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### 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
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/
```
Associated Topics:
College Definitions
College Statistics
High School Definitions
High School Statistics

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