In this dissertation a new distribution, “Transmuted modified Burr III distribution” is proposed. The proposed distribution is more flexible and tractable than its parent Burr III distribution. The proposed distribution generalizes some of the well known distributions which already exist in literature. Further we try to develop some of the characteristic properties of the proposed distribution. We try to characterize the proposed distribution using conditional negative moments and reliability measure of transmuted modified Bur III (TMB III) distribution is derived. Using stereographic projection we derived semicircular stereographic TMB III distribution for circular or directional data. Some of the characteristic measures are derived of semicircular stereographic TMB III distribution such as characteristic function, mean direction, trigonometric moments, central trigonometric moments, resultant length, variance, skewness and kurtosis. In last to elaborate the application of the proposed distribution we use real data set. Using maximum likelihood technique we estimate parameters of proposed distribution among all its sub models. Likelihood ratio criterion such as AIC (Akaike Information Criterion), CAIC (Consistent Akaike Information Criterion) and BIC (Bayesian Information Criterion), are computed using SAS to find the best model among all its sub models and we come to know that the proposed model provide us best fit.