Definitions of Range

Date: 10/20/2003 at 15:37:55
From: glockhart@satx.rr.com
Subject: Definition of Range

I can find only one definition of range in the math dictionaries - 
the difference between the smallest and the largest number in a set.  
We are always talking about "They range in age from, or they range in 
height, or they range in weight, or they range in size, etc".  If 
the only defintion of range is the difference, why do we say "They 
range..."?

Date: 10/20/2003 at 22:40:07
From: Doctor Peterson
Subject: Re: Definition of Range

Thanks for writing to Dr. Math.

Math definitions generally give only the specific technical sense of 
words; your use of "range from this to that" is a common-language 
sense, which can be found in ordinary dictionaries, and doesn't need 
a special definition. In fact, if you look up "range" in, for 
example, Merriam-Webster (m-w.com), you find the technical definition 
of the noun as 7c, among other related uses within mathematics:

  7 a : a sequence, series, or scale between limits (a wide range
        of patterns)
    b : the limits of a series: the distance or extent between
        possible extremes
    c : the difference between the least and greatest values of an
        attribute or of the variable of a frequency distribution
  8 a : the set of values a function may take on
    b : the class of admissible values of a variable

Your use as a verb is

  5 : to change or differ within limits

In math, technical uses and common uses coexist; we are, after all, 
speaking English, with special words used only where needed. Here the 
range as the difference between maximum and minimum is, more 
specifically, a "statistic", a single number used to indicate one 
aspect of the behavior of some variable; that needs precise 
definition, as the other uses do not.

Why do we use this single number, when a broader sense (a pair of 
limits, or a set of numbers) gives more information?

If you think about it, you will recognize that the whole idea of 
statistics is to boil down large amounts of information to single 
numbers (mean, standard deviation, and so on). True, the range alone 
doesn't tell you much; the same can be said of the median. But 
together, they give a useful picture of how the variable varies. For 
some purposes, that is just what we want; for others, the set of 
possible values is of more interest. It all depends on what you want 
to do with the data.

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


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