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This paper introduces a new distribution called the power exponentiated Weibull distribution by generalizing the Weibull distribution using the power exponentiated family of distributions. The statistical aspects of the proposed distribution such as the probability density function, cumulative distribution function, and reliability measures are studied. The mixture representation of the distribution is studied. The distribution includes the alpha power Weibull distribution and the power exponentiated exponential distribution as special cases. Statistical properties such as moments, stress strength reliability, Renyi entropy, quantiles, income inequality measures, stochastic ordering, order statistics, etc., are derived. The parameter values of the distribution are obtained using the maximum likelihood estimation method. The maximum likelihood estimates have been inferred by the Monte Carlo simulation study on the proposed distribution, and the adequacy of the proposed distribution is shown by modeling it on real-life cancer-related datasets. The measures such as Akaike information criteria, Bayesian information criteria, Hannan–Quinn information criterion, Anderson–Darling statistic, k-s statistic, p-value, etc., shows that the proposed distribution can provide a better fit compared to the exponentiated power Weibull distribution, the exponentiated exponential distribution, the alpha power Weibull distribution, the Weibull distribution and the exponential distribution. [ABSTRACT FROM AUTHOR]