We consider skew products over Bernoulli shifts, whose fibred dynamics is given by diffeomorphisms of the interval. We study the statistical and/or non-statistical behavior, referring to convergence and/or non-convergence, almost everywhere, of the Birkhoff average, respectively. We employ the Schwarzian derivative of the fiber maps and the arc-sine law to identify conditions under which these skew products exhibit these types of behavior. We identify distinct types of behavior according to the Schwarzian derivative. When the Schwarzian derivative is negative, the skew product has intermingled basins. Conversely, when the Schwarzian derivative is positive, the skew product has a physical measure. Finally, when the Schwarzian derivative is zero, the skew product has non-statistical behavior. In the latter scenario, we establish a connection between non-statistical behavior and the arc-sine law that allows us to obtain results in other settings independent of the sign of the Schwarzian derivative.