  A central assumption in our analysis of cosmic structure is that cosmological
perturbations have zero ensemble mean. This property is one of the consequences
of statistically homogeneity, the invariance of correlation functions under
spatial translations. In this article we explore whether cosmological
perturbations indeed have zero mean, and thus test one aspect of statistical
homogeneity. We carry out a classical test of the zero mean hypothesis against
a class of alternatives in which perturbations have non-vanishing means, but
homogeneous and isotropic covariances. Apart from Gaussianity, our test does
not make any additional assumptions about the nature of the perturbations and
is thus rather generic and model-independent. The test statistic we employ is
essentially Student's t statistic, applied to appropriately masked,
foreground-cleaned cosmic microwave background anisotropy maps produced by the
WMAP mission. We find evidence for a non-zero mean in a particular range of
multipoles, but the evidence against the zero mean hypothesis goes away when we
correct for multiple testing. We also place constraints on the mean of the
temperature multipoles as a function of angular scale. On angular scales
smaller than four degrees, a non-zero mean has to be at least an order of
magnitude smaller than the standard deviation of the temperature anisotropies.
