Given a population of size *N* and mean *µ* (it
is recalled here that all populations are considered to be normalized), we
define a Progressive Sampling Approach (PSA) as a sequence of *N*
independent random samples of size *n*, where *n* takes values
1,2,..., *N*. The mean of each sample with size *n* will estimate the
population mean at an accuracy level *A _{n}* defined by expression
(2.6). It is evident that samples with size

In order to examine the way the accuracy *A* varies
throughout a PSA we first consider the independent variable:

(3.1) |

representing the proportion of sample size to population size.
Since *x* varies between
and 1, and *A* between
0 and 1, all plots of *A *on *x* (referred to as *A*-curves),
will always be contained in the standard rectangle formed by the coordinates
(, 0) and (1, 1). Figure 3.1
illustrates such an *A*-curve for a skewed population with size
*N=50*. The small graph inside the plot describes the distribution of the
population.

The curve formed by the accuracy *A* has a
"hyperbolic-type" shape (it is not a true hyperbola as it intersects the axes
rather than having them as asymptotes), with two distinct and clearly visible
patterns of growth. The first pattern shows that *A *is expected to be low
when the sampling proportion *x *is near its origin, and that it increases
sharply as *x* starts increasing. The second pattern shows that at some
critical value of *x* (that will be later discussed), the accuracy *A
*reaches a breakpoint after which its growth becomes much steadier and slower
until it finally reaches its maximum value 1.

When plotting the accuracy *A* throughout a PSA the area
under the *A-*curve provides a measure of the overall mean accuracy
of the PSA in question.
This is because the difference between two successive values of the variable
is always
and the area under the
accuracy curve can be expressed as:

(3.2) |

** Fig. 3.1. Illustration of a
typical plot of accuracy A against the sample proportion
x=n/N**

In general, accuracy growth throughout a PSA follows the
"hyperbolic-type" pattern illustrated in Figure 3.1. However, there exist other
possible paths, albeit of lower frequency, that do not necessarily have that
shape. Our aim now is to prove that all possible *A-*curves of a normalized
population of size *N*, have a geometrical boundary specific to that
population.

To any given population there corresponds a "Worst Possible
PSA" which is uniquely defined. This PSA consists of the "worst" possible (or
most "unlucky") samples with size 1,2,...,N. To construct it we first assume
that the elements of a
normalized population of size *N *and mean *m*, have
been ranked in ascending order so that:

It is evident that the worst sample of size 1 will be either
or
and, using the expression
(2.6) for accuracy, the worst *A* for the sample proportion
will be the minimum of
and . Likewise for
the sample proportion
*
*the worst accuracy will correspond either to the subset
or to . Generally,
for any sample proportion
the corresponding worst accuracy, now denoted by *W(x),* will be formed
either by the first *n* or the last *n* ranked elements.

In can thus be concluded that for any given population of size
*N*, all *A-*curves will be found above the curve formed by the worst
accuracy *W(x)* determined by the approach described above.

Figures 3.2-3.5 illustrate examples of *A(x)* and
*W(x)* curves for four different populations with size *N=200*. The
plots are accompanied by smaller diagrams describing the distribution of the
populations. In each case the region of all feasible PSA's is to be found above
the worst accuracy curve *W(x).*

The analytical model describing "hyperbolic-type" accuracy curves is briefly discussed in Section 6.1.

** Fig. 3.2. Example of A-
and W-curves for a flat population**

** Fig. 3.3. Example of A-
and W-curves for a convex and symmetrical population**

** Fig. 3.4. Example of A-
and W-curves for a convex and skewed population**

** Fig. 3.5. Example of A-
and W-curves for a concave population**