The Lorenz curve (LC) is one of the most important graphic tool for the measurement of income inequality, which was introduced by Max Otto Lorenz for the first time in 1905 and since then it has been studied in various fields and is now fully recognized as a subject used in scientific research. The Lorenz order is a partial order and a simple tool for comparing the alteration of non-negative random variables and is important in many fields including economics, social studies and reliability. The developments of the Lorenz curve and Lorenz order is applied in many areas of scientific fields which we have many more features of these concepts in statistics, relying on its application in the economy. 
In this article, first of all we introduce beta distribution family. Then we have shown the some special cases by letting the parameters to have specific values. More features and characteristics of Lorenz order for income distribution family are made and we compare these orders theoretically for the different distributions.
Random variables like the Lorenz curves have an easier hierarchical Lorenz order. This made us to explain the characteristics of them in a part of the article. Finally, we estimate Lorenz curves and Gini coefficient of households in Iran in 1384 and we discuss ordering the earnings data in South Khorasan province in terms of rural and urban income data.