<p>A new two-parameter over-dispersed discrete distribution named ‘A new quasi Poisson-Sujatha distribution’, the Poisson compound of new quasi Sujatha distribution, has been proposed. The moment-based descriptive measures have been derived and discussed. Some other important statistical properties of the distribution have been discussed. The method of moments and the method of maximum likelihood have been discussed for the estimation of parameters. The goodness of fit of the proposed distribution has been presented with four examples of count data. The proposed distribution provides better fit than other over-dispersed discrete distributions.</p>