In a secondary analysis of data taken from a publicly available database, we examined cognitive performance, postural sway, and relations between them for four groups: younger and older individuals with versus without a recent history of falls. Our objective was to compare linear versus nonlinear measures of postural activity as post-hoc predictors of cognitive performance and falling. We evaluated standing body sway in 147 participants (18 to 85 years-old) over 60 seconds, separately with eyes-open and with eyes-closed. We evaluated cognitive performance using portions of the Trail Making Test. We evaluated postural activity in terms of standard deviation, velocity, and amplitude of the CoP. Separately, we used detrended fluctuation analysis (DFA) to examine the complexity of CoP displacements. Using analysis of variance, we conducted separate analyses of cognitive performance and postural activity comparing Younger and Older Adults and Non-fallers and Fallers, taking into account Vision (eyes-closed vs. open) and the direction of postural movements (AP vs. ML) while also controlling for participants' characteristics. We used moderation analyses to evaluate whether relationships between Trail Making Test scores and the linear and nonlinear outcomes were moderated by Age group or Fall status. For postural activity, only DFA differed between Non-fallers and Fallers. Older adults exhibited increased complexity associated with better processing speed function, while fallers show an opposite association, relying on processing speed to increase postural rigidity instead of facilitating adaptive control of balance. We conclude that DFA can provide information regarding postural activity and cognitive performance that cannot be obtained from more traditional, linear measures of postural activity, and that DFA may be a valuable tool for assessing fall risk.