Sorry to say that it's been a while since my math and statistics
courses and I'm trying to avoid any erroneous concept ("lying with
statistics") errors.

I am trying to calculate some percentages (no rounding applied in
the example below) from two sets of tallies.  I know pretty well how to
handle (a.) and (b.).  The (c.) percentage is over 100%.  The (d.)
percentage is a negative value. The (m.) percentage expresses the sum
of the (a.) through (d.) values.

a.  120 is 83% of 144
b.  195 is 93% of 208
c.  182 is 189% of 96
d.  -91 is -124% of 73

m. 406 is 77% of 521 

My question is about the effect of the exception (c.) or (d.) values
upon (m.).  [I actually have about sixty values almost all of which
are like (a.) and (b.).]

Specifically my first question is: what is the term that should be
used to express the (m.) 77% value.  Can it be termed the arithmetic mean
percentage of the (a.) through (d.) values, or should it be called something else?

Second question: is the (d.) calculation correct and/or does its [and
(c.)s for that matter] inclusion in the set of numbers create a
situation where the (m.) percentage value could be criticized.  In
other words, are any mathematical incongruities [probably nonrelevant
but, for example, a sin such as dividing by zero] being introduced
into the (m.) percentage that someone could take issue with?

Hi, drpangloss-ga:

Although the percentage found in (m) is not a "straight" arithmetic
mean of the percentages found in (a) through (d), it is a "weighted"
average (or weighted arithmetic mean) in which the percentages (a)
through (d) are weighted according to their corresponding "bases":

        144*(83%)+208*(93%)+96*(189%)+73*(-124%)
77% =  ------------------------------------------
                144 + 208 + 96 + 73

I could quibble about your statement that you did not apply rounding,
since the rounding rule you seem to have used is "truncation toward
zero".  But I checked your arithmetic, esp. in (c),(d), and (m), and
found everything was at least consistently rounded by that rule.

Along the lines of mathematical improprieties, I'd beware of using
negative weights in a calculation like this.  But it seems that all
your weights (denominators of the intermediate calculations) will be
positive, so I think it's okay to describe (m) as a weighted mean.

regards, mathtalk-ga

Thanks for addressing my question perfectly and providing a most
complete answer.  I especially appreciate that you additionally caught
my sloppy use of the term "no rounding" and provided appropriate
terminology!