We consider a human contact social network that has connections through cellphone addresses. To construct such a social network, we use real call records provided by a large carrier, and connect to each other if there exists a call record between any two cellphone users. Due to its huge amount of data, we down-sample it in a way that the smallest-degree nodes are removed, in turn, from the network. For a moderate size of the networks we show that the degree distribution of the network follows a power-law distribution via linear regression analysis, implying the so-called scale-free property. We finally suggest some alternative measures to analyze a social network.