Going for gold
Looking at the Olympic Medal table, can you see how the data is organised? Could the results be presented differently to give another nation the top place?
Problem
Here are the top ten nations in the table of medal winners for the 2016 Olympic Games:
| Position | Flag | Nation | Gold | Silver | Bronze | Total |
| 1 | Image
| United States | 46 | 37 | 38 | 121 |
| 2 | Image
| Great Britain | 27 | 23 | 17 | 67 |
| 3 | Image
| China | 26 | 18 | 26 | 70 |
| 4 | Image
| Russia | 19 | 17 | 20 | 56 |
| 5 | Image
| Germany | 17 | 10 | 15 | 42 |
| 6 | Image
| Japan | 12 | 8 | 21 | 41 |
| 7 | Image
| France | 10 | 18 | 14 | 42 |
| 8 | Image
| South Korea | 9 | 3 | 9 | 21 |
| 9 | Image
| Italy | 8 | 12 | 8 | 28 |
| 10 | Image
| Australia | 8 | 11 | 10 | 29 |
Is your own nation in the list? If not, find out where your nation ranked and compare it with the nations in this table.
How do you think the positions have been decided?
Could the positions be decided in a different way? How would this affect other results in the table?
What do you think the best way of deciding the positions would be? Why?
If you'd like to have a go at another problem involving sporting leaderboards, take a look at Climbing conundrum.
Getting Started
Look at the totals. Any surprises there?
Look at the numbers in each medal category. What do you notice?
Student Solutions
Ms. McCarthy from St. Francis Xavier in the United States sent in the following contribution:
We determined that the gold medal totals were used to determine what place each of the 10 countries listed came in. If there was a tie, they went to the silver medal count to choose which team placed higher.
Another way to order the list would be to go with the total medal count. While it wouldn't change the top few, it would certainly change some others, especially Australia.
We think that no matter how you manipulate the results, USA would always come out on top because they won the most medals in every category.
We had a number of very good ideas coming from Roseville College in Australia.
Firstly from Mia:
1. Is your own nation in the list? Yes my nation is Australia.
2. How do you think the positions have been decided? The positions have been decided by the number of gold medals won by each country.
3. Could the results be presented differently to give another nation the top place? The results could be presented in order of the total number of medals won, but America would still be in first place.
4. How would this affect other results in the table? Some countries will move down the table and some countries will move up the table like Australia. Australia would move to 6th position.
Ella wrote:
The way that the Olympics place countries is by the amount of gold medals your country wins. Another way you can order the top 10 countries is by alphabetical order. (I never said it would be fair). Another way is the total amount of medals your country wins.
The way it would affect the scores on the table is that the countries that have a first letter in the alphabet that is far down in the alphabet would be low in the ranking. It would take the fun of the Olympics for those countries because they have no chance of winning.
Then my second reason is that we should place the countries by how many medals they win altogether not how many gold medals they win. This is because a country may be able to easily get heaps of bronze and silver medals but still lose because they have very few gold medals. If they order countries according to how many medals they have altogether it would give other countries a chance of winning.
Imogen had these thoughts:
The positions have been decided based on how many Gold medals the country won. If the ranking was based on how many Bronze medals were won, Russian Fed. would be in the top place and every other team would be in a different position. Australia and France would both share 6th place.
Hannah also answered the questions:
The country that has the most amount of medals in total could be at the top.
The country that tried the hardest could be at the top.
The country that went in the most events could be at the top.
How would this affect other results in the table?
Countries who might not have got into the top 10 might get into the one above.
Alice wrote the following:
1. Yes, Australia is on the table.
2. By the number of gold medals.
3. Yes, If the countries where ranked by total number of medals but the USA would still be on top.
4. Ranks would swap around and maybe even new countries would appear on the table.
Ashley showed how the resuts might be changed:
Australia came tenth on the list with 7 gold medals, 16 silver medals and 12 bronze medals. I think the positions have been decided by the number of gold medals the country has: for example, the United States has 46 gold medals and China has 38. The United States came 1st and China came 2nd. If there was a tie in gold medals they would look on how many silver ones they have, like they have done for Italy and Hungary. You can change it by looking at the total medal tally which would change the positions on the leaderboard.
Leaderboard with total instead of gold medals:
United States, 104
China, 88
Russia, 85
Great Britain, 65
Germany, 44
Australia, 35
France, 34
South Korea, 28
Italy, 28
Hungary, 17
South Korea came before Italy because they had more gold medals as well as
totals.
Thank you, Roseville students for those excellent solutions.
Abhishek from Singapore International School, Mumbai in India wrote:
I believe that the positions have been decided by the number of gold medals first, then silver, then bronze (in descending order).
I believe gold is first because Great Britian is ahead of Russia even though Russia is ahead on silver and bronze.
Silver is second because Germany is ahead of France.
Could the results be presented differently to give another nation the top place?
I tried calculating the median of the gold, silver and bronze for each country, but the 1st place remains the same, that is US which is 29 (29, 29, 46) because the median comes ahead of China (23, 27, 38).
If you rank by the highest number of bronze medals, Russia comes first. South Korea comes down dramatically on most bronze medals. Australia comes up by three places.
Thank you for all the contributions. We hope to see you sending in solutions to other NRICH activities.
Teachers' Resources
Going for Gold
Here are the top ten nations in the table of medal winners for the 2016 Olympic Games:
| Position | Flag | Nation | Gold | Silver | Bronze | Total |
| 1 | Image
| United States | 46 | 37 | 38 | 121 |
| 2 | Image
| Great Britain | 27 | 23 | 17 | 67 |
| 3 | Image
| China | 26 | 18 | 26 | 70 |
| 4 | Image
| Russia | 19 | 17 | 20 | 56 |
| 5 | Image
| Germany | 17 | 10 | 15 | 42 |
| 6 | Image
| Japan | 12 | 8 | 21 | 41 |
| 7 | Image
| France | 10 | 18 | 14 | 42 |
| 8 | Image
| South Korea | 9 | 3 | 9 | 21 |
| 9 | Image
| Italy | 8 | 12 | 8 | 28 |
| 10 | Image
| Australia | 8 | 11 | 10 | 29 |
Is your own nation in the list?
If not find the data on the internet and compare it with the table.
How do you think the positions have been decided?
Could the results be presented differently to give another nation the top place?
How would this affect other results in the table?
Why do this problem?
This problem is intended to encourage children to develop their skills in data analysis, and to be critical of the way in which data are presented to them.
Possible approach
Present the children with the table to look at and invite them to tell you about what it shows. Explain where it comes from and why it might have been made. Encourage the children to look at the variation in the numbers of medals of different types or the totals or both.
Key questions
How are the nations ordered?
Are there nations that have the same numbers of gold medals? Of silver medals? Of bronze medals?
Are they next to each other in the table? Why do you think this is so?
Do you think the system for ordering the nations is fair? Why?
Possible extension
Encourage the children to think creatively about scoring systems such as 3 points for a gold, 2 for a silver and 1 for a bronze, and investigate the impact this would have on the order.
Further investigations might help the children to offer each nation advice about the sports they should focus on to maximise their position in the table. They would need to find and examine further data to do this.
Can they work out a system that is 'fair' but puts their favourite nation at the top?
It might be interesting to consider other factors that affect perception of performance at the Olympics, such as population and wealth - see this Plus article for further discussion.
Possible support
Encourage children to tell you and each other about the meaning of the data in the table. You could begin by asking straightforward questions such as:
How many gold medals does China have?
Which country is top of the table?
Once they have a good idea of the meaning of the data, they may be able to go on to consider the main task.