Topic 2 of 4

Errors, accuracy and precision

Repeated readings can agree closely and still be displaced from the correct value. Identify what caused a limitation before choosing how to improve the measurement.

Error: The difference between a measured value and a reference or true value. The true value, and hence the exact error, is often unknown.
Uncertainty: The doubt associated with a result, often expressed as an estimated range around it. An uncertainty is not automatically a known correction.
Accuracy: Closeness to the true or accepted reference value.
Precision: Closeness of agreement between repeated measurements under the stated conditions. A narrow scatter indicates greater precision.

Compare supplied repeated length readings

Each dot is one reading. All three rows use the same horizontal scale. The dashed reference is 20.0 mm; the row height only separates the sets.

Sets 1 and 2 have equally small scatter here, but set 2 is displaced from the reference. Set 3 has a mean close to the reference with less precise individual readings. Averaging does not automatically remove a persistent bias.

Each set uses the same scale and a supplied reference of 20.0 mm. The two narrow sets are equally precise, but one is displaced. The wider set has a mean close to the reference while its individual readings scatter more.

The narrow displaced set has mean 20.6 mm, compared with the reference 20.0 mm. Repeating this biased procedure many times could produce a very consistent mean near the wrong value. More display digits would not repair the cause.

Systematic effects follow a pattern

A systematic error shifts readings in a consistent way under the stated conditions. A zero offset adds a fixed amount; a calibration scale error may instead multiply readings by a factor.

Correcting a zero offset

Use the sign of the empty reading

A balance reads +0.20 g with no load and 24.80 g with an object. If the zero offset remains unchanged, corrected mass = displayed mass - offset = 24.80 - 0.20 = 24.60 g.

This removes the known additive offset. Resolution, random fluctuations and other calibration effects can still contribute uncertainty.

A ruler that consistently indicates lengths 2% too high has a scale-factor error. It would indicate 51 cm for a true 50 cm length and 102 cm for a true 100 cm length. Subtracting one fixed number of centimetres cannot correct both. Under this supplied model, divide each indicated length by 1.02.

Random variation produces scatter

Random effects make repeated readings vary. For suitable independent repeats, using a mean reduces the effect of random variation on the estimated value. Keep the observed spread as evidence of uncertainty; averaging does not make it vanish or remove a persistent calibration error.

Read and calculate a mean

Two symbols for the same arithmetic mean

The arithmetic mean of readings x can be written <x> or x, pronounced x bar. The symbol Σ means sum the specified readings. Here n counts readings and is dimensionless:

<x> = x = (Σx_i)/n

For the supplied 19.8, 19.9, 20.0, 20.1 and 20.2 mm set, the sum is 100.0 mm and n = 5. Thus <x> = 100.0/5 = 20.0 mm. It retains the readings' unit. This mean does not erase their spread or establish that every reading is exactly 20.0 mm.

In this notation <x> encloses a quantity to indicate its mean. A statement such as x < y instead uses < to mean less than.

Describe the actual mechanism. A stopwatch user who consistently starts late, with no matching delay when stopping, biases the interval low. Variation in reaction time from trial to trial also causes random scatter. The word "human" alone does not tell you which kind of error is present.

Repeating measurements is helpful for random variation. A stable zero error needs a zero check and correction; a wrong scale factor needs calibration. An improvement should address the identified effect.

Optional check A balance displays -0.30 g with no load and 18.20 g with an object. The offset stays unchanged. Which conclusion is justified?

Choose range, resolution and a suitable method

Resolution is the smallest change an instrument indicates. It is one consideration when judging uncertainty, but it is not automatically the total uncertainty of a reading or procedure. Use a supplied uncertainty or justify an estimate for the actual method.

Instrument choice

Measure a rod about 120 mm long

The target is an uncertainty no greater than 0.1 mm. Use the following supplied specifications.

Micrometer: range 0 to 25 mm: Display step 0.001 mm; stated uncertainty ±0.005 mm. It cannot span the rod, despite its fine resolution.
Rule: range 0 to 300 mm: Divisions 1 mm; estimated uncertainty for the complete length method ±1 mm. The range is sufficient, but the stated method misses the target.
Calipers: range 0 to 150 mm: Display step 0.01 mm; stated uncertainty for this aligned measurement ±0.05 mm. Both the range and uncertainty meet the target.

Choose the calipers, check the zero and hold the rod square to the jaws without compressing it. The 0.01 mm display step does not justify replacing the supplied ±0.05 mm uncertainty with ±0.01 mm.

There is no universal rule that every measurement has uncertainty equal to half its smallest division. Reading two ends, positioning the object, reaction time and calibration can change what is justified.

For a repeated motion timed by hand, timing many complete cycles can reduce the fractional effect of a similar start/stop timing uncertainty. Count accurately and use the same reference point and direction. A longer interval does not correct an incorrect clock calibration.

When a result is calculated from measurements, carry the relevant uncertainties into the derived result.