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There are two classifications of spacing equations which will be discussed here. Some equations estimate the proximity of the paste to the voids, and others estimate the proximity of the voids to one another. Although this may seem a subtle distinction, it will be shown that the mathematical relationships that characterize these concepts have different behaviors.

Also,
any reasonable concept of "spacing" should address the fact that there
must exist a distribution of distances which characterize the
spacing.
Clearly, some regions of the paste are closer to an air void
than other regions, and some voids have nearer neighbors than others. This
characteristic can be represented by a distribution of distances,
as depicted in Fig. 1 for a distance *s*.
In this figure, the probability density function (PDF) is a normalized function
with unit area under its curve. This function represents the fraction
of spacings found in the interval *[s,s+ds]* for some
differential element *ds*. The associated cumulative distribution function
(CDF) is the integral of the PDF. This function increases monotonically from
zero to unity and represents the fraction of spacings less than *s*.

**Figure 1:** An idealized representation of a spacing cumulative
distribution function (CDF) and the associated probability density
function (PDF) for some distance *s*. The dashed lines demonstrate how
to determine the 50 * th* and the 95 * th*
percentiles from CDF data.

An illustration of using the CDF is also shown in Fig. 1.
Two horizontal dashed lines intercept the ordinate axis at the 50 * th*
and 95 * th* percentiles. These lines intercept the CDF at **s** values of
1.95 and 3.1, respectively; 50 % of the spacings are less
than 1.95, and 95 % are less than 3.1.
In theory, the CDF only asymptotes to unity, and to capture all of the
spacings, *s* must increase to infinity. In practice, however, the quantity
*s* can only increase to the size of the
system. Therefore, the concept of a maximum spacing is an ill-defined
quantity.
In this experiment,
the 50 * th* and the 95 * th* percentiles of the spacing
distributions will be used
to characterize both the measured and the estimated values since these
percentiles are intuitive to one's concept of spacing and protected
paste.