<p>Pythagorean fuzzy set (PyFS) is a highly effective method to capture uncertainty and characterize fuzzy information in uncertain environments. However, many existing distance measures tend to act directly on the membership and non-membership of set to quantify the dissimilarity between two PyFSs, leading to counter-intuitive results. To address this problem, the expected characteristics of the distance measure for PyFS are first provided, which can assess the performance of distance measurement methods. Meanwhile, we define the expected boundary of PyFS, which serve as the crucial metrics for measuring the inherit uncertainty of a PyFS. Second, based on the requirements of the expected characteristics and the expected boundary, a new Pythagorean fuzzy Chi-squared distance called PFCD is proposed. Theoretical proofs and experimental simulations illustrate that the PFCD satisfies all expected characteristics of distance measure, including boundedness, separability, triangle inequality, symmetry, monotonicity, complementarity, and non-counter intuition, which ensure that all PyFSs form a metric space under the PFCD, thus providing a robust framework for quantifying differences. The PFCD can not only output reasonable measurement results but also reduce the reliance on the original data. Furthermore, through the comparative analysis, it is testified that the PFCD can be effectively applied to medical diagnosis, drug screening for COVID-19, and clustering analysis. Finally, the extended form of PFCD (CPFCD) to deal with continuous PyFS is defined, and simulation example further shows the effectiveness under continuous PyFS environment.</p>