Precision, accuracy, and uncertainty

In brief

Four archers each fire five arrows at a target. One archer's arrows land in a tight cluster in the upper left corner, far from the bullseye. Another's arrows scatter widely but center on the bullseye on average. A third clusters tightly right on the bullseye. A fourth scatters widely and off to one side. All four are shooting, but only one of them is doing well in the way that matters for physics: hitting close to the true target consistently. The first archer is precise but not accurate. The second is accurate on average but not precise. Confusing these two failures, or reporting a result without saying how much either one might be present, is one of the most common ways a measurement misleads. This entry treats precision, accuracy, and the honest number that captures both, uncertainty, as three distinct things that every physical result must address.

The full treatment

First look: the archery target

Return to the four archers. If you only saw the average position of each archer's arrows, you would call archers two and three both good, since both center near the bullseye. But archer three's shots are grouped tightly, so you can trust the next shot to land near the last ones, while archer two's shots are scattered, so a single arrow tells you little about where the next one will go. Precision is about the spread of repeated attempts, how tightly they cluster together, regardless of whether that cluster is in the right place. Accuracy is about whether the center of that cluster sits at the true target, regardless of how tight the cluster is.

Building the idea: repeat, compare, and quantify the spread

Translate this to measurement. Suppose you time the same pendulum swing five times with a stopwatch, getting 2.03, 2.05, 2.02, 2.04, and 2.06 seconds. These five readings cluster tightly, within about two hundredths of a second of each other, so your method is precise. Whether it is also accurate depends on something the repeated readings alone cannot tell you: whether your stopwatch itself runs at the correct rate, and whether your reaction time introduces a consistent early or late press. A precise method can still be systematically wrong, if, for instance, your stopwatch runs slow, every reading would cluster tightly around a value that is consistently too small. This is why precision is judged from the spread of repeated measurements, typically summarized as a standard deviation, while accuracy can only be judged by comparison against an independently known or trusted true value, which is not always available.

The formal model: expressing uncertainty

Because the true value is often unknown, physics reports a measurement not as a bare number but as a best estimate plus an uncertainty: x equals x-bar plus or minus delta-x, where x-bar is the average of repeated readings and delta-x is a stated range, often one standard deviation, within which the true value is expected to lie with reasonable confidence. For the pendulum times above, x-bar is 2.04 seconds and delta-x, computed from how the five readings scatter around that average, might be about 0.01 seconds, so the result is reported as 2.04 plus or minus 0.01 seconds. This single expression carries both precision, through the size of delta-x, and an honest admission that the true value is not claimed to be known exactly. When several measured quantities are combined in a calculation, their uncertainties propagate: if z depends on measured x and y, the uncertainty in z grows from the uncertainties in x and y according to rules derived from calculus, so a result calculated from several imprecise inputs is never more certain than those inputs allow, no matter how many digits a calculator displays.

Significant figures as a crude proxy

Before formal uncertainty became routine, physicists used significant figures, the number of meaningful digits in a reported value, as a rough stand in for precision. Writing "2.04 seconds" rather than "2.043827 seconds" signals that the last digit is uncertain and further digits are not meaningful. This convention is useful but crude: it cannot distinguish a case where the true uncertainty is 0.01 from one where it is 0.04, and it says nothing at all about accuracy. Stating an explicit delta-x is always more informative than counting digits, and reduces to the same idea handled honestly.

Lineage

The distinction between the closeness of repeated trials and closeness to truth is at least as old as the practice of repeated astronomical observation, where early astronomers noticed that repeated readings of the same star's position scattered around a value rather than agreeing exactly, and treated the average as the best estimate. Carl Friedrich Gauss formalized the mathematics of how such scatter behaves in the early nineteenth century, work that underlies the modern standard deviation. The vocabulary of precision versus accuracy as distinct, testable properties, and the systematic propagation of uncertainty through a calculation, was consolidated for physics students by texts such as John Taylor's error analysis and standardized internationally through the Guide to the Expression of Uncertainty in Measurement.

The strongest case for it

Separating precision from accuracy, and attaching an explicit uncertainty to every result, is what lets two independent experiments be compared honestly. If one laboratory reports the speed of light as 2.998 times ten to the eighth, plus or minus 0.001 times ten to the eighth, meters per second and another reports a value within that stated range, the two experiments agree, even though their central values differ slightly. Without stated uncertainty, any two numbers that are not identical would appear to disagree, which is both misleading and impossible to satisfy given that no measurement is perfectly repeatable. This framework is also what allows physics to detect a genuine discrepancy: when two careful measurements, each with honestly small uncertainty, still fail to overlap, something real is being missed, and that gap has repeatedly been the first clue to a new effect.

The strongest case against it

The framework has real limits and common failure points. Precision can always be improved by more careful repetition of the same flawed method, but this does nothing for accuracy if the method has a hidden systematic bias, such as a stopwatch that runs slow or a ruler that has stretched. No amount of repeated measurement detects a systematic error, only an independent method or a known reference can. A widespread misconception is treating a small uncertainty as proof of correctness, when it only proves the method is repeatable, not that it is unbiased. Another misconception is quoting a result to many decimal places as if that alone signals rigor, when digits beyond the actual uncertainty are noise dressed as precision. Finally, uncertainty estimates themselves are models of how a method behaves and are only as good as the assumptions behind them, so a stated delta-x is a claim that can itself be wrong.

Where it stands now

The distinction between precision and accuracy, and the practice of stating measurement uncertainty explicitly, is universal and uncontested in experimental physics, formalized internationally through the Guide to the Expression of Uncertainty in Measurement. What remains an active, ordinary part of experimental work is the harder task of identifying and bounding systematic errors in any specific setup, which is a matter of judgment and cross checking rather than a fixed formula, and where careful experimenters continue to disagree case by case.

Test yourself

You measure the length of a metal rod five times with the same ruler and get 15.2, 15.3, 15.2, 15.1, and 15.3 centimeters. Report the length as a best estimate with an uncertainty, and state whether your result is precise, accurate, both, or neither, given only this information. Then a colleague tells you the ruler was manufactured with a printing error that makes every centimeter mark 2 percent too short. Explain what changes and what does not change in your reported result, and which of precision or accuracy that new fact affects.