  To combine the proportions from different studies for meta-analysis, Freeman
and Tukey double arcsine tranformation can be useful for normalization and
variance stabilization. The inverse function of the double arcsine
transformation has been also derived in the literature to recover the original
scale of the proportion after aggregation. In this brief note, we present the
domain and range of the inverse double arcsine transformation both analytically
and graphically. We notice an erratic behavior in the mathematical formula for
the inverse double arcsine tranformation at both limits of its domain, and
propose approximation methods for both small and large samples. We also propose
a simple accuracy measure, the maximum percent error (MPE), of the large sample
approximation, which can be used to determine the sample size that would
provide a certain accuracy level, and conversely to determine the accuracy
level of the approximation given a sample size.
