mathematics//statistics//precision and accuracy

Two different ways a measurement can be good. **Precision** is how much the measurements resemble each other, their dispersion, measured by \(\sigma\). **Trueness** is how close their mean comes to the truth, measured by the bias \(\mathbb E[z]-x\). ISO 5725 calls the combination of both **accuracy**. A narrower bell means a more precise sensor, not necessarily a more exact one.

Two different ways a measurement can be good. Precision is how much the measurements resemble each other, their dispersion, measured by σ\sigmaσ. Trueness is how close their mean comes to the truth, measured by the bias E[z]−x\mathbb E[z]-xE[z]−x. ISO 5725 calls the combination of both accuracy. A narrower bell means a more precise sensor, not necessarily a more exact one.

The variance is blind to bias. A sensor can have σ=0.1\sigma=0.1σ=0.1 m and sit 3 m from the truth, always, with admirable consistency: a very determined stopped clock. The four combinations are the four targets of the classic picture, tight and centred, tight and off, loose and centred, loose and off, and only the dispersion shows up in a variance.

The real-world version is a change of datum. Coordinates read from an old Spanish map (ED50) and from a GPS (ETRS89) differ by a shift of the order of 200 m on the mainland. The GPS is precise, the map is precise, and one of them speaks another language: excellent precision, no trueness.

The two errors are fixed by different means. Averaging more readings shrinks the random part and leaves the bias untouched; calibration removes the bias and leaves the random part untouched.

What a filter needs done about a bias, and why it cannot find one alone, is sensor bias.