 relationship if their readings are, in effect, random selections from the available pool. To illustrate, consider two randomly generated “witnesses” with ten variation units where each variation unit has two equally probable readings:
 The distance between these two is 0.7, corresponding to seven out of ten disagreements. If the same kind of trial is performed many times then the resulting distribution of disagreements conforms to the binomial distribution:
 Plotting the distribution for ten trials and a probability of disagreement of 0.5 shows the nature of the distribution:
 Any number of disagreements between zero and ten out of ten is possible, but numbers near the centre of the range are more likely to occur than those in the upper (i.e. right-hand) or lower (i.e. left-hand) “tails” of the distribution.
 Given a knowledge of the distribution of distances, it is possible to calculate how often a particular distance is likely to occur by chance. By choosing a sufficiently small probability level, one can assert that any distances with an even smaller probability of chance occurrence are significant from a statistical point of view. This small probability is called an alpha level, and for this kind of study is often chosen to be 0.05 (i.e. 5%). Once the alpha level is selected, it is possible to calculate corresponding minimum and maximum distances called critical limits. The probability of a smaller or greater distance occurring by chance is then less than or equal to the alpha level. Thus, choosing an alpha level of 0.05 means that distances more extreme than the critical limits have less than or equal to one chance in twenty of occurring.
 The following table gives critical limits of distance calculated for various numbers of variation units by assuming that each one has two equally probable readings:
|Number||Lower limit||Upper limit|
|10||0.200 (2/10)||0.800 (8/10)|
|50||0.360 (18/50)||0.640 (32/50)|
|100||0.400 (40/100)||0.600 (60/100)|
|500||0.456 (228/500)||0.544 (272/500)|
|1000||0.469 (469/1000)||0.531 (531/1000)|
 The relative width of the central range between lower and upper limits increases as the number of variation units decreases. This central range corresponds to distances which are not considered to be significant. If the number of variation units is five or less, the central range expands to cover all possible distances. Consequently, it is a waste of time to calculate a distance for such a small sample size. The same applies if one prefers to calculate a percentage of agreement rather than a distance.
 Real variation units do not conform to the special case of two equally probable readings. Instead, each variation unit may have more than two readings, and each reading has its own probability of occurrence. Estimating critical values of distance is more difficult in these circumstances but is still achievable. The first step is to estimate the probability of random disagreement for each variation unit as follows:
Estimate the probability of each reading in the variation unit by counting the number of witnesses that support the reading (ri) then dividing by the total number of witnesses (w) reported for the variation unit.
Take the complement of the sum of squares of probabilities of readings to estimate the probability of random disagreement. That is, P(disagree) = 1 - Σ(ri/w)2.
 To elaborate, if two readings are chosen at random from within a variation unit then the chance of choosing the same reading twice is the square of its probability of being selected once. The squared probabilities are summed to take account of all possible ways in which two witnesses can have the same reading: both having the first reading, both having the second, or both having the third, and so on up to however many readings are contained in the variation unit being examined. The sum of squared probabilities is the estimated probability of agreement. The estimated probability of disagreement is then obtained by subtracting the probability of agreement from one. As a concrete example, the first variation unit in the UBS4 