robotics//sensor//sensor calibration//noise characterization//normality assumption

Is it legitimate to assume a bell? Short answer: often yes, and for three very different reasons. Long answer: it depends on your data, and you have to look at them.

Is it legitimate to assume a bell? Short answer: often yes, and for three very different reasons. Long answer: it depends on your data, and you have to look at them.

The physical reason: noise is usually the sum of many small independent causes, and such sums tend to a normal (central limit theorem).

The epistemological reason, maximum entropy: if all you know about your noise is its mean and its variance, the normal is the distribution that invents the least. Choosing any other shape would claim information you do not have; with two numbers, it is the honest choice.

The practical reason, without shame: it is convenient. A Gaussian through a linear transformation stays Gaussian, and the product of two Gaussians is Gaussian too, which is why the equations of a Kalman filter come out closed, exact and cheap, five lines instead of a cluster.

In the real world it stops being a bell when one cause dominates (a GPS in a narrow street receives signals bounced off buildings, and the bounced path is always longer: heavy tails and a positive bias), with quantization (a converter of few bits can only say certain values, and the noise comes in steps), at physical limits (a distance cannot be negative, a saturated sensor sticks at its maximum), with discrete faults (a flipped bit, a corrupt packet: a giant outlier now and then), with several regimes (the sensor alternates between two modes, and the histogram comes out bimodal), and when precision changes over time with the temperature or the satellite geometry, a mixture of bells of different widths that has heavy tails as a whole even if every instant is Gaussian.

So it depends on the data collected, and looking at them is part of noise characterization. The precise statistical question, whether these data could come from a normal, is one of goodness of fit. When how the noise behaves over time also matters, one enters stochastic processes, and for inertial sensors the standard tool has its own name, the Allan variance (autocorrelation).