In this paper, an orness measure to reflect the or-like degree of the quasi-arithmetic mean operator is proposed. With the generating function representation method, some properties of a quasi-arithmetic mean, associated with its orness measure, are analyzed. Then, the paper focuses on two kinds of parameterized quasi-arithmetic means with exponential function and power function generators respectively. Some properties of them are examined and compared. As the aggregation value for any set always monotonically increases with the orness level, the orness level can be used as a control parameter to represent the decision maker's preference. A generic method to get the quasi-arithmetic mean operator which makes the aggregation result consistent with its orness level is also examined. Finally, the properties of the quasi-arithmetic mean and that of the exponential and power function generators are extended to the weighted quasi-arithmetic mean by considering the importance of the aggregated elements