On 3 Jun 2003 at 12:07, R. Allan Reese wrote:

> A student has submitted work using the phrase "margin of error".  From
> checking a few web pages, this appears widely used as a synonym for
> confidence interval.  Am I alone in finding this a sloppy misuse?

My first reaction on reading this was as follows.  Sometimes I use
this expression in an informal sense, in seeking to explain to
students and health professionals what information a confidence
interval conveys.  However, a while ago I came across its use in a
more formal sense, in relation to some kind of opinion or electoral
sampling data, as a sort of halfway house between quoting the
standard error and the confidence interval.  Supposing that out of n
individuals questioned, r answered in the affirmative, the resulting
proportion, p=r/n, was then assigned a margin of error 1/sqrt(n).
This clearly comes from the Wald 95% confidence interval p +/-
z*sqrt(p(1-p)/n), but uses 2 (instead of 1.96) for z, and substitutes
the most conservative case value, 0.25, for p(1-p).  From my
recollection the method wasn't referenced, suggesting it was in
common use in this field of application.  The idea of a flat-width CI
for a proportion is a curious one.  I haven't seen any evaluation of
its coverage properties, but from this point of view it will be
better than the Wald interval which is pretty abysmal.  It avoids
Wald's zero width intervals when p = 0 or 1, but conversely increases
the incidence of intervals with lower limit below 0 or above 1.  I
guess that the relevant user community take for granted that n is
always in the hundreds at least and p is not close to 0 or 1, and all
they're after is a very rough, computationally simple appreciation of
the implications of different sample sizes.  But it does all suggest
sloppiness.

Then, following Allan Reese's example, I checked the web.   The
second link Google identified is from the ASA series, "What is a
survey?"

http://www.amstat.org/sections/srms/brochures/margin.pdf

This has clearly made the use of the term respectable, but isn't
prescriptive as to how to calculate it.  They do say, "In sampling,
to try an estimate a population proportion - such as in telephone
polls - a sample of 100 will produce a margin of error of no more
than about 10 percent, a sample of error of 500 will produce a margin
of error of no more than about 4.5 percent, and a sample of size
1,000 will produce a margin of error of no more than about 3
percent."  The practice of using 1/sqrt(n) clearly parallels this,
and users have simply neglected the proviso "no more than" - with the
consequence of sometimes imputing a grossly unnecessarily large
degree of imprecision.  Furthermore, there is a real risk that
someone out there may have applied this approach to assess the
sampling uncertainty of some quantity other than a proportion ...

Robert G. Newcombe
University of Wales College of Medicine
Cardiff.

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