Aggregation operators for linguistic variables usually assume a uniform and symmetrical distribution of the linguistic terms that define the variable. This paper de- fines the Induced Unbalanced Linguistic Ordered Weighted Average (IULOWA). This aggregator takes into account the fuzzy membership functions of the terms during the aggregation operations of the pairs of terms. There is no restriction on the form of the mem- bership functions of the terms, which can be triangular or trapezoidal, non-symmetrical and non-equally dis- tributed. Moreover, the paper proposes to use the speci- ficity and fuzziness measures of the terms to induce the order of the arguments, providing some examples of this criterion in decision making.