We extend recent advances in computing variance-covariance matrices from genetic distances to a sequence method of phylogenetic analysis. These matrices, together with other statistical properties of corrected sequence spectra, are studied as a foundation for more powerful and testable methods in phylogenetics. We start with ($) over cap s, a vector of the proportion of sites in a sequence of length c showing each of the possible character-state patterns for t taxa. Hadamard conjugations are then used to calculate ($) over cap gamma, a vector of the the support for bipartitions, or splits, in the data, after correcting for all implied multiple changes. These corrections are made independently of any tree and are illustrated with Cavender's two-character-state model. Each entry in ($) over cap gamma(($) over cap gamma(0) excluded) that is not associated with an edge on the tree that generated the data is an invariant (sensu Cavender) with an expected value of 0 as the number of sites c-->infinity. Under an independent identically distributed model (sites are independent and identically distributed), vector ($) over cap s is a random sample from a scaled multinomial distribution. Starting from this point, we illustrate the derivation of V[($) over cap gamma], the variance-covariance matrix of ($) over cap gamma. The bias induced by the delta method, a convenient approximation in deriving V[($) over cap gamma], is evaluated for both population and sample variance-covariance matrices. It is found to be acceptable in the first case and very good in the second. Likewise bias in ($) over cap gamma due to a logarithmic transform and to short sequences is also acceptable. We infer the marginal distributions of entries in ($) over cap gamma. Simulations with illustrative values of c and lambda (the rate per site) show how ($) over cap gamma tends to multivariate normal as c-->infinity. Our results extend naturally to four-color (nucleotide) spectra.