A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino-Southern Oscillation. The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finitelength time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red ...