This paper illustrates the importance of independent, component-wise stochastic scaling values, from both a theoretical and empirical perspective. It is shown that a swarm employing scalar stochasticity in the particle update equation is unable to express every point in the search space if the problem dimensionality is sufficiently large in comparison with the swarm size. The theoretical result is emphasized by an empirical experiment which shows that a swarm using scalar stochasticity performs significantly worse when the optimum is not in the span of its initial positions. It is also demonstrated that even when the problem dimensionality allows a scalar swarm to reach the optimum, a swarm with component-wise stochasticity significantly outperforms the scalar swarm. This result is extended by considering different degrees of stochasticity, in which groups of components share the same stochastic scalar. It is demonstrated on a large range of benchmark functions that swarms with dimensional coupling (including scalar swarms in the most extreme case) perform significantly worse than a swarm with component-wise stochasticity. The paper also shows that, contrary to previous results in the field, a swarm with component-wise stochasticity is not biased towards the subspace within which it is initialized. The misconception is shown to have arisen in the previous literature due to overzealous normalization when measuring swarm movement, which is corrected in this paper.