Weak-measurement induced quantum discord or super quantum discord (SQD) is a generalization of the normal quantum discord and is defined as the difference between quantum mutual information and classical correlation obtained by weak measurements in a given quantum system. This correlation is an information-theoretic measure and is, in general, different from entanglement-separability measures such as entanglement. Super quantum discord may be nonzero even for certain separable states. So far, SQD has been calculated explicitly only for a limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for SQD for X states, a seven real-parameter family of two-qubit states and investigate its monogamy properties. The monogamy behaviour of SQD depends on the measurement strength. The formalism can be easily extended to N-qubit X states.