In the field of optics, the design of a lens often involves analyses of the variability in an optical parameter (such as lens focal length) under manufacturing conditions, which is done in order to identify sources of high variability that may cause the optical parameter to be out of its specification. In this interdisciplinary paper, the author shares his experience in lens design, where analyses of variability are often performed through examining the partial derivatives in a multivariable Taylor series expansion of an optical parameter to second order when interactions are expected. It is shown algebraically that when the second-order expansion is applied to express a sum of squares, negative terms in the sum can occur. This instance of negative-valued “variance-like components” is interesting pedagogically in applied mathematics and statistics, as well as from an engineering standpoint, as it suggests that, based on theory, the variance of a function of random variables may not necessarily be comprised of a simple sum of positive-valued squared deviations comprising its total variance.
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