Detecting multimodality in empirical distributions is a fundamental problem in statistics and data analysis, with applications ranging from clustering to the study of complex systems. In practice, however, assessing departures from unimodality in a consistent and comparable way remains challenging. Widely used methods such as Hartigan and Hartigan's Dip Test illustrate these difficulties, as the interpretation of their statistics depends strongly on sample size, requires calibration to determine significance, and, for large samples, exhibit increasing sensitivity, leading to rejection of unimodality for arbitrarily small deviations from the null. We introduce Z-Dip, a standardized measure of multimodality that addresses these limitations. By treating the Dip statistic as a random variable under the null hypothesis of unimodality and standardizing its observed value, the proposed approach yields scores that are directly comparable across datasets of different sizes. Using simulation-based calibration, we derive a universal decision threshold that closely reproduces classical Dip Test decisions without requiring sample-size-specific adjustments. Extensive validation on simulated data and on more than 88,000 empirical opinion distributions shows near-perfect agreement with the classical Dip Test while providing a more interpretable and comparable measure of modality. Finally, we propose a downsampling-based correction that mitigates residual sensitivity in extremely large samples. Open-source software and reference tables are provided to facilitate practical adoption.

翻译：检测经验分布中的多模态性是统计学和数据分析中的一个基本问题，其应用范围从聚类到复杂系统研究。然而，在实践中，以一致且可比的方式评估单模态性的偏离仍具挑战性。广泛使用的方法（如Hartigan和Hartigan的Dip检验）揭示了这些困难，因为这些方法的统计量解释强烈依赖于样本量，需要校准以确定显著性，并且对于大样本，其敏感性会逐渐增加，导致对任意小的零假设偏离都会拒绝单模态性。我们引入Z-Dip，一种标准化的多模态性度量，以解决这些局限性。通过将Dip统计量视为单模态性零假设下的随机变量并对其观测值进行标准化，所提出的方法能够生成在大小不同的数据集之间直接可比的得分。利用基于模拟的校准，我们推导出一个通用决策阈值，该阈值无需针对特定样本量的调整即可紧密复现经典Dip检验的决策。在模拟数据和超过88,000个经验意见分布上的广泛验证表明，Z-Dip与经典Dip检验具有近乎完美的一致性，同时提供了更具可解释性和可比性的模态性度量。最后，我们提出一种基于降采样的修正方法，以减轻极大样本中的残余敏感性。提供了开源软件和参考表格以促进实际应用。