This paper describes a new algorithm for calculating exact statistics on directional data and its application to pattern processing. Although information about directional characteristics is practically useful in image processing, e.g. texture analysis or color segmentation, dominant information is not always extracted as exact statistics on directional data. The main reason is concerned with periodicity inherent in directional data. For example, an expectation of a random variable X is defined as ∫xp(x)dx, where p(x) is a probability density function of X; therefore, when a random direction D is distributed only at 170[]and 170[] with same probability density, the expectation of D leads to 0[] if nothing about the periodicity is considered. We would, however, expect that the exact expectation of D should be 180[]. To overcome the problem, we, at first, define a directional distance in such a form that can introduce the periodicity. Then, we propose an idea of defining directional statistics by a problem of minimizing an arithmetic mean of squared directional distances to each sample direction. Because the periodicity is introduced to the directional distance definition, the directional statistics are calculated as the exact statistics on directional data. Although the introduced periodicity might cause the minimization to be complex, we can compensate the complexity by introducing recurrence formulas; consequently, dominant information can efficiently be extracted as the directional statistics from those data. Experiments on their applications to pattern processing show that the proposed algorithm works well in detecting (1) divergent points of distorted vector field patterns with noise and (2) moving directions from translational movement vector fields.