From the data we can see that 50% mortality falls between dosages of 60 and 70%. By interpolating linearly between these two points, we can estimate LC50 as 62.0%. Similarly, LCi0 falls between 40 and 50% concentration, and by interpolation we find LCi0 to be 44.5%. However, we would expect the latter value to be too high, because the response curve in this region will be curved upward. Better accuracy can be obtained using the method of moments. First, it is necessary to compute the tolerance distribution using equation (19.1). This gives the results shown in the sixth and seventh columns of Table 19.4 [d and T(d)] and plotted in Figure 19.2b. Note that the dosages correspond to the midpoints between those in the first column.
Equations (19.2) and (19.3) require the sums of T, dT, and d2T. These values are tabulated in the last three columns of Table 19.4, and the sums are the numbers in the bottom row. Thus, the mean of the distribution is dmean = 5.911/9.374 = 63.1%, and s = (3.894/9.374 - 0.6312)1/2 = 13.3%. Thus, LC50 = 63.1%, and from equations (19.5) and (19.6), LC10 = 42.6 and LC01 = 30.9%. You can see that in this case, the interpolation estimates of LC50 and LC10 were not far off and that the LC10 was overestimated, as expected. The curves in Figure 19.2a and b are the normal distribution with the mean and standard deviation from this example.
Toxicology retains a curious legacy from precomputer days. To simplify hand calculations involving the normal distribution, toxicologists avoided the use of negative numbers by the expedient of adding an arbitrary value of 5 to the standard deviations. The resulting units are called probits. Thus, —2 standard deviations from the mean is the same as +3 probits (see Table 19.3). The median, or LD50 (0 standard deviations) becomes 5 probits. This has become standard practice and remains in use today.
The assumption that the tolerance distribution is normal is not based on fundamental principles, and a number of other distributions have been proposed. The best known is the logit, in which the following equation replaces equation (19.2):
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