Since there is, in principle, no reason why third parties should not pay individuals for the use of their data, we introduce a realistic market that would allow these payments to be made while taking into account the privacy attitude of the participants. And since it is usually important to use unbiased samples to obtain credible statistical results, we examine the properties that such a market should have and suggest a mechanism that compensates those individuals that participate according to their risk attitudes. Equally important, we show that this mechanism also benefits buyers, as they pay less for the data than they would if they compensated all individuals with the same maximum fee that the most concerned ones expect.