Aggregation operators for linguistic variables usually assume a uniform and symmetrical distribution of the linguistic terms that define the variable. A well-known aggregation operator is the Linguistic Ordered Weighted Average (LOWA), which has been extensively applied. However, there are some problems where an unbalanced set of linguistic terms is more appropriate to describe the objects. In this paper we define the Unbalanced Linguistic Ordered Weighted Average (ULOWA) on the basis of the LOWA operator. ULOWA takes into account the fuzzy membership functions of the terms during the aggregation process. There is no restriction on the form of the membership functions of the terms, which can be triangular or trapezoidal, non symmetrical and non equally distributed. The paper demonstrates the properties of ULOWA. Finally, a comparison of this operator with some other aggregation operators for unbalanced sets of terms is done.