Comparisons of frequency distributions often invoke the concept of shift to describe directional changes in properties such as the mean. In the present study, we sought to define shift as a property in and of itself. Specifically, we define distributional shift (DS) as the concentration of frequencies away from the discrete class having the greatest value (e.g., the right-most bin of a histogram). We derive a measure of DS using the normalized sum of exponentiated cumulative frequencies. We then define relative distributional shift (RDS) as the difference in DS between two distributions, revealing the magnitude and direction by which one distribution is concentrated to lesser or greater discrete classes relative to another. We find that RDS is highly related to popular measures that, while based on the comparison of frequency distributions, do not explicitly consider shift. While RDS provides a useful complement to other comparative measures, DS allows shift to be quantified as a property of individual distributions, similar in concept to a statistical moment.

翻译：频率分布的比较常通过偏移概念来描述均值等属性的方向性变化。本研究旨在将偏移界定为一种独立属性。具体而言，我们将分布偏移（DS）定义为频率向除最大值所在离散类别（如直方图最右侧区间）之外区域的集中程度，利用指数化累积频率的归一化求和导出了DS的度量方法。进而定义相对分布偏移（RDS）为两个分布间DS的差值，揭示一个分布相对于另一个分布向较小或较大离散类别集中的程度与方向。研究发现RDS与若干基于频率分布比较但未明确考虑偏移的常用度量高度相关。RDS为其他比较度量提供了有益补充，而DS则能如统计矩概念般将偏移量化为单个分布的内在属性。