Figure 1. Distribution of pivot[N] from the Cricklewood experiment (M = N). Solid circles include all the data, whereas open circles refer to N < 5, and.

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Figure 1. Distribution of pivot[N] from the Cricklewood experiment (M = N). Solid circles include all the data, whereas open circles refer to N < 5, and ‘x’ to N > 25 and indicate rough independence of the distribution on N (UK Crown Copyright, 1982; reproduced from Ogden, 1982, by permission). From: Confidence Intervals for Asbestos Fiber Counts: Approximate Negative Binomial Distribution Ann Work Expo Health. 2017;61(2):237-247. doi:10.1093/annweh/wxw020 Ann Work Expo Health | Published by Oxford University Press on behalf of the British Occupational Hygiene Society 2017.

Figure 7. Negative binomial confidence interval levels at s = 40% Figure 7. Negative binomial confidence interval levels at s = 40%. Solid curves are approximate limits on the discrete variation [equation (18)]. From: Confidence Intervals for Asbestos Fiber Counts: Approximate Negative Binomial Distribution Ann Work Expo Health. 2017;61(2):237-247. doi:10.1093/annweh/wxw020 Ann Work Expo Health | Published by Oxford University Press on behalf of the British Occupational Hygiene Society 2017.

Figure 2. Normalized negative binomial quantiles— (nβ−N)/N+s2N2 —for large-N true relative standard deviation s = 20% at levels β = 5 and 95% plotted versus population mean count N. Exact values (blue); approximations [red, equation (A21)] shifted slightly left for graphical visibility; smooth curve [red, equation (A22)] neglecting the discontinuities of the discrete values. From: Confidence Intervals for Asbestos Fiber Counts: Approximate Negative Binomial Distribution Ann Work Expo Health. 2017;61(2):237-247. doi:10.1093/annweh/wxw020 Ann Work Expo Health | Published by Oxford University Press on behalf of the British Occupational Hygiene Society 2017.

Figure 3. Normalized negative binomial quantiles at s = 40% at levels β = 5 and 95% plotted versus population mean count N. Exact values (blue); approximations (red). From: Confidence Intervals for Asbestos Fiber Counts: Approximate Negative Binomial Distribution Ann Work Expo Health. 2017;61(2):237-247. doi:10.1093/annweh/wxw020 Ann Work Expo Health | Published by Oxford University Press on behalf of the British Occupational Hygiene Society 2017.

Figure 4. Normalized Poisson distribution quantiles at levels β = 5 and 95% plotted versus population mean count N. Exact values (blue); approximations (red). From: Confidence Intervals for Asbestos Fiber Counts: Approximate Negative Binomial Distribution Ann Work Expo Health. 2017;61(2):237-247. doi:10.1093/annweh/wxw020 Ann Work Expo Health | Published by Oxford University Press on behalf of the British Occupational Hygiene Society 2017.

Figure 5. Normalized 95% quantiles versus the mean N at several values of true relative standard deviation s. The horizontal lines represent asymptotic values lβ at N→∞ . From: Confidence Intervals for Asbestos Fiber Counts: Approximate Negative Binomial Distribution Ann Work Expo Health. 2017;61(2):237-247. doi:10.1093/annweh/wxw020 Ann Work Expo Health | Published by Oxford University Press on behalf of the British Occupational Hygiene Society 2017.

Figure 6. Negative binomial confidence interval levels at s = 20% at nominal single-sided levels = 5% and two-sided interval with nominal level = 10%. Solid curves are approximate limits on the discrete variation [equation (18)]. From: Confidence Intervals for Asbestos Fiber Counts: Approximate Negative Binomial Distribution Ann Work Expo Health. 2017;61(2):237-247. doi:10.1093/annweh/wxw020 Ann Work Expo Health | Published by Oxford University Press on behalf of the British Occupational Hygiene Society 2017.

Figure 8. Relative [i.e. (nβ−n)/n×100% ] confidence limits at the 95% level at intra-lab imprecision s = 0.20. From: Confidence Intervals for Asbestos Fiber Counts: Approximate Negative Binomial Distribution Ann Work Expo Health. 2017;61(2):237-247. doi:10.1093/annweh/wxw020 Ann Work Expo Health | Published by Oxford University Press on behalf of the British Occupational Hygiene Society 2017.

Figure 9. Cumulative normal and negative binomial distributions of background counts in 100 fields, assuming the mean value N′i=2.5 mm−2 as published in NIOSH 7400 and intra-counter variability = 20%. The LOD values indicated refer to bias-less signals from a single filter, i.e., with 2.5 mm<sup>2</sup> subtracted out. From: Confidence Intervals for Asbestos Fiber Counts: Approximate Negative Binomial Distribution Ann Work Expo Health. 2017;61(2):237-247. doi:10.1093/annweh/wxw020 Ann Work Expo Health | Published by Oxford University Press on behalf of the British Occupational Hygiene Society 2017.