On Sun, 24 Jun 2001, Melady Preece wrote in part:
> I am teaching educational statistics for the first time, and although I
> can go on at length about complex statistical techniques, I find myself
> at a loss with this multiple choice question in my test bank. I
> understand why the range of (b) is smaller than (a) and (c), but I
> can't figure out how to prove that it is smaller than (d).
> 1. Which of the following classes had the smallest range in IQ scores?
>
> A) Class A has a mean IQ of 106 and a standard deviation of ll.
> B) Class B has an IQ range from 93 to 119.
> C) Class C has a mean IQ of 110 with a variance of 200.
> D) Class D has a median IQ of 100 with Q1 = 90 and Q3 = 110.
>
> The test bank says the answer is b.
Right. Since you're happy that range(B) < range(A) and
range(B) < range(C), I'll focus on (B) vs. (D).
In (B), the entire _range_ is from 93 to 119: 26 (or 27,
depending on how you choose to define "range") points.
In (D), the central half of the distribution is from 90 to 110:
the interquartile range (IQR) is 20 points, symmetric about the median;
the full range must therefore be greater than 20. Now, _if_ the
distribution is normal (which may be what we were to assume from the
allegation that these are IQ scores; although as Dennis has pointed out,
ille non sequitur -- unless these are rather large classes AND NOT
SELECTED BY I.Q. (or by any variable strongly related to I.Q.)), then 10
points from Q1 to median (or from median to Q3) represents 0.67 standard
deviation, which implies a standard deviation of about 15, which is
larger than the standard deviation in (A) and slightly larger than that
in (C).
However, we need not invoke the normal distribution. We observe
that the distribution in (D) is at least approximately symmetric (insofar
as the quartiles are equidistant from the median). If we may assume also
that the distribution is unimodal (which I should think reasonable), it
then follows (from the "tailing off" of distributions as one approaches
the extremes) that the distance from minimum to Q1 (and the distance from
Q3 to maximum) is greater than the distance from Q1 to median (or median
to Q3). This implies that the range of the distribution exceeds twice
the interquartile range: that is, range(D) > 2*20 = 40. Since the
range in (B) is only 26, clearly the range of (B) is less than the range
of (D).
If any part of this argument remains unclear, I'd be happy to attack it
again. A rough sketch should make things pretty obvious, but it's a bit
of a nuisance to draw pictures in ASCII characters!
--DFB.
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Donald F. Burrill [EMAIL PROTECTED]
184 Nashua Road, Bedford, NH 03110 603-471-7128
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