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Who's the Best?

Stage: 3 and 4 Challenge Level: Challenge Level:1

In this problem, we needed to find a way to use the Olympic medal tally in order to work out which country is the most athletic. 
Joe and Matthew, from Chatham Grammar School, both considered the total number of medals which countries have won in previous Olympics. Matthew thought that because the 2008 Olympics contained the greatest number of events, the top five medal winning countries from this Olympics would be the most athletic.


Matthew then ranked these five countries according to the number and type of medals won using a weighted points system:

he gave $15$ points for gold, $10$ points for silver and $5$ points for bronze, and then divided the sum of these scores by the number of Olympic games the country had participated in.


Country # Olympics attended Total # gold medals Total # silver medals Total # bronze medals Total weighted score Total weighted score / # Olympics attended
USA 27 946 765 662 25 137 931
Russia 4 108 97 110 3 724 931
China 7 163 117 106 4 144 592
Germany 25 374 439 469 12 350 494
Great Britain 28 220 277 269 7 420 265


Because the USA and Russia had the same (and highest) result, Matthew then recalculated the USA weighted score for the last 4 Olympic games only, when both the USA and Russia competed:
For the past $4$ Olympics, the USA had $153$ gold medals, $133$ silver medals and $120$ bronze medals.
So the new total weighted score is
and the average over the four Olympics is
Therefore the USA is the most athletic nation.
Elliott, from Wilson's School, considered the number of medals won but also wanted to take into account the population size of each country. He first listed the countries according to the number of gold medals achieved in the 2008 Olympics:
Elliott then calculated a weighted score from the medal tally: $$ \text{Total Score} = \Big( 3 \times \text{# Gold medals} + 2 \times \text{# Silver medals} + 1 \times \text{# Bronze medals} \Big)\times 10^7 $$
He multiplied the weighted score by $10^7$ so that the Overall Results would contain reasonable sized numbers to divide by the population: $$\text{Overall Result} = \text{Total Score} \div \text{Population} $$
The Overall Results have been rounded to the nearest whole number:
From this, he ranked the countries according to their overall score, and concluded that Jamica is the most athletic nation.
Well done to Will and Ellie from Roundwood Park School who used the same method as Elliot.
Olie and Dan from Baston Primary School, and Owen and Jack from Chatham Grammar School, all started to think about using population size and also suggested that the wealth of countries should be taken into account when trying to identify the most athletic country - after all, a rich country may be very successful in the Olympic Games because a lot of money has been spent training and preparing their top sports men and women. Jack and Owen thought about using the Human Development Index to indicate how wealthy and developed a country is.
Can you suggest a mathematical formula to find the most athletic nation using information about the number of medals, population size and each country's wealth? Or perhaps you could consider other data, such as sporting event results, population age, prevalence of obesity, and investment in Olympic althetes to determine which country is the most athletic.