mathematics//statistics//normal distribution

The bell curve of measurement errors, and when it is fair to assume it. The **normal distribution** is the bell curve \(\mathcal N(\mu,\sigma^2)\), fixed entirely by a mean and a variance. Measurement errors are so often assumed to be normal that it is worth knowing why, and when not to. Think of each measurement as a flash: a torch fired at the truth lights a spot, but the light never lands as a clean point, it blurs around. Fire it many times and the flashes pile up into the shape of the noise. The shape is the kind of noise; its width, more exactly, is how much.

The bell curve of measurement errors, and when it is fair to assume it. The normal distribution is the bell curve N(μ,σ2)\mathcal N(\mu,\sigma^2)N(μ,σ2), fixed entirely by a mean and a variance. Measurement errors are so often assumed to be normal that it is worth knowing why, and when not to. Think of each measurement as a flash: a torch fired at the truth lights a spot, but the light never lands as a clean point, it blurs around. Fire it many times and the flashes pile up into the shape of the noise. The shape is the kind of noise; its width, more exactly, is how much.

n2000 mean of z0.55 sample σ1.46 bias0.55 Readings of a true value x = 0 pile into a histogram with a normal curve fitted over it. Gaussian noise fits the curve; multipath noise (2000 readings shown) leans right and biases the mean by 0.55.

Each flash is one reading of the same true value; watch the histogram fill and the fitted bell settle over it, then switch the kind of noise and watch the bell stop fitting.

Noise is so often a bell because it is usually a sum of many small independent causes (central limit theorem); whether a given sensor's noise may be treated as one is decided with its data (normality assumption).

Each step of the bell measures one σ\sigmaσ. Within ±1σ\pm1\sigma±1σ lies 68.27 % (one reading in three falls outside), within ±2σ\pm2\sigma±2σ 95.45 % (one in 22 outside), within ±3σ\pm3\sigma±3σ 99.73 % (one in 370). A GPS with σ=5\sigma=5σ=5 m puts two readings in three within 5 m of the truth and nineteen in twenty within 10 m. Beyond three sigmas is the usual threshold for this is not noise, something is happening.