Statistics

A normalized Stokes parameter (P etc) is `the difference between 2 intensity values, divided by their sum' and, since this quotient comprises parameters which are not statistically independent, repeated measurements do not yield a Gaussian distribution. The non-normality becomes apparent at small numbers of detected photons, a situation one might easily encounter in CCD polarimetry. Other non-Gaussian distributions are sent to plague us, notably those of degree of polarization and polarization angle in situations of low signal to noise. Clarke and Stewart (reference in Appendix I) treat these questions in detail; careful interpretation of data is especially required in low signal-to-noise situations. Whether full statistical treatment is required as a result of non-normality can be judged from the following rules of thumb:

• When the number of detected photons per elementary detector (i.e. after on-chip binning, but before software binning) is less than about 1000, the distribution of the normalized Stokes parameters is noticeably skew and this must be allowed for in proper binning or averaging procedures.
• Degree of polarization is always positive. When P is less than about 5 times the standard error of the normalized Stokes parameters, straightforward averaging of P produces biased results.
• Confidence intervals, for estimates of P from a sample of observations, are asymmetric in the domain P/
• Although the determined values of polarization angle are unbiased, their symmetric distribution about the mean is kurtose and confidence intervals do not correspond to those for a normal distribution. Attention to this detail is only warranted in the domain P/
• When is not equal to (as may be the case when the ISIS standard recipe is used under conditions of varying extinction or pressure of time near sunrise), and confidence intervals of degree and angle of polarization will depend on the relative values of (in other words on what the polarization angle happens to be).
• A prescription for a (statistically) maximally effective observing campaign is given by Clarke and Stewart in their last section; however, add your own considerations of what fraction of time to spend on calibration of zeropoint, angle and degree of polarization.