Statistical Considerations Regarding BC Variations
I appreciate the discussion, and I apologize that I cannot address every question asked in the desired level of detail. In some cases, time constraints prevent a longer discussion. In other cases, I am limited by consideration of my co-authors and the approval process of my employer regarding internal review and approval for public release of data. Rest assured, our unpublished data is in various stages of manuscript preparation, peer review, and approval processes.
We should recall that if the shot-to-shot random measurement errors are +/- X% for a given bullet, that the uncertainty in the mean will be +/-X%/sqrt(n - 1) for a sample size of n. In other words, if the shot-to-shot measurement errors are +/- 5% for a sample size of 5 shots, then the uncertainty in the mean BC will be +/- 2.5%. One wound need to average BCs from a sample size of 26 shots to reduce the uncertainty in the mean to 1%.
It was aptly pointed out that few shooters choose lead tipped bullets for long range hunting, so that it makes more sense to confine the discussion to match, hollow point, and plastic tipped bullets. Recall that near Mach 1.75, the drag coefficient of the Berger .264 caliber 140 grain VLD (Litz p. 402) varies from roughly 0.27 to 0.31 which is about 13%, or +/- 6.5%. Dividing by sqrt(n - 1) for a sample size of 10 suggests an uncertainty in the mean of +/- 2.2%. Consider also that near Mach 2.4, the .257 caliber 115 grain VLD (Litz p. 390) shows a shot-to-shot variation of close to 10%, or +/- 5%. The uncertainty in the mean for a sample size of four shots is then close to +/- 2.9%. Likewise, near Mach 2.3, the Barnes 115 grain TTSX (Litz p. 397) shows about a 10% variation in drag coefficient, which for a sample size of four shots also yields an uncertainty in BC close to +/- 2.9%. (There is some ambiguity between the label, the drawing, and p. 316 whether this is the tipped TTSX or the hollow point TSX.)
Much like the Litz data, our data also tends to show greater shot-to-shot variations in lead tipped bullets than hollow point match designs and plastic-tipped designs. However, just as in the Litz data, the shot-to-shot variations in drag prevent us from determining the average BC to 1% or better in many hollow point and plastic tipped designs. In our earlier published paper (
http://arxiv.org/pdf/0705.0389), the uncertainties in the mean BC vary from 0.5% for the 40 grain .224 VMAX to 11% for the 110 grain .308 VMAX.
In the earlier paper, we report the measured the G1 BC of the 115 grain .257 caliber VLD to be 0.419 with a 1% uncertainty in the mean. The published reply by Bryan Litz suggested that an old lot of bullets yielded dimensional variations that were responsible for the 13% difference between our measurements and measurements on a more recent lot as published by Litz. Presumably there is a range of lot-to-lot variations from different production lines, and perhaps we accidentally stumbled upon an unusually large lot-to-lot variation, and the vast majority of lots from most companies are within 5% of each other.
However, since there is very little published data on lot-to-lot variations, we should recognize that optimistic estimates of lot-to-lot variations are little more than an educated guess. Such hypotheses should be subject to validation by empirical testing rather than being relied upon too heavily. Comparing the Sierra and Litz data certainly suggests that either rifle-to-rifle or lot-to-lot variations over 5% and even over 10% are not rare events. As one ballistics expert quotes at his web site "In theory, there is no difference between theory and practice. But in practice, there is." (Yogi Berra)
Aye!
Michael
