Summary: Niemi [Am Polit Sci Rev 63:48–49] proposed a simple measure of the cohesiveness of a group of \(n\) voters’ preferences that reflects the proximity of their preferences to single-peakedness. For three-candidate elections, this measure, \(k\), reduces to the minimum number of voters who rank one of the candidates as being least preferred. The current study develops closed form representations for the conditional probability, PASW\((n,IAC |k)\), that all weighted scoring rules will elect the Condorcet winner in an election, given a specified value of \(k\). Results show a very strong relationship between PASW\((n,IAC |k)\) and \(k\), such that the determination of the voting rule to be used in an election becomes significantly less critical relative to the likelihood of electing the Condorcet winner as voters in a society have more structured preferences. As voters’ preferences become more unstructured as measured by their distance from single-peakedness, it becomes much more likely that different voting rules will select different winners.