Fibonacci, Lucas, Catalan, and all asymptotically exponentially increasing positive sequences have counting functions (number of elements that do not exceed a large number y) that are asymptotically proportional to the logarithm of y, a slowly varying function. For all such sequences, the variance of the elements that do not exceed y is asymptotically proportional to the product of three factors: the logarithm of the largest sequence element a(n) that does not exceed y; an explicit function of the asymptotic ratio of successive sequence elements; and the square of the mean of the elements that do not exceed y. The variance function of an integer sequence has number-theoretic interest because it distinguishes integer sequences according to the form of their variance function. The variance function is also important in the analysis of variance. Number-theoretic examples make it possible to analyze the variance function of well specified processes observed without error.