The variance of the unbiased focal-plane estimates of image centroids from a point source is calculated for arbitrary strength of scattering using path-integral results. The Markov approximation and narrow-angular scattering are assumed. The telescope aperture is a circular disk for centroid variance and two identical circular disks for the variance of the difference between two centroids. The atmospheric turbulence described by the Hill spectrum are assumed to be homogeneous along the propagation path. When there are irradiance fluctuations, the variance of the unbiased estimate of image centroids depends on the choice of the origin of the weighting vector. The variance of the centroid of a single image depends on three physical parameters related to the Fresnel scale, inner scale, and strength of scattering. The variance of the separation between two images from two separated telescope apertures depends on the same parameters in addition to the normalized separation of the apertures. The important contributions from propagation distance and spectral wavenumber are identified for the different parameter regimes. The behavior of centroid variances are presented for typical boundary layer conditions and aperture sizes.