By Pearn W. L., Lin G. H.

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**Example text**

10 shows a histological section from the ovarian follicle of a mouse. The shapes visible in the Figure are plane sections of individual granulosa cells. It is of interest to know the number of granulosa cells in a follicle, and their average size. 10 Thin section of granulosa cells in murine ovarian follicle. Optical microscope, transmitted light. 09 mm). R. Nyengaard, University of Aarhus, Denmark. Stereologists emphasise that, to avoid serious methodological errors, it is important to distinguish the three-dimensional particles from the flat shapes visible in plane sections, which are called profiles.

Ellipsoids with major axes a1 , a2 , a3 satisfying a1 = a2 < a3 ) or oblate spheroids (a1 = a2 > a3 ). 30) remain true for ellipsoids if the diameter of an elliptical profile is taken as the © 2005 by Chapman & Hall/CRC geometric mean of its major and minor axes, and the diameter of the ellipsoid is the largest diameter (as just defined) of any plane section parallel to the same set of planes. However, Cruz-Orive [107, 108] later showed that a general size-shape distribution of ellipsoids is unidentifiable from plane sections.

In words, © 2005 by Chapman & Hall/CRC each profile diameter S is obtained by selecting a sphere diameter T according to the diameter-weighted distribution, then multiplying T by a random fraction R. 33) πΓ( 2k + 1) 2Γ( 2k + 23 ) . and c2 = 32 . 31). Note that E[T k ] is the kth moment of diameter in the weighted distribution of sphere size: E[T k ] = = Z ∞ 0 1 µ t k dF1 (t) Z ∞ 0 t k+1 dF(t). Thus, weighted moments of particle size can easily be estimated from the section profiles. The sample mean of observed profile diameters, multiplied by 4/π, is an unbiased estimator of the diameter-weighted mean particle diameter E [T ] = Z ∞ 0 t dF1 (t) = 1 µ Z ∞ 0 t 2 dF(t).

### A Bayesian-like estimator of the process capability index Cpmk by Pearn W. L., Lin G. H.

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