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SCI199Y: October 24, 1995
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\noindent{\bf Required for next week}
\begin{itemize}
\item Short project 2 (with extension allowed until November 7)
\end{itemize}

\noindent{\bf More on the polls}

The following table is taken from the Globe \& Mail, Wednesday, October 18
and Saturday, October 21.  It seems to be the definitive summary of the polls,
to date, although it's not 100\% consistent with earlier published 
stories.\footnote{The Compas poll of Sept.~11-14 was earlier reported has 
having a sample size
of 500.  The Leger \& Leger poll of Sept.~25-28 was earlier reported as
covering the period Sept.~21-25.  }


\begin{tabular}{lrrlll}
Polling&number&&&&Undec/\\
Company&polled&date&No&Yes&Won't Say\\ \hline
L\&L&959&Sept. 7-8&42.9&43.8&13\\
SOM&1003&8-12&45.0&37.0&18\\
Compas&959&11-14&40.0&36.0&24\\
Createc&1004&15-199&46.2&39.8&15\\
Crop&2020&20-25&47&39&14\\
SOM&1820&19-25&48&39&13\\
L\&L&1006&25-28&45.1&43.8&11\\
L\&L&1015&Oct. 1-3&44&43&13\\
Gallup&1013&10-12&43&39&18\\
L\&L&1002&8-12&42&45&13\\
SOM&981&13-16&43.4&42.9&13.6\\
L\&L&1005&16-20&42.3&45.7&12.1
\end{tabular}

The numbers that make the headlines have the
undecideds allocated to either yes or no.  There was an excellent
discussion of this in the G\&M on October 18 (A5).  SOM allocates the
undecideds in the percentages observed in the decided voters;
approximately 50-50.  Leger \& Leger allocates
them based on their answers to other questions in the poll and on
demographic factors; about 70\% to no.
This results in a substantial boost to the reported
No vote.  The poll corresponding to the last line in the above table was
reported as a headline:  ``Yes 50.2, No 49.8, poll suggests".
(The poll of size 2020 has a margin of
error of approximately 2.2\%; the poll of size 1820 has a margin of error
of approximately 2.3\%.  The other polls have a margin of error of 3.1\%)

\bigskip

\noindent{\bf In other news this week...}
\begin{itemize}
\item ``Bouchard slip helps to boost dollar": G\&M Oct.~17,B1
\item ``Dollar plunges on fears of Yes vote": G\&M Oct.~21,A1
\item ``Poll pratfall of '80 not applicable now": G\&M Oct.~17,A4.
``On May 18, 1980, two days before the first referendum, Dimanche-Matin
published the following banner headline: Yes 52\%, No 48\%.  The
ultimate result was very different: Yes 40\%, No 60\%."  The article
goes on to say that ``polling has become far more sophisticated since 1980.
In the infamous Dimanche-Matin survey, 24 per cent of people described
themselves as undecided.  And this 48 hours before The Day?  No way.
Then the prediction was exacerbated by dividing the undecided evenly
between the two camps.  In fact, more than 80 per cent of the undecided voted
No."
\item ``Trade winds that blow across Canada": G\&M Oct.~16.
Nice graphic showing interprovincial and international trade balances,
by province.  Ontario has a huge overall trade surplus (twice the
closest contender, Alberta) entirely created by interprovincial trade surplus.
(Ontario has the largest international trade deficit of all the provinces.)
\item ``Council bans smoking in Toronto food courts"
Councillor Peter Tabuns, chairman of the city's Board of Health,
is quoted as saying: ``With more than 3,000 Canadians dying each year of
second-hand smoke, it is certainly a public health issue".  Where do you
think this number came from?
\item ``When racial categories make no sense": G\&M Oct.~21,D8.
Statistics Canada will ask a question on race in the next census
(see handout, Sept.~12):  this article describes biologists' viewpoints
that race does not account for a meaningful proportion of genetic variability
between individuals.  The main research referred to is a study of
Richard Lewontin carried out in 1972.
\end{itemize}

\noindent{\bf What is the bell curve anyway}

One way to describe a measurement that varies in a population is to
quote the frequency of each possible measurement in that
population.  (Think of the students lined up on the lawn of Penn. State,
according to their height.)  With many types of measurements, as you
get a larger and larger population, with frequencies measured
on a finer and finer grid, the plot of the frequencies will start to
look like a curve of a very predictable form.  (Think of getting
5 times as many students lined up by height classes, and taking a photo
from an airplane.  Then 500 times as many students, seen from the space
shuttle...) 

The frequency curve has a particular mathematical expression, in standard
form, as:
$$
f(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}
$$
and looks like this:

%\begin{figure}
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%\epsffile{psnormalcurve}}
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It is a surprising, but true, fact that a wide variety of potential
measurements tend to follow this curve, once the measurements are
converted to standard units.  It is a theorem, known as the central
limit theorem, that the frequency distribution of measurements that
are {\it averages} will always follow this curve, as the number of things
being averaged increases.  (With a bit of a stretch, you could imagine
a particular person's height being determined by a number of effects that
are averaged: genetic makeup, pre-natal nutrition, post-natal nutrition, etc..
IQ's are typically computed by averaging over a number of test items.)

The frequency curve given above has the property that about 2/3 of its mass
is contained in the interval $(-1,+1)$ and about 95\% of it mass is contained in
the interval $(-2,+2)$.  In other words, 95\% of the measurements, heights,
IQ scores, whatever, will fall within 2 standard units of the center point.
The center point is called the mean, and the standard unit is called the
standard deviation.

In fact, my analogy above to looking at the students lined up by height,
from space, doesn't quite work, unless we think of lining up either all
females, or all males.  Even from space we'd probably see two humps.

Even when measurements do have a frequency distribution that is exactly
described by the bell curve, they can vary enough to look surprising.
On p.172 of {\it VDQI}, Tufte shows 12 sets of 50 measurements from the
bell curve.  What is plotted is the actual frequency distribution
for each set of 50 measurements.  

 
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