On the distribution of the largest part of small integers

Balanced groups ranked set samples method (BGRSS) is suggested for estimating the population mean with samples of size m=3k where (k=1,2,...). The BGRSS sample mean is considered as an estimator of the population mean. It is found that the BGRSS produces unbiased estimators with smaller variance than the commonly used simple random sampling (SRS) for symmetric distributions considered in this study. For asymmetric distributions that we considered, the BGRSS estimators have a small bias. A real data set is used to illustrate the BGRSS method.