Lessons taught; Lessons learnt

Last time, we explored how changing the relative values of silver and bronze altered the 2010 Winter Olympic medal table.

This time we concentrate on the top three countries: Canada, Germany, and the USA.

Here is an applet that lets us assign different values to silver and bronze (by moving the blue dot), and so find the scores, and the rankings, of the countries. The initial setting values a silver at 20% of a gold, and a bronze at 20% of a silver. The results, and the dotted curves, are explored below.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

[Source: 2010-top-three places.ggb.]

As you will have found if you played with the applet, the dotted lines indicate where two of the countries swap position in the rankings. Starting on the left (no or low value for silver), the rankings are the IOC ones:

1. Canada
2. Germany
3. USA

As we increase the value of a silver, the first transition swaps Germany and the USA, giving

1. Canada
2. USA
3. Germany

Next, USA and Canada swap places, giving

1. USA
2. Canada
3. Germany

and finally, with silver valued highly, Canada and Germany swap positions, giving the North American-style ranking:

1. USA
2. Germany
3. Canada

Note that none of these options have Germany leading -- the top country is always either Canada or the USA. This answers one of the questions posted in the last post: it is not possible to find any sensible set of values which lead to Germany winning. I say 'sensible' because it is possible to put Germany first if you allocate a negative value to a bronze, but this would not be an option many people would agree with!

The dotted curves are found by setting the 'scores' of pairs of countries equal to each other. If we call 'x' the value of silver relative to gold (from 0 to 1), and 'y' the value of bronze relative to silver (from 0 to 1), then

• Canada = .
• Germany = .
• USA = .

and we have, for example, Canada and the USA's score equal when . Rearranging gives , and the other two curves are derived similarly.

... or, fun with pentagons:

The new version of Geogebra has several nice new features which make it much more useful for a range of statistical uses. Firstly, it comes with a simple spreadsheet-style view, which allows you to enter and manipulate data in a grid of cells, similar to a spreadsheet. Secondly, it has a number of new statistical functions, covering a range of data creation, summation, and visualisation options.

Here is an applet which demonstrates a couple of these new features. It takes 50 random values, generated to fit a Normal(8,4) distribution, and plots a box-and-whisker plot and a histogram.

All this just uses what are now built-in features of Geogebra. My contribution is to make the histogram interactive: move the blue points on the x-axis around to alter the class boundaries. This lets you explore the ways that small changes to the class intervals can sometimes have large effects on the histogram.

Enable Java to see this Geogebra applet.

(Source: adjustable_histogram.ggb.)

Double click on a cell in the spreadsheet view to change its value. Also, just as in Excel, press F9 while in the spreadsheet view to regenerate all the random numbers. You may see the purple x-axis points move: they have been constrained to always be below the minimum and above the maximum value.