The below is an excerpt from an email that summarizes how to quote astrometric errors when one has knowledge of both random (statistical) and systematic errors in their measurements. The formalism is general enough to be applied to other areas. ============================================================================== From: Frank Masci Subject: astrometric bias revisited Date: February 3, 2012 12:45:34 PM PST To: Roc Cutri CASE 1: ------- This assumes any astrometric bias "b" is stochastic over the sky and one has estimated its approximate dispersion (sqrt[] ~ sigma_b) over many fields. If so, and assuming this component is ~independent of the internal RMS (statistical) error w.r.t. astrometric matches (call it sigma_r), then it's ok to RSS the two to quote the total 1-sigma error: sigma_tot = +/- sqrt(sigma_b^2 + sigma_r^2). Then for a *blind* position measurement x_obs anywhere on the sky (from the catalog), the probability that the "true" position x_true lies in the range: x_obs - sigma_tot < x_true < x_obs + sigma_tot, will be ~68% if the total error (b+r) is distributed as Gaussian, otherwise, it will depend on the shape of the underlying distribution. Note: the above is similar to how source-confusion noise is treated and estimated - i.e., stochastically. As you well know, like confusion noise, averaging more data will not reduce sigma_b. CASE 2: ------- On the other hand, if the bias is simply a constant offset "b" with fixed direction (+ or -), or maybe several constants over the sky, then the best one can do when quoting the total error on a measurement x_obs is to specify separate lower and upper confidence limits that incorporate "b" for that sky region: sigma_tot_low = b - sigma_r sigma_tot_upp = b + sigma_r This means there's a 68% chance (for Gaussian random errors r) that the "true" position will lie somewhere in the range: x_obs + b - sigma_r < x_true < x_obs + b + sigma_r, where the sign of "b" matters. Note that we could have first de-biased our raw measurement by adding/subtracting "b" so that all we are left with is the statistical error (+/- sigma_r) on our effective measurement "x_obs + b". CASE 3: ------- This is the same as CASE 2 except that the actual bias estimate has its own random error sigma_br when estimated for a sky region. This is not to be confused with sigma_b in CASE 1 which was a stochastic estimate for the whole sky (or many regions). Therefore, here we're assuming "b" is stochastic for a region with mean or some robust estimate of the "true b". The error in this **mean estimate** is then sigma_br, or ~ RMS/sqrt(N) where RMS refers to the sample used to estimate "b" and the sqrt(N) assumes the measurements are approximately independent and identically distributed (i.i.d). By "identically distributed", we mean that each measurement came from the same underlying population with a given variance. Otherwise, and for the case where **each** measurement came from a Gaussian population, inverse noise-variance weighting is suggested in order to obtain an optimal (maximal signal-to-noise) estimate of with sigma_br. This uncertainty can then be RSS'd with the overall statistical component sigma_r to quote the full error range as in CASE 2: sigma_tot_low = - sqrt(sigma_br^2 + sigma_r^2) sigma_tot_upp = + sqrt(sigma_br^2 + sigma_r^2) Care must be exercised here since these expressions assume errors in the bias estimate and the overall random component are independent. This generally won't be true since usually one would estimate the bias, its error, and the overall random error from the same input measurements. It's always best to 'err' on the conservative side and account for any positive correlations if possible. That's all there is to it.