In this paper we consider the effect of element positioning errors on the pattern of a uniform linear array. An error in positioning each element in the longitudinal direction relative to the previous one is assumed to occur with a Gaussian distribution of zero mean and variance σ2. An important effect of this type of error is that although the main beam remains practically unchanged, there is a suppression of the grating lobes whose expected levels can be made less than the first sidelobe by increasing the number of elements and/or σ. In the side-lobe region, far from the mainlobe, the mean and variance of the pattern approach a constant whose value is inversely proportional to the number of elements. This permits one to use larger element spacings that give higher resolution without grating-lobe ambiguities. An example is given in connection with the mapping of two incoherent point sources. Expressions are derived and graphs are given for the mean and Standard deviation of the field pattern and for the mean and standard deviation of the power pattern. The theory was verified statistically by means of a computer that generated Gaussian random numbers. It also provided relative frequency histograms of power at the first grating lobe.