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The variance of a spectral estimate of a stochastic process is essential to the formulation and performance of a spectral classifier. The overall variance of a spectral estimate originates from two sources: the within-class spectral uncertainty and the variance introduced in the spectral estimation procedure. In this paper, we derive the total variance of a periodogram-based spectral estimate under some assumptions. Using this result, we formulate a linear spectral classifier based on Fisher's separability metric. The classifier is used to classify two oceanographic processes: ocean convection versus internal waves.