Mean phase error and the map-correlation coefficient

In judging the effectiveness of methods of solving crystal structures, or in phase refinement and development, two criteria are commonly used. The first is the mean phase error, which may be weighted in some way, and the second is the map correlation coefficient which describes the similarity of a map with estimated phases to that with true phases. It is shown that these two measures are directly related and that given the individual phase errors the map correlation coefficient may be found without the need to calculate a map. Various aspects of this connection are examined, including the map correlation coefficient when weights are used for calculating maps and the conditions under which phase extension leads to maps with a higher map correlation coefficient - which involves a balance between the advantage of employing more data and the disadvantage that the extra data may have a higher average phase error.