Accuracy, precision, uncertainty and meaningful digits

Ask two different questions

Accuracy concerns agreement with the true quantity value. In practice that value is not known exactly, so reference values and their uncertainties help us investigate error. The formal metrology term accuracy is qualitative; a specification’s numerical “accuracy” limit needs its own definition. Avoid treating the word as a complete uncertainty statement.

Precision concerns how closely repeated indications or measured values agree under stated conditions. A tight cluster can be consistently displaced from a reference. A wide cluster can happen to average close to a reference while individual readings vary greatly. The diagram compares two constructed sets against the same assigned reference. It distinguishes cluster spread from cluster position.

Random variation makes readings differ from one another. A systematic effect may shift many readings in the same direction. More repetitions can help evaluate random variation, but cannot by themselves reveal or remove every systematic effect. A reference comparison, a changed method or a changed setup may reveal something that repetition alone misses.

Describe what remains uncertain about the result

Measurement uncertainty characterises the dispersion of quantity values attributed to the measurand using the available information. It is not a confession that the operator made a mistake, and it is not the actual error, which is generally unknown. Correct a recognised effect when appropriate, then evaluate the uncertainty that remains in the corrected result.

Sources can include variation among readings, reference-standard uncertainty, resolution, alignment, environmental influences and uncertainty in a correction. A Type A evaluation uses statistical analysis of observations. A Type B evaluation uses other information, such as calibration reports or instrument specifications. These labels describe evaluation methods; they do not mean “random” and “systematic” respectively.

A standard uncertainty u is expressed on a standard-deviation scale, a statistical measure of spread in the quantity’s own unit. An expanded uncertainty U multiplies a combined standard uncertainty by a stated coverage factor k: U = k × u. An interval written with ± needs its meaning stated. A factor k = 2 gives approximately 95 % coverage only when the distribution and reliability of the uncertainty estimate justify that interpretation; it is not a universal guarantee.

Match numerical detail to the evidence

Extra calculator digits are not extra measurement knowledge. Keep guard digits during calculations, then round the reported value to the same decimal place as its stated uncertainty. One or two significant digits in uncertainty are commonly useful; the reporting convention and the purpose govern the choice. Preserve the unrounded calculation in the record.

Tolerance states what a design allows. Uncertainty states what is known about a measurement. A part allowed between 49.90 mm and 50.10 mm does not automatically pass because a display says 50.09 mm. The uncertainty and an agreed acceptance rule affect the decision, especially near a limit. A wider tolerance does not improve the instrument, and a smaller uncertainty does not change the design requirement.

Report a supplied uncertainty clearly

For an invented length measurement, suppose a completed uncertainty evaluation gives corrected value 50.023 mm and combined standard uncertainty u = 0.040 mm. Assume an approximately normal distribution and a reliable standard uncertainty. With coverage factor k = 2, U = 2 × 0.040 mm = 0.080 mm.

A suitable rounded report is “50.02 mm ± 0.08 mm, expanded uncertainty, k = 2, approximately 95 % coverage under the stated normal-distribution assumptions.” Its displayed interval is 49.94 mm to 50.10 mm. The supplied uncertainty and distribution are assumptions of this example, not things inferred from the number of displayed digits.

Try the reasoning

1. The three readings 10.19, 10.20 and 10.21 mm are very close together. Have they established good accuracy?

   Show the answer and reasoning

   No. They establish a small spread under those conditions. Their displacement from the assigned 10.00 mm reference suggests an effect to investigate.

2. Would averaging a thousand readings automatically remove a constant zero offset?

   Show the answer and reasoning

   No. Averaging can reduce the random contribution under suitable independence and stability assumptions. A constant offset remains unless identified and corrected.

3. Why is the example reported as 50.02 ± 0.08 mm rather than 50.023000 ± 0.08 mm?

   Show the answer and reasoning

   The uncertainty is stated to hundredths of a millimetre. Extra decimal places in the reported value imply detail the uncertainty does not support. The calculation can retain its guard digits.

What to carry forward

A tight cluster, a small error and a small uncertainty are different ideas. Report the value, unit and meaning of uncertainty together.