If you're not counting photons or marbles or poll responses and the like, but 
rather are getting a distribution of answers to a question, you can't use Poisson 
statistics!  For example, you want to know the height of an American adult.  So 
you ask 89 adults how tall they are.  You might get a range of answers like this:

  Ht. (inches)    Number of people
  ------------    ----------------
           60     ooo
           61     oo
           62     oooo
           63     oooo
           64     ooooo ooooo oooo
           65     ooooo oo
           66     ooooo ooooo ooo
           67     ooooo o
           68     ooooo ooooo o
           69     ooooo ooo
           70     oo
           71     oo
           72     oo
           73     oooo
           74     o
           75     ooo
           76     oo
           77
           78     o

You could just take the average, that is the arithmetic mean; add up
all the heights, then divided by the number of people in the survey.
You'll get a number like 5'7" or so.  How accurately does that number
represent the height of an American adult?

Well, it may be close to an average, but clearly there are a LOT of
people who aren't 5'7" (or, 67" if we just use inches).  What we realize here is 
that this distribution is more like a "bell curve" with a peak around 66" or so; 
and the width of the "bell curve" halfway between the base and the peak tells you 
how widely distributed the spread is.

Let's use a more contrived example.  I ask the students in a class to measure the 
brightness of the Moon.  This is the distribution of answers:

 12                  oo
                    oooo
                    oooo
  9                ooooo
                   ooooooo
                  ooooooooo
  6              oooooooooo
                oooooooooooo
               ooooooooooooooo
  3          o ooooooooooooooo
         o o ooooooooooooooooooo o
        oooooooooooooooooooooooooooo
 -|--------|---------|---------|----------|-
 90       100       110       120        130  (flux units)

It looks like there is a bell-shaped curve, with a maximum of 12 units.  The width 
of the bell curve at half the maximum (in this case, 6 units) runs from about 106 
to 117 units; and the center of the bell curve looks to be about 110.5 units.  So 
in this case, the most likely value of the true brightness of the moon is 110.5 
flux units, and the margin of error is plus or minus 5.5 flux units.  This means 
that the FULL WIDTH AT HALF MAXIMUM of this distribution is 11 units (106 to 117), 
and the margin of error is half of that width in each direction.

Notice there are irregularities and slight asymmetry in the bell curve. Data from 
real-world observations are always likely to be imperfect; it is up to us, the 
analyzers of the data, to spot the peak, the FWHM, and interpret the data 
effectively.