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In this newsletter:

* Latest news
* Browse with Plus
* Maths in a minute
* Live maths

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Latest news

* Electoral impossibilities
Where democracy meets maths
http://plus.maths.org/latestnews/jan-apr10/election/index.html?nl=0

* Medical research plagued by bad reporting
Lack of statistical detail leads to wrong conclusions
http://plus.maths.org/latestnews/jan-apr10/reporting/index.html?nl=0

* Abel Prize 2010 goes to John T. Tate
A Galileo of number theory
http://plus.maths.org/latestnews/jan-apr10/abel10/index.html?nl=0

* Further evidence for Arctic melt-down
The state of the Arctic
http://plus.maths.org/latestnews/jan-apr10/arctic/index.html?nl=0

Plus... read more on the Plus blog
http://plus.maths.org/blog?nl=14

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Browse with Plus  -  Maths in the New York Times

Steven Strogatz, mathematician and great popular maths writer, is  
currently a contributor to the New York Times Opinionator blog. From  
geometric square dancing to differential calculus, his posts are  
easily digestable and interesting.

http://opinionator.blogs.nytimes.com/author/steven-strogatz/

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Maths in a minute - The central limit theorem

The central idea of applied statistics is that you can say something  
about a whole population by looking at a smaller sample. Without this  
idea there wouldn't be opinion polls, the social sciences would be  
stuffed, and there would be no way of testing new medical drugs, or  
the safety of bridges, etc, etc. It's the central limit theorem  
that is to a large extent responsible for the fact that we can do all  
these things and get a grip on the uncertainties involved.

Suppose that you want to know the average weight of the population in  
the UK. You go out and measure the weight of, say, 100 people whom  
you've randomly chosen and work out the average for this group - call  
this the "sample average". Now the sample average is supposed to give  
you a good idea of the nation's average. But what if you happened to  
pick only fat people for your sample, or only very skinny ones? 

To get an idea of how representative your average is likely to be, you need 
to know something about how the average weight of 100-people-samples 
varies over the population: if you took lots and lots of samples of size 
100 and worked out the average weight for each, then how variable would 
this set of numbers be? And what would its average (the average of 
averages) be compared to the true average weight in the population?

For example, suppose you know that if you took lots and lots of 100- 
people-samples and wrote down the average weight of each sample, you'd get 
all values from 10kg to 300kg in equal proportion. Then this would tell you 
that your method of estimating the overall average by taking one sample of 
a 100 people isn't a very good one, because there's too much variability - 
you're just as likely to get any of the possible values, and you don't know 
which one is closest to the true average weight in the population.

So how can we say anything about the distribution of 100-people-averages - 
called the "sampling distribution" - when we don't know anything about the
distribution of weight across the population? This is where the central 
limit theorem comes in: it says that for a big enough sample (usually 
sampling 30 people is good enough) your sampling distribution is 
approximated by a normal distribution - this is the distribution with the 
famous bell shape.
 
The mean of this normal distribution (the average of averages corresponding 
to the tip of the bell) is the same as the mean in the population (the 
average weight of the population). The variance of this normal 
distribution, that is how much it varies about the mean (indicated by the 
width of the bell), depends on the sample size: the larger the sample, the 
smaller the variance. There's an equation which gives the exact 
relationship.

So if your sample size is big enough (100 would certainly do since it's 
bigger than 30), then the relatively small variance of the normal sampling 
distribution means that the average weight you observe is close to the mean 
of that normal distribution (since the bell is quite narrow). And since the 
mean of that normal distribution is equal to the true average weight across 
the population, your observed average is a good approximation of the true 
average.

You can make all this precise, for example you can say exactly how  
confident you are that the true average is within a certain distance  
of your sample average, and you can also use the result to calculate  
how large a sample you need to get an estimate of a given accuracy.  
It's the central limit theorem that lends precision to the art of  
statistical inference, and it's also behind the fact that the normal  
distribution is so ubiquitous.

The central limit theorem is actually a bit more general than we've let on 
here. For a precise statement, see 
http://mathworld.wolfram.com/CentralLimitTheorem.html

Related ideas are covered in the following Plus article:
http://plus.maths.org/latestnews/jan-apr07/abel07/index.html


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Live Maths

The Romantic Economist addresses the limits of knowledge in markets

The relationship between theory and fact is particularly problematic  
when interpreting the complex, creative and socially-constructed world  
of markets. In this free lecture Richard Bronk from the London School  
of Economics and Political Science will explore how theories frame our  
vision and how prices and institutions transmit information, before  
examining the vital distinction between measurable risk and radical  
uncertainty.

When: 28th April 2010, 6pm
Where: Gresham College, Barnard's Inn Hall, London EC1N 2HH
Entry: free
More info: http://www.gresham.ac.uk/event.asp?PageId=45&EventId=1021


Indra's Pearls: Geometry and Symmetry

Felix Klein, one of the great nineteenth-century geometers, discovered in 
mathematics an idea prefigured in a Buddhist myth: the heaven of Indra 
contained a net of pearls, each of which was reflected in its neighbour, so 
that the whole Universe was mirrored in each pearl. Klein studied 
infinitely repeated reflections and was led in his imagination to 
remarkable forms with hitherto unknown symmetries. Join Caroline Series on 
a journey from basic mathematical ideas to simple algorithms whose 
repetition creates delicate fractal filigrees which are only now beginning 
to be explored fully. (Also see Series' Plus article on the same topic: 
http://plus.maths.org/issue43/features/serieswright/
  )

When: 4th of May 2010,  6pm
Where: London Wall, London EC2
Entry: free
More info: http://www.gresham.ac.uk/event.asp?PageId=45&EventId=998


Is there anybody out there?

As we face the extinction of many species on our own planet, we are turning 
to the skies to look for other signs of life. But how do we avoid our 
actions from affecting this new frontier? Join a group of astrobiologists 
and meteorite researchers to debate how our actions in other worlds could 
affect life that is as yet undiscovered, and decide whether we should be 
preparing to meet ET now, rather than later. (One of the astrobiologists 
taking part is Plus author Lewis Dartnell, see his article on alien life: 
http://plus.maths.org/issue51/features/dartnell/index.html
  ).

When: 29th of 2010, 7-9pm Where: Attenborough Studio, The Natural  
History Museum, Cromwell Road, London, SW7 5BD
Entry: £6 More info: http://shrunk.net/7560c9c5

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Happy reading from the Plus team!

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