31 (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 apparatus of Hebrews (Heb 1:3) reports the readings of 66 witnesses. Of these, 24 have the first reading, 6 the second, and 36 the third. The estimated probabilities of the readings are then 0.364 (24/66), 0.091 (6/66), and 0.545 (36/66), respectively. The estimated probability of random agreement is (24/66)2 + (6/66)2 + (36/66)2 = 0.438, giving an estimated probability of disagreement of 1 - 0.438 = 0.562.
 Once the probability of random disagreement is known for each variation unit, it is possible to calculate critical values of distance. To obtain the values by an exact calculation is an intensive computing exercise; performing such a calculation for the 44 variation units of Hebrews presented in the UBS4 apparatus would take a typical computer months to complete. Happily, one can instead use a “Monte Carlo” calculation to obtain the critical limits in a short time. This kind of calculation simulates a large number of trials using a program that generates pseudo-random numbers. Such a calculation that employs enough trials (say, 10,000) has a good chance of producing the same critical values as would be obtained by an exact calculation. To give an example, the estimated critical distances obtained by a Monte Carlo calculation for the set of variation units where the readings of both P46 and B are reported are 0.174 (4/23) and 0.565 (13/23), respectively. The calculated distance between P46 and B (0.130) is less than the lower critical distance (0.174) so it can be regarded as statistically significant. That is, there is less than or equal to one chance in twenty that such a distance will occur if readings are randomly chosen from variation units where both P46 and B are reported.
 The UBS4 apparatus provides 44 variation units for Hebrews. This is certainly not a comprehensive account of all variation among extant witnesses. To identify all of the variation among these witnesses would require them all to be collated, and this would still miss variations that once occurred among witnesses now lost. Extant witne