Abstract In this paper, discrete alpha power inverted Kumaraswamy distribution is proposed. The general approach of discretization of a continuous distribution is used to derive discrete alpha power inverted Kumaraswamy distribution. Its probability mass function has different shapes including decreasing, increasing, upside-down bathtub and unimodal. The proposed distribution has three parameters and its hazard rate function has several shapes. Moreover, the proposed distribution can be used to analyze over and under dispersed count data. Some properties of the proposed distribution are studied including the quantiles, mean residual life, mean time between failures and mean time to failure, R nyi entropy, non-central moments, central moments, standard moments and order statistics. Maximum likelihood estimation is considered under Type-II censored samples for estimating the model parameters, survival, hazard rate and alternative hazard rate functions. Also, confidence intervals are constructed. Monte Carlo simulation is applied to demonstrate the theoretical results of the maximum likelihood estimates and confidence intervals. Finally, two real data sets are presented to examine the precision of the maximum likelihood estimates and to ensure the simulated results.