ents a selection of sample sizes and corresponding relative widths obtained using a probability of disagreement of 0.5 and cumulative probabilities of 0.025 and 0.975:
|Sample size||Lower limit||Upper limit||Relative width|
 In summary, this section covers three statistical questions:
What range of distances between a pair of witnesses is likely to occur if readings are chosen at random from the available pool? If the actual distance is outside this range then it can be considered to be significant in a statistical sense.
What confidence interval applies to a distance between two witnesses estimated from a sample of variation units? One can be confident but not certain that that the actual distance is within an interval defined by critical limits calculated from a random sample of the entire population.
What is the minimum sample size required to obtain an acceptable relative width for the confidence interval calculated from a sample of variation units?
 A Monte Carlo calculation provides estimates for critical values which allow one to decide whether a distance between two witnesses is statistically significant. The binomial distribution is useful for calculating the confidence interval which applies to the distance between two witnesses estimated from a sample of variation units. The binomial distribution can also be used to estimate the minimum sample size required to obtain a given relative width for the confidence interval.
 A distance matrix is a table of distances between pairs of objects, which, in the present case, are textual witnesses. The distance matrix is square and symmetrical about its diagonal because its rows and columns relate to the same set of objects. The diagonal is comprised entirely of zeros because the distance from every object to itself is zero.
 The larger the sample size, the more reliable a distance estimate based on the sample. The relative width of the confidence interval associated with an estimated distance between two witnesses increases as the number of variation units in the sample decreases. A distance calculated from a sample of five or less variation units is entirely worthless because the associated confidence interval covers all possible distances. Unfortunately, increasing the sample size excludes witnesses that do not have a full complement of variation units. It is therefore necessary to reach a compromise which produces a confidence interval of tolerable width while not excluding too many interesting witnesses. For calculations related to this article, a minimum sample size of twelve has been used, corresponding to a relative interval width of just over 50% (actually 53.8%). The main reason for choosing this rather than a larger sample size is to include Codex Vaticanus (B) in the results.
 Three distance matrices are used to produce the results which follow:
 The first two are contrived examples that will be used to illustrate certain aspects of multivariate analysis. The third is derived from the previously mentioned data matrix which is based on the UBS4 apparatus of Hebrews. In constructing this distance matrix, a constraint was introduced to ensure that each distance was calculated from at least twelve variation units. Due to the added constraint, only 54 objects are represented.
 Multidimensional scaling (MDS) takes a distance matrix and derives a configuration of points in a space with a specified number of dimensions so that distances between the derived points are a best match to the original distances. As Venables and Ripley say, “We can think of 'squeezing' a high-dimensional point cloud into a small number of dimensions ... whilst preserving as well as possible the inter-point distances.” The configuration found by a popular variety of multidimensional scaling called “classical” MDS minimises the sum of squared differences between the original and derived distances. In general, the maximum number of dimensions required to perfectly reproduce the original distances is one less than the number of objects. However, a reasonably good approximation can sometimes be ob