The one-way analysis of variance is concerned with comparing the locations of several one-dimensional samples. This paper gives a simple unified semi-graphical and semi-analytical approach to the problem based on approximation intervals for the locations of the samples. The intervals are standardized for normally distributed data to achieve given coverage probabilities. No assumptions are placed on the sample sizes or the sample scales. All questions concerning the relationships between the locations of the samples are reduced to questions concerning the approximating intervals. The procedure is stable in that small perturbations either of the data or the standardizing model lead to only small changes in the analysis.