Accuracy, Precision, and Uncertainty: What Each One Actually Measures

Accuracy is how close a measurement is to the true value. Precision is how close repeated measurements are to each other — regardless of whether they’re anywhere near the truth. Uncertainty is a calculated, honest range around a reported value, stating how far off it might plausibly be. These are three different properties, a measurement can have any combination of them, and significant figures only ever gesture vaguely at one — precision. A number with a lot of digits can still be flatly wrong.

This is the concept that everything else in this silo builds on — Error Propagation is about calculating with these ideas, this page is about actually understanding them first. It matters more than it looks: “precision” and “accuracy” get used interchangeably in casual speech, but they describe genuinely different failure modes with genuinely different fixes. A miscalibrated instrument can produce beautifully consistent, high-sig-fig, completely wrong numbers all day long. Knowing which problem you actually have determines whether the fix is “calibrate the equipment” or “take more measurements” — and those are not interchangeable solutions.

<!– BLOCK: B02 – Inline Mini-Calculator –> <!– DEV NOTE: Embed the Percent Error Calculator tool here (Tier 2 tool #8 in the site blueprint). Shortcode: [sfc_percent_error_calculator default_measured=”24.7″ default_true=”25.0″]. Pairs naturally with this page since percent error is the most common way accuracy actually gets quantified and reported. –>

[Live Percent Error Calculator embeds here] — Enter a measured value and an accepted/true value to calculate percent error, with correct sig fig reporting on the result.

Three Different Properties

Accuracy: closeness to the true value. If the true length of an object is 10.00 cm and your measurement reads 10.01 cm, that’s accurate — it’s close to reality. Accuracy is about correctness relative to something external and (ideally) known.

Precision: closeness of repeated measurements to each other. If you measure the same object five times and get 10.01, 10.02, 10.01, 10.00, 10.01 cm, that’s precise — the numbers agree with each other tightly, regardless of whether they’re actually close to the object’s true length. Precision is entirely internal to the measurement process; it says nothing on its own about correctness.

These are independent. A measurement process can be accurate without being precise (scattered, but centered on the truth), precise without being accurate (tightly clustered, but centered on the wrong value), both, or neither. See Example 1.

Uncertainty: a calculated, honest range of doubt. Where accuracy and precision describe qualities of a measurement process, uncertainty is a specific number attached to a specific result — typically written as a value ± a range, ideally with a stated confidence level. It answers “how far off might this reported number plausibly be?” It is not the same thing as “how many significant figures I happened to write down” — sig figs are a rough, informal stand-in for precision; a real uncertainty is a deliberately calculated quantity. See Example 4.

What actually causes each problem: systematic vs. random error. This is the part most classroom treatments skip, and it’s the part that determines what you actually do about a bad measurement.

• Systematic error is a consistent bias in one direction — a scale that was never zeroed, a ruler with a worn end, a flawed procedure applied the same way every time. It damages trueness (and therefore accuracy), and critically, it does not average out. Every single reading is wrong in the same direction by roughly the same amount, no matter how many times you repeat the measurement.
• Random error is unpredictable scatter — estimating between the smallest marked lines on a ruler, small environmental fluctuations, ordinary human variability in reading an instrument. It damages precision, and it does average out: because random errors are roughly as likely to be too high as too low, taking the mean of many repeated readings makes the average more reliable even though any single reading might still be off.

This is the single most useful practical takeaway on this page: taking more measurements and averaging them fixes a precision problem. It does nothing for an accuracy problem. If your instrument is miscalibrated, averaging a thousand readings from it just gives you an extremely precise, extremely wrong number.

A formal refinement, for anyone working to a real standard. The classroom version above treats accuracy and precision as two independent axes — which is the right mental model for everyday use. Formal metrology, however, defines “accuracy” more narrowly: the international vocabulary of metrology (the VIM, jointly maintained by the BIPM, ISO, and six other international bodies) treats accuracy as the combination of trueness and precision, not a separate third thing. In this stricter usage, a measurement process that’s true on average but wildly scattered isn’t fully “accurate” even though a lucky individual reading might land near the truth — it has good trueness and poor precision, and “accuracy” requires both. This distinction rarely matters for classroom work, but it matters in exactly the industrial and lab-conformance contexts where this site’s ASTM E29 guide also applies.

Worked Examples

Example 1 — All four combinations

True value: 10.00 cm. Four sets of four repeated measurements:

Set B is the dangerous one: every reading agrees with every other reading almost exactly, which feels trustworthy, and every single one of them is wrong.

Example 2 — A closer look at “precise but wrong”

Set B from Example 1: 12.50, 12.51, 12.49, 12.50 cm, against a true value of 10.00 cm.

The spread here is tiny — deviations from the mean of 0, +0.01, −0.01, and 0 cm give a sample standard deviation of roughly 0.008 cm, well under a hundredth of a centimeter. By any statistical measure, this is an excellent, highly precise data set. It is also wrong by 2.50 cm, a 25% error relative to the true value. Writing “12.50 cm” — three significant figures, looking every bit as authoritative as Set A’s correct answer — communicates nothing about this 2.50 cm gap. Sig figs describe precision. They cannot, on their own, tell you whether the instrument was ever calibrated correctly.

Example 3 — Calculating percent error

Calculation: measured value 24.7 g, true/accepted value 25.0 g

Percent error = |measured − true| / true × 100% = |24.7 − 25.0| / 25.0 × 100% = 0.3 / 25.0 × 100%

Answer: 1.2%. This is the standard way accuracy actually gets quantified and reported — a single number describing how far off a result is, as a proportion of the accepted value.

Example 4 — Reporting a value with its uncertainty

Raw data: a calculated value of 12.7346 cm with a calculated uncertainty of 0.0623 cm

Standard metrology guidance (the GUM, JCGM 100:2008, §7.2.6) is to round the uncertainty first — typically to 1 or 2 significant figures — and then match the reported value’s decimal places to it, not the other way around.

Round 0.0623 to 2 significant figures → 0.062 (3 decimal places). Round 12.7346 to the same 3 decimal places → 12.735.

Answer: 12.735 ± 0.062 cm. Reporting more decimal places on the value than the uncertainty justifies — say, 12.7346 ± 0.062 — implies a false precision the data doesn’t actually support.

Example 5 — Why averaging doesn’t fix everything

Scenario: a student measures a table’s length five times with a metal ruler whose zero mark is worn down by 0.5 cm (unknown to the student): 152.3, 152.4, 152.2, 152.3, 152.5 cm.

The small differences between these five readings — 152.2 up to 152.5 cm — are random error: ordinary variation in how precisely a ruler can be read by eye. The consistent 0.5 cm the entire set is offset from the table’s true length is systematic error, caused by the worn zero mark, and it affects every single reading the same way.

Averaging the five readings gives a mean of 152.34 cm — a more reliable, more precise central estimate than any single reading, because averaging cancels out random scatter. It does nothing about the 0.5 cm systematic offset. That error is baked into every measurement this ruler will ever produce, no matter how many times the table gets measured with it.

Where This Still Trips People Up

• A precise-looking number is not the same as a correct one. Three, four, even six sig figs can accompany a measurement that’s simply wrong — see Example 2. Precision and correctness are independent properties.
• “Just take more measurements” only fixes half the problem. It reduces random error beautifully. It does nothing for systematic error — see Example 5. If results are consistently biased, the fix is recalibration or procedural correction, not more repetitions of the same flawed method.
• Casual English blurs accuracy and precision together — “a precision instrument” is often assumed to also mean an accurate one, but precision alone says nothing about whether the instrument was ever calibrated against a true reference.
• An uncertainty isn’t the same as “the last digit is a guess.” Some intro courses use “assume ±1 in the last significant figure” as a rough shortcut, and it’s a reasonable estimate for a single unlabeled reading — but it’s a simplification, not a substitute for an actually calculated uncertainty like the one in Example 4.
• A single lucky reading can look accurate from a bad process. Set C in Example 1 has a correct mean, but any individual measurement drawn from it could be off by half a centimeter. Accuracy of the average is not the same guarantee as accuracy of any one result.

How “Accuracy” Gets Defined Differently

The classroom version is the right mental model for day-to-day sig fig work — it’s simpler and it’s what Example 1’s four-quadrant table above uses. The VIM version matters once you’re working to an actual standard (calibration certificates, lab accreditation, conformance testing), where “accurate” is a defined term with real consequences attached to it.

Where the Definitions Come From

The formal split between trueness, precision, and accuracy comes from the VIM — the International Vocabulary of Metrology, JCGM 200:2012 — jointly maintained by the BIPM, ISO, IEC, and five other international standards bodies specifically so that “accuracy” means the same thing to a lab in Jakarta as it does to one in Geneva. The VIM’s own guidance is explicit that trueness reflects systematic error and is unrelated to random error, which is exactly the causal split covered in this page’s rule statement above. The companion document, the GUM (JCGM 100:2008), governs how the uncertainty number itself gets calculated and rounded once trueness and precision are understood — the mechanics of that calculation are covered in Error Propagation.

Common Mistakes

1. Assuming more decimal places or sig figs means a more accurate result. They only ever imply more precision — see Example 2.
2. Trying to fix a systematic error by averaging more readings. Averaging only helps with random error — see Example 5.
3. Reporting a value’s precision beyond what its uncertainty justifies. If the uncertainty is ±0.06, reporting the value to five decimal places is false precision — see Example 4.
4. Confusing “error” with “uncertainty.” Error is the actual (usually unknowable) difference between a measurement and the truth. Uncertainty is a defensible, calculated estimate of how large that error plausibly is — the two words get used interchangeably in casual speech but mean different things.
5. Treating a single accurate-looking reading as proof the whole process is reliable, when it could simply be a lucky draw from a wide, imprecise scatter.
6. Using “precision instrument” and “accurate instrument” as synonyms — a precise instrument that’s never been calibrated can produce confidently wrong numbers indefinitely.

Practice Problems

Concept: Accuracy vs. precision

Q1. A set of measurements clusters tightly together but far from the true value. This demonstrates: A) High accuracy, low precision B) High precision, low accuracy C) High accuracy and precision D) Low accuracy and precision Answer: B.

Q2. A set of measurements scatters widely but averages out close to the true value. This demonstrates: A) High accuracy, low precision B) High precision, low accuracy C) High accuracy and precision D) Low accuracy and precision Answer: A.

Concept: Formal (VIM) definitions

Q3. In the formal metrology vocabulary (VIM), “accuracy” is best described as: A) A synonym for precision B) A combination of trueness and precision C) Only about repeated measurements agreeing D) Only about the number of significant figures used Answer: B.

Q4. “Trueness,” in the VIM sense, relates most directly to: A) Random error B) Systematic error (bias) C) Rounding error D) Sig fig count Answer: B.

Concept: Systematic vs. random error

Q5. Averaging many repeated measurements primarily helps reduce: A) Systematic error B) Random error C) Both equally D) Neither Answer: B.

Q6. A scale that is never zeroed before use, and so reads 50 g high on every single measurement, is exhibiting: A) Random error B) Systematic error C) Rounding error D) High precision, no accuracy problem at all Answer: B.

Concept: Percent error

Q7. A measured value is 24.7 g against a true value of 25.0 g. What is the percent error? A) 0.3% B) 1.2% C) 3.0% D) 12% Answer: B) 1.2%.

Q8. A measured value is 48 m/s against a true value of 50 m/s. What is the percent error? A) 2% B) 4% C) 96% D) 0.04% Answer: B) 4%.

Concept: Reporting uncertainty

Q9. Per standard metrology guidance, when reporting a value together with its uncertainty, which is rounded first? A) The value B) The uncertainty C) Both, independently and separately D) Neither needs rounding Answer: B.

Q10. An uncertainty is calculated as 0.0623 units. Rounded to 2 significant figures, it becomes: A) 0.06 B) 0.062 C) 0.0623 D) 0.1 Answer: B) 0.062.

The Bullseye, With Real Numbers

• Top-left (accurate + precise): Set A — 9.98, 10.01, 9.99, 10.02 → tight cluster, centered on the bullseye (true value 10.00)
• Top-right (precise, not accurate): Set B — 12.50, 12.51, 12.49, 12.50 → tight cluster, off-center by 2.50
• Bottom-left (accurate on average, not precise): Set C — 9.20, 10.65, 9.85, 10.30 → scattered, but centered on the bullseye
• Bottom-right (neither): Set D — 13.50, 9.50, 12.50, 10.50 → scattered and off-center

Quick Reference

Continue Learning

Related fundamentals:

• Significant Figures: The Complete Guide
• Error Propagation: How Uncertainty Moves Through a Calculation

Go deeper on one topic at a time:

• Absolute Error vs. Relative Error vs. Percent Error
• How to Calculate Percent Error (and Report It Correctly)
• What Measurement Uncertainty Actually Is
• How to Read Uncertainty From a Ruler, Burette, or Digital Meter
• Propagating Uncertainty Through Addition and Multiplication
• How Many Sig Figs Should an Uncertainty Have?
• Reporting Mean and Standard Deviation With Correct Precision
• Why Significant Figures Are a Simplification of Real Uncertainty

Tools:

• Percent Error Calculator
• Absolute & Relative Uncertainty Calculator

Sources and Further Reading

• JCGM 200:2012, International Vocabulary of Metrology — Basic and General Concepts and Associated Terms (VIM), jointly published by the BIPM, ISO, IEC, and five other international bodies — the primary source for the trueness/precision/accuracy relationship described throughout this page. (bipm.org)
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results — NIST’s implementation guide of the GUM, cited here for the uncertainty-rounding rule used in Example 4 (reused from our rounding pillar, where it’s also cited for the same underlying GUM guidance). (emtoolbox.nist.gov)
• Cherry Biotech, Accuracy and Precision in Measurements — a clear secondary explainer connecting VIM terminology to the parallel statistical vocabulary (bias and variability) and to ISO 5725. (cherrybiotech.com)

Review and Methodology

Reviewed by: [Pending — reviewer assignment required before publication] Last reviewed: [Pending] Methodology: Definitions and the trueness/precision/accuracy relationship are cross-checked directly against BIPM’s own VIM documentation (see Sources), not inferred from secondary explainers. 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.