We show several variants of concentration inequalities on the sphere stated as sub-Gaussian estimates with optimal constants. For a Lipschitz function, we give one-sided and two-sided bounds for deviation from the median as well as from the mean. For example, we show that if \(\mu \) is the normalized surface measure on \(S^{n-1}\) with \(n\geqslant 3\) , \(f : S^{n-1} \to \mathbb {R}\) is 1-Lipschitz, M is the median of f, and \(t >0\) , then \(\mu \big (f \geqslant M +t\big ) \leqslant \frac 12 e^{-nt^2/2}\) . If M is the mean of f, we have a two-sided bound \(\mu \big (|f - M| \geqslant t\big ) \leqslant e^{-nt^2/2}\) . Consequently, if \(\gamma \) is the standard Gaussian measure on \(\mathbb {R}^n\) and \(f : \mathbb {R}^{n} \to \mathbb {R}\) (again, 1-Lipschitz, with the mean equal to M), then \(\gamma \big (|f - M| \geqslant t\big ) \leqslant e^{-t^2/2}\) . These bounds are slightly better and arguably more elegant than those available elsewhere in the literature.