We present a new method for detecting superdiffusive behaviour and for determining rates of superdiffusion in time series data. Our method applies equally to stochastic and deterministic time series data (with no prior knowledge required of the nature of the data) and relies on one realisation (ie one sample path) of the process. Linear drift effects are automatically removed without any preprocessing. We show numerical results for time series constructed from i.i.d. $α$-stable random variables and from deterministic weakly chaotic maps. We compare our method with the standard method of estimating the growth rate of the mean-square displacement as well as the $p$-variation method, maximum likelihood, quantile matching and linear regression of the empirical characteristic function.