Stacking : The Square-Root Law

If \(f\) is a function of two Gaussian variables, \[y = f(x_1,x_2)\]

the variance propagates according to the norm of partial derivatives weighted with covariances NIST engineering statistics handbook: \[\sigma_y = \sqrt{\bigg(\frac{\partial{y}}{\partial{x_1}}\bigg)^2\sigma^2_{x_1} + \bigg(\frac{\partial{y}}{\partial{x_2}}\bigg)^2\sigma^2_{x_2} + ... + \bigg(\frac{\partial{y}}{\partial{x_1}}\bigg).\bigg(\frac{\partial{y}}{\partial{x_2}}\bigg)\sigma_{x_1x_2}^2}\]

For a \(n\) repeated experiments with noise of constant variance throughout, the equation simplifies to \[y = \sum_n{x_n};\] \[\sigma_y^2 = \sum_n{\sigma_n} = n\sigma^2;\]

The variance of the noise is reduced by a \(\sqrt{n}\) factor, the square-root of the stack-order : \[\sigma_y = \sqrt{n}.\sigma;\]