The intermeans parameter has some similarity with the
interquartile range, where one subtracts the median of the left hand
part from the median of the right hand part, both parts being separated
by the median of the population. Here we use the means and subtract
the mean of the left part from the mean of the right part, both parts
being separated by the mean of the population, the result being afterwards
multiplied by 
. Therefore 
appears
to be a measure of the dispersion of the distribution.
From now on, all 
are relative to the mean,
and subscripts will be omitted. It can be seen that 
and 
. Therefore, the points
are ever situated as in Figure 3
and the relation 
holds.
Since using to describe the dispersion seems
to be new, we have undertaken a comparison between this parameter
and the usual one, namely the standard deviation 
. For any
distribution, we have:
. In Table 1,
the ratio 
is given for various distributions.
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It could be noticed that the ratio is quite the
same for all triangular distributions, whatever their skewness. For
the lognormal distribution, the ratio varies. For
the wildest ones, i.e. for 
, one has
. But, in the practical cases, i.e.
when the demand has only one chance over thousand to go outside the
interval 
, the relation 
holds.
In order to extract from historical data, the
sampling distribution of this parameter must be described, and compared
with the sampling distribution of the variance. Let 
be a
-sized sample, obtained by independent random drawings from
the distribution 
.
It is well-known that 
is biased, while
verifies:
is the fourth centered moment. Defining 
as times the squared coefficient of variation of the estimator
, we obtain:
The first point is that the quantity 
is well defined, even if some points of the sample are near or equal
to the sample mean 
. If it happens that 
, you
can split at will 
into a "left part" and
a "right part" according to 
and ever obtain: 
.
Exact results have been obtained for the "two Dirac's"
and the uniform distributions. They are given in Table 2.
The first relation 
was expected since a limited
sample is unlikely to catch all the dispersion of the population (for
the same reason, 
). But now, the bias
depends on and an unbiased estimator 
cannot be
obtained in a "distribution free" manner.
The second relation shows that the uncertainty around decrease
in 
since 
.
This mimics the well known behavior of the sample estimators for both
the mean and the variance. For others distributions, direct computations
aren't so easy and, at the present moment, only results obtained by
simulations are available.
For each of Table 3 and each of the
four sample size 
, we have drawn 
random samples. For each sample, the quantities 
and 
have been determined. Then 
and 
have been estimated from the 
obtained values, leading to an estimation
of 
. For the Gaussian,
we have obtained respectively 
and have summarized the results as 
. The same has been
done with , while the values given (in the last column)
for are proven results. For the lognormal distributions,
the results depends on the shape parameter 
(here 
has been used).
In any case, it appears that bias is 
i.e. is
a second order effect compared to the 
uncertainty and that doesn't behave worse than .
