Indeed, the (non-central) chi-squared distribution associated to a random point in the interval [-1, 1] is the same as the distribution of the length-squared of a random point in the d-cube. By the law of large numbers, this distribution concentrates itself in a narrow band around d times the standard deviation squared (σ2) of the original derivation. This illuminates the chi-squared distribution and also illustrates that most of the volume of the d-cube concentrates near the surface of a sphere of radius σ d {\displaystyle \sigma {\sqrt {d}}} .