List of links to frequency distributions
All these plots are "probability plots" whose horizontal axes are concentrations or log concentrations of Sb and whose vertical axes represent cumulative probabilities (normal or uniform). Straight lines represent coherent distributions, i.e., modes.
14 points (Guinn's test bullets)
The first two plots are the same as in the monograph, but with lines added to show the possible presence of two modes. The normal probability plot shows a good fit to the lower mode, which presumably represents virgin ore with some effect from scrap ore, and a less-good fit to the upper more, which presumably represents diluted scrap lead previously hardened with high concentrations of Sb. The log-normal probability plot is the one that Larry used for his probability calculations, with a single line through all the points. That still seems to be a reasonable approach, far more so than trying to fit a single line through the normal plot. Here we investigate whether it lends itself to two lines, and conclude that it works better at the upper end than at the lower end.
The last two plots are attempts to see whether a uniform distribution (even spread between 0 and 1200 ppm of Sb) can explain the points. The uniform plot shows that the points as a whole do not fit a single line. Instead, the lower eight fit tightly to one line and the upper six somewhat more loosely to another. The log-uniform plot also shows two lines, but with the opposite tightness of fit. Interestingly, it appears that the uniform lines fit as well or better than the normal lines.
Upper points (of the 14 test bullets)
The second of each pair of plots fits the upper mode better. This leads us to think that the upper mode is logarithmic.
Lower points (of the 14 test bullets)
The lower points are definitely fit worse by log functions than by linear functions. This we consider the lower points to be broadly linear.
12 quarters of three bullets (heterogeneity test)
The interesting thing about these four plots is how similar they are to each other. We suspect that the fundamental reason is the narrow range of Sb in the 12 quarters—a factor of only three vs. 50 to 60 for the 14 bullets. The narrower the range, the more easily it can be fit by any distribution. The three lines on these plots are strictly preliminary—there may be only two zones.
16 points (Guinn's 14 plus unfired round and Walker fragment)
The upper points are the same here as in the 14 points because the two added have the lowest concentrations of Sb (15 and 17 ppm). The extra points seem to make a cluster at the low end. Whether this represents a mode remains unclear. The expanded set of lower points is still fit better by a uniform function than by a normal one.
Views of the uniform distribution
How do real uniform distributions appear on the kinds of
probability plots discussed above? To address this question, we took 100 uniform
points (1, 1, 1, etc.), formed the cumulative distribution (1, 2, 3, etc.), and
plotted the latter against various probability scales on the vertical axis
(normal, log-normal, log-uniform). We did this for 100 points and 10 points. The
results showed that the normal probability plot appears S-shaped (for lack of a
better word), with a broad linear zone in the center and positive and negative
tails at the top and bottom, respectively. We were interested to note that the
actual distributions, at least part of which he have independently come to think
may may be uniform or nearly so, shows this lower tail, a linear zone in the
middle, and at least the beginnings of a positive tail at the upper end. The
log-normal plot of the uniform data appear quasi-exponential, with a steep tail
at the top. We do not see this kind of tail at the upper end of our actual
log-normal plots. A similar shape was seen for the log-uniform plot. Taken
together, these results reinforce our impression that the lower ends of our
distributions may be uniform (or normal), whereas the upper ends are logarithmic
100 points viewed on a normal probability plot
100 points viewed on a log-normal probability plot
100 points viewed on a log-uniform plot
10 points viewed on a normal probability plot
10 points viewed on a log-normal probability plot
10 points viewed on a log-uniform plot
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