DEFAULT, a new lightweight cryptosystem presented at Asiacrypt in 2021, is designed to protect the information security of Internet of Things (IoT) devices, such as microchips, microcontrollers, and sensors. Based on the ciphertext-only attack assumption, the statistical fault analysis of the DEFAULT cipher with the algebraic relationship is proposed. The statistical fault analysis uses the random nibble-oriented fault model. It not only combines statistical distributions of the intermediate states before and after the fault injections but also takes advantage of the algebraic relationship and novel distinguishers, including Anderson Darling test-square Euclidean imbalance (AD-SEI), Anderson Darling test-maximum likelihood estimate (AD-MLE), and Anderson Darling test-Hamming weight (AD-HW). The analysis requires at least 1344 faults to achieve the reliability of 99% in the recovery of the 128-bit secret key of DEFAULT. The theoretical analysis and experimental results show that the DEFAULT lightweight cryptosystem is not resistant to the statistical fault attack based on the algebraic relationship. This study provides an important reference for the security analysis of the other lightweight cryptosystems.