact Numbers and Why They Never Limit Precision

Related subjects:

• Significant Figures in Chemistry
• Significant Figures Constants Reference
• Significant Figures in Engineering Drawings and Tolerances
• Significant Figures in Astronomy

Exam-specific guidance:

• Significant Figures Rules for AP Physics and AP Chemistry
• Significant Figures for GCSE and A-Level Sciences

Tools:

• Error Propagation Calculator
• Percent Error Calculator

Sources and Further Reading

• BIPM, The International System of Units (SI): Defining Constants — the primary source for which seven physical constants carry exactly zero uncertainty since the 2019 redefinition, cited throughout Example 2. (bipm.org)
• Wikipedia, Elementary Charge — cross-referenced for the exact 2019-defined value of e used in Example 2, consistent with multiple independent technical sources checked during drafting. (en.wikipedia.org)
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results — reused from earlier pillars, relevant here for the reading-uncertainty and reporting conventions applied throughout. (emtoolbox.nist.gov)

Review and Methodology

Reviewed by: [Pending — reviewer assignment required before publication] Last reviewed: [Pending] Methodology: The exact/measured constants distinction is drawn directly from BIPM’s own SI documentation, not a secondary summary. Every worked example was independently recomputed during drafting. Calculator results referenced on this page use an arbitrary-precision decimal engine, not native floating-point math, validated against the site’s versioned regression fixture set.

Changelog

v1.0 — Initial draft completed, 2026-08-10.