For completely randomized designs the experimental units are assigned to treatments entirely at random. Hence, for example, if an experiment is examining the effects of 4 different treatments (e.g. medical procedures, application of fertilizers to sample plots, or additives to a chemical process) and we wish to test each treatment on 8 units, we have a total of 32 units x treatments to test. Each of the 32 tests is then assigned a number, from 1 to 32, and units are assigned to treatments by random number selection. A similar approach applies where a single factor is being studied, with a number of fixed levels of that factor. For example, four different fertilizer concentrations might be applied, with 8 plots being used per level, again giving a total of 32 plots or runs to use for the experiment. There are a very large number of possible random arrangements that might be selected in these examples. The specific assignment made can be created manually or programmatically. In the latter case, the software used will often allow the design to be stored and associated with the data obtained when the experiment is run, thereby integrating the subsequent analysis with the design process.
Let yij represent the data obtained from the experiment (the measured outcome or result) conducted on the jth replicate that receives the ith treatment; let Ti be the effect attributable to the ith treatment and let e denote residual error, unexplained by other factors. Then the statistical model for this kind of experiment is of the form:
This model states that the measured response is a simple linear function of the overall mean value for all the data, plus a treatment effect, plus some residual error. The overall mean is estimated from the mean of all the sample data, whilst the treatment mean values are estimated from the means for each treatment group. The treatment effects are estimated as the difference between the overall mean and the individual treatments means. Analysis of the data is typically achieved using one-way or single-factor analysis of variance (ANOVA).