Systematic sampling of points in continuous space is widely used in microscopy and spatial surveys. Classical theory provides asymptotic expressions for the variance of estimators based on systematic sampling as the grid spacing decreases. However, the classical theory assumes the sample grid is exactly periodic; real physical sampling procedures may introduce errors in the placement of the sample points. This paper studies the effect of errors in sample positioning on the variance of estimators. First we sketch a general approach to variance analysis using point process methods. We then analyse three different models for the error process, calculate exact small-sample variances, and derive asymptotic variances. Errors in the placement of sample points can lead to substantial inflation of the variance, dampening of Zitterbewegung effects, and a slower order of convergence. This suggests that current practice in some areas of microscopy may be based on over-optimistic predictions of estimator accuracy.