Range - Difference or Difference + 1?

Date: 08/05/2003 at 17:42:34
From: Pat
Subject: Range

I just received a response from the ESP Crayfish science unit to a 
question I had asked about RANGE. I had asked why they computed RANGE 
by subtracting the smallest number from the largest number and adding 
one - since in the math text that we use, they compute RANGE by just 
subtracting the smallest from the largest number. Is range the same 
as 'difference' or not? I feel that RANGE should be taught 
consistently and that one of these two processes must be incorrect. I 
feel that if range is the same as difference, then the math way is 
correct, but if range should include the smallest and largest numbers, 
then the science way is correct. I am a teacher and want to be 
teaching it to my students correctly.

If I want to find the range of ages of students in my class using the 
math text, I would subtract 8 from 10 and get a range of two. But the 
science kit states that we should subtract 8 from 10 and add 1, 
arriving at a range of 3. Which is correct and why? Thank you!

Date: 08/06/2003 at 12:05:36
From: Doctor Peterson
Subject: Re: Range

Hi, Pat.

What was their response? I'm curious!

I'm not a statistician, so I can't answer this from my own experience; 
but I searched the Web a bit to get a feel for this issue, and found 
that in fact the range is not defined consistently. It is sometimes 
defined as the actual interval ("from 3 to 12"), sometimes as the 
difference between lowest and highest ("12-3 = 9"), and sometimes as 
one more than that ("12-3+1 = 10").

Here is one example:

   Statistical Survival Kit for HRD Practitioners 
   http://www.internetraining.com/Statkit/StatKit.htm 

  Range is the difference between the highest and lowest scores.
  You should only use the range to describe interval or ratio level
  data. To calculate the range, subtract the lowest score from the
  highest score

  Note: In some statistics books, they will define range as the
  High Score minus the Low Score, Plus one (1). This is an
  inclusive measure of range, rather than a measure of the
  difference between two scores. For example: the inclusive range
  for data ranging from 6 to 10 would be 5.

  For our purposes, we will define the range as the difference
  between the highest and lowest scores.

I have seen these referred to as "exclusive range" and "inclusive 
range," as here:

   Descriptive Statistics (Notes)
   http://www.positivepractices.com/ResearchandEvaluation/
DescriptiveStatisticsNote.html

  -- inclusive range: (XH-XL)+1
  -- exclusive range: (XH-XL)

Why would this be? It appears that the inclusive definition is used 
when the data consist of whole numbers; we don't add 1 when the data 
can be arbitrary real numbers (including fractions). I can see two 
reasons for this. First, it gives the size of the range, in the sense 
of the NUMBER of values between the highest and lowest, inclusive. 
Second, in some cases we can think of each whole number as 
representing the interval from 1/2 less to 1/2 more (that is, all 
numbers that round to the given whole number). Doing this extends the 
range by 1/2 on either end, adding 1 to the range. There are probably 
some purposes for which this is appropriate mathematically, though I 
have not taken time to find an example.

So both definitions are in use, both make sense in their own context, 
and it really doesn't matter which you use as long as you are 
consistent, since you will only be comparing ranges measured the same 
way. It's unfortunate that there are different definitions for the 
same term (though that is quite common not only in English in 
general, but even within mathematics); what's really unfortunate is 
that no one seems to explain this, and you (and the students) are 
left confused.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/