Accuracy, Precision, Resolution, and Sensitivity

Accuracy is how close a measurement is to the true value, precision is how close repeated measurements are to each other, resolution is the smallest change in a quantity that an instrument can show, and sensitivity is the change in an instrument’s output per unit change in the quantity measured. In brief: accuracy is limited by systematic errors, precision by random errors, resolution by the scale or digital display of the instrument, and sensitivity by its design. A measurement can be precise but inaccurate (consistently off), accurate on average but imprecise (scattered), or both accurate and precise.

These four terms are often confused in everyday language, but they have distinct meanings in physics. This guide defines each term, shows how they are related, compares them in a table, works through numerical examples with a balance, rulers and a meter, and corrects common mistakes.

What are accuracy and precision?

Accuracy describes the closeness of a result to the accepted or true value. It is affected by systematic errors (calibration, zero errors, flaws in the method) and is measured by the percentage error: (measured value − accepted value) ÷ accepted value × 100 percent. Precision describes the closeness of repeated measurements to each other, whatever their closeness to the true value. It is affected by random errors and is indicated by the spread (range or standard deviation) of the readings. Think of darts on a board: all close together but away from the bullseye is precise but not accurate; scattered around the bullseye is accurate on average but not precise; clustered at the bullseye is both. Note that “precision” is also used for the number of significant figures given in a result. A result quoted with many digits suggests precision, but not necessarily accuracy. In the metrological vocabulary (VIM), “trueness” is the closeness of the mean to the true value, and “accuracy” combines trueness and precision, but in school physics the simpler meaning above is used.

What is resolution?

The resolution of an instrument is the smallest change in the quantity that it can detect or display. For an analog scale it is the smallest division (often with an estimate of half a division), for a digital instrument it is the last digit. Examples: a meter ruler with millimeter marks, resolution 1 mm; a vernier caliper, 0.1 mm or 0.05 mm; a micrometer screw gauge, 0.01 mm; a digital stopwatch, 0.01 s; a digital voltmeter showing 1.234 V, 0.001 V. The resolution sets a lower limit on the uncertainty of a single reading (often taken as ± one smallest division for an analog scale or ± one unit in the last digit for a digital display, though a careful reading of an analog scale may justify half a division). A finer resolution gives a smaller percentage uncertainty for the same quantity, so a micrometer is better than a ruler for the diameter of a wire. However, a good resolution does not ensure accuracy: an instrument with a zero error or a poor calibration can have a fine resolution and still be wrong. And for a human-timed measurement, the reaction time (about 0.2 s) is much greater than the 0.01 s resolution of the stopwatch, so the effective uncertainty is larger than the resolution.

What is sensitivity?

Sensitivity is the ratio of the change in the output (reading, deflection) of an instrument to the change in the input quantity: sensitivity = Δoutput ÷ Δinput. A galvanometer that deflects 30 mm for a current of 0.5 mA has a sensitivity of 60 mm/mA; one that deflects only 12 mm for the same current has a sensitivity of 24 mm/mA. A more sensitive instrument responds more strongly to small changes, which makes it possible to detect small quantities and so can improve the effective resolution. A sensitive balance responds to very small changes in mass; a sensitive thermometer shows a large change in the output for each degree. High sensitivity is not always better: a very sensitive instrument may be more affected by noise and interference, and it may have a smaller range. A linear instrument has the same sensitivity over its range (a uniform scale). Related terms: the range (the span of values that it can measure) and the threshold (the smallest input that gives a detectable output).

Terms compared

Worked example: comparing instruments and measurements

1. Accurate or precise? A standard mass of 50.0 g is measured five times on two balances. Balance A reads 49.9, 50.1, 50.0, 50.0, 50.0 g (mean 50.0 g, spread 0.2 g). Balance B reads 51.2, 51.3, 51.2, 51.3, 51.2 g (mean 51.24 g, spread 0.1 g). Balance B is more precise (smaller spread) but less accurate: its percentage error is (51.24 − 50.0) ÷ 50.0 × 100 = 2.5 percent, a systematic error. Balance A is both accurate and slightly less precise.
2. Resolution and percentage uncertainty. A plate is about 2.50 mm thick. With a ruler of 1 mm resolution, taking ±1 mm, the percentage uncertainty is 1 ÷ 2.50 = 40 percent. With a micrometer of 0.01 mm resolution, the percentage uncertainty is 0.01 ÷ 2.50 = 0.4 percent. The micrometer’s resolution is 100 times finer than that of the ruler.
3. Sensitivity. Meter 1 deflects 30 mm for a current of 0.5 mA: sensitivity 30 ÷ 0.5 = 60 mm/mA. Meter 2 deflects 12 mm for 0.5 mA: 12 ÷ 0.5 = 24 mm/mA. Meter 1 is 60 ÷ 24 = 2.5 times as sensitive, so a current change of 0.1 mA moves its pointer 6 mm, against 2.4 mm for meter 2.
4. Resolution is not accuracy. A digital thermometer shows 0.1 °C but is not calibrated: in melting ice it reads 1.4 °C, and in steam at standard pressure it reads 101.4 °C. The constant offset of +1.4 °C (a systematic error) makes the readings inaccurate despite the fine resolution; subtracting 1.4 °C corrects them.
5. Precision in significant figures. A calculator gives g = 9.81237 m/s² from data that are only good to two significant figures. Quoting 9.81237 suggests a precision that the data do not have; the result should be given as 9.8 m/s².

How did the terminology develop?

The standardization of measurement terms is the work of metrology. The international vocabulary of metrology (VIM), published by the Joint Committee for Guides in Metrology (JCGM) in 1984 and revised in later editions, defines measurement accuracy, precision (the closeness of agreement between repeated measurements), resolution and sensitivity. The ISO 5725 standard (from 1994) introduced “trueness” and “precision” as components of “accuracy”. Many school curricula still use the simpler definitions.

Which mistakes are common?

Mistake 1: Using accuracy and precision as synonyms

A precise set of results can be inaccurate. Check both the scatter and the closeness to the accepted value.

Mistake 2: Assuming that more digits means more accuracy

Digits show the resolution, not correctness. A miscalibrated digital meter can show many digits and be wrong.

Mistake 3: Confusing resolution with sensitivity

Resolution is the smallest change displayed, in the units of the quantity; sensitivity is the output change per unit input.

Mistake 4: Quoting results with more significant figures than the data justify

The result of a calculation should not have more significant figures than the least precise data used.