Should Medal Counts Be Adjusted for Relative Population Size? In my humble opinion, yes

The Olympic Games are competitions between athletes in individual or team events and not between countries …

     —International Olympic Committee, Olympic Charter, Chapter 1, section 6

Despite the aversion of the IOC to nationalism and ranking of countries, you cannot visit a web site devoted to the Olympics without finding a medal count by country, usually ranked by some weighting of gold, silver, and bronze medals and their totals.  

Fair enough, but such rankings are ludicrous if there is no attempt to adjust for at least the relative population sizes of the countries, if not GDP or both.   The Guardian newspaper attempted to do both for the 2012 Summer Olympics, creating an interesting alternate ranking system based on population (by this system, Grenada, Jamaica, and the Bahamas won those Olympics).  

What I present here is a crude population-adjusted medal rank by total medals per million citizens and total weighted medals (using a 4:2:1 gold:sivler:bronze weighting) per million citizens for the 2014 Winter Olympics so far (as of 2/12/2014).  The results are intriguing:

Norway jumps from #3 to #1 when its population of 5 million is taken into account.

Switzerland surges to #2 from #6 when its 8 million population is counted.  

Tiny Slovakia takes third, winning .370 weighted medals per million citizens.

Canada drops from #2 to #6.

The United States plunges from #5 to #18 when its weighted medal count is divided by its 313.9 million citizens.

China and Great Britain are the last-ranking of the medal-scoring countries.  However, this is not as bad as it sounds since only 24 of 96 competing countries (25%) won medals so far.

Why Should Population Matter?

The laws of probability determine that - all things being equal - a country with a larger population is likely to send a team to the Olympics that is faster, quicker, more competitive than a country that is smaller.  

You need not be a mathematician to understand this principle.   Imagine a simple running race.  You select the winner from a small village of 100 people.  Each of these winners goes on to compete in a regional championship of 100 villages.   The probability of winning the village race is 1 in 100.  Assuming it is a fair race, you are selecting the 99th percentile of runner by definition.  But in the regional championship, you are selecting the top winner of each race, meaning you are selecting the top runner out of 10,000 (100 x 100) runners.   If these regional winners were all to compete in a national competition of 100 regions, let's say, you would be selecting the top runner of 1,000,000 runners (100 x 100 x 100).   This runner would be by definition faster than 999,999 runners in that country, putting her in the top .0001 % of runners.  

Imagine another country that has 100 times the population of the first country.  This country is able to hold a fourth competition in which the fastest runner of 100,000,000 (100 x 100 x 100 x 100) is selected.   Again, assuming no differences (or insignificant ones) in athletic endowment by ethnicity, the probability that the faster runner of the second country is faster than the fastest runner of the first is 100:1.   

Another thought experiment that illustrates this principle.   Imagine that you are playing a game of pick-up soccer.  You and an opponent choose the best players from two different pools of potential players.  Does it not make sense that if the pool from which you could recruit was ten times the size of your opponents that your probability of finding some extraordinary players would be far higher? (For a more detailed mathematical explanation, see the Appendix)

Crude Medal Rank by Country (Unadjusted for Population)

So calculating medals per capita (or per million citizens to make the math easier) yields some interesting results.  

First, the crude totals, unadjusted by population size, as of today (2/13/2014):

- source:  20 Minutes, Tableau des Médailles

By a scoring system that weights gold more than silver and bronze, Germany is on top, followed by Canada, Norway, and the Netherlands.  The United States is at #5 and appears to be dominating Switzerland.

Top 6 Countries When Population Is Taken Into Account

But when adjusted for population size, the rankings of the top 6 are quite different:

Norway, with only 5 million people, jumps to the top, with a total of 2.39 medals per million inhabitants (.797 gold medals per million).  

The Netherlands with 16.8 million people, takes second place with .596 medals per million inhabitants.

Switzerland (population 8 million) jumps to third, with half a medal per every million citizens, including .375 gold medals per million.  

Germany (81.9 million) drops to 5th place from 1st, and the United States (313.9 million) is a distant 6th, with only .029 medals per million citizens (.01 gold medals per million).  

What this should remind us all is that before we in the United States do too much self-congratulatory back-slapping, we should remember that given our relative population size (almost 63 times that of Norway and over 39 times that of Switzerland), all things being equal, we should be winning many more medals than we are, if this were a true head-to-head competition of our population's athletic prowess versus that of the rest of the world.  

Ranking by Total Medals Per Million Citizens, All Medal-Winning Countries

It gets worse for large countries when you include all countries that have won a medal so far in these Olympics, ranking by total medals won per country per million citizens.

Rank	Country	G	S	B	Total	Pop (m)	G	S	B	Total pm	Reverse Weighted 4:2:1 pm
1	NOR	4	3	5	12	5.0	79.7%	59.8%	99.6%	239.1%	134.5%
2	SLO	1	1	2	4	5.4	18.5%	18.5%	37.0%	73.9%	37.0%
3	NED	4	2	4	10	16.8	23.9%	11.9%	23.9%	59.6%	35.8%
4	AUT	1	4	0	5	8.5	11.8%	47.3%	0.0%	59.1%	35.5%
5	SUI	3	0	1	4	8.0	37.5%	0.0%	12.5%	50.0%	40.6%
6	LAT	0	0	1	1	2.0	0.0%	0.0%	49.4%	49.4%	12.3%
7	SWE	0	3	1	4	9.5	0.0%	31.5%	10.5%	42.0%	18.4%
8	CAN	4	4	2	10	34.9	11.5%	11.5%	5.7%	28.7%	18.6%
9	CZE	0	2	1	3	10.5	0.0%	19.0%	9.5%	28.5%	11.9%
10	SVK	1	0	0	1	5.4	18.5%	0.0%	0.0%	18.5%	18.5%
11	FIN	0	1	0	1	5.4	0.0%	18.5%	0.0%	18.5%	9.2%
12	BLR	1	0	0	1	9.5	10.6%	0.0%	0.0%	10.6%	10.6%
13	GER	6	1	1	8	81.9	7.3%	1.2%	1.2%	9.8%	8.2%
14	RUS	2	4	3	9	143.5	1.4%	2.8%	2.1%	6.3%	3.3%
15	FRA	1	0	2	3	65.7	1.5%	0.0%	3.0%	4.6%	2.3%
16	AUS	0	1	0	1	22.7	0.0%	4.4%	0.0%	4.4%	2.2%
17	ITA	0	1	1	2	60.9	0.0%	1.6%	1.6%	3.3%	1.2%
18	USA	3	1	5	9	313.9	1.0%	0.3%	1.6%	2.9%	1.5%
19	POL	1	0	0	1	38.5	2.6%	0.0%	0.0%	2.6%	2.6%
20	JPN	0	2	1	3	127.6	0.0%	1.6%	0.8%	2.4%	1.0%
21	UKR	0	0	1	1	45.6	0.0%	0.0%	2.2%	2.2%	0.5%
22	KOR	1	0	0	1	50.0	2.0%	0.0%	0.0%	2.0%	2.0%
23	GBR	0	0	1	1	63.2	0.0%	0.0%	1.6%	1.6%	0.4%
24	CHN	0	1	0	1	1,351.4	0.0%	0.1%	0.0%	0.1%	0.0%

Norway still dominates with 2.39 medals per million citizens, but tiny Slovakia (population 5.4 million) pulls into the second spot with .739 medals per million.  

Austria (8.5 million) with .591 medals per million citizens nudges ahead of Switzerland.

Tiny Latvia (population 2 million) with .494 medals per million is right on Switzerland's heels.

Slovakia and Finland may only have one a single medal each, but for their population size, this is phenomenal, pulling them to 10th and 11th place respectively.

The host country's 9 medals do not seem so impressive when population is taken into account.  Russia (population 143.5 million) is 14th with .063 medals per million citizens, behind Germany (.098) and ahead of France (.046).  

But the United States (313.9 million population) on a per capita basis is doing even worse (#18), 4 places behind Russia, and just ahead of Poland (.026) and Japan (.024).

Great Britain and China pull up the rear with .016 and .001 medals per million citizens, respectively.  Ouch.

Total Medals Weighted 4:2:1 Per Million Citizens

To be fair, gold medals should count more than silver, and silver more than bronze.  Although the IOC refuses to get involved in ranking countries or medals, the formula that makes the most sense to me is one proposed by the New York Times in which gold medals are twice as valuable as silver, which in turn are twice as valuable as bronze.   This 4:2:1 system gives a reverse weighted total (equal to the gold medals added to half of the total silver medals added to one-fourth of the total bronze medals) for each country, from which a per capita weighted total can be derived.

Rank	Country	G	S	B	Total	Pop (m)	G	S	B	Total pm	Reverse Weighted 4:2:1 pm
1	NOR	4	3	5	12	5.0	79.7%	59.8%	99.6%	239.1%	134.5%
2	SUI	3	0	1	4	8.0	37.5%	0.0%	12.5%	50.0%	40.6%
3	SLO	1	1	2	4	5.4	18.5%	18.5%	37.0%	73.9%	37.0%
4	NED	4	2	4	10	16.8	23.9%	11.9%	23.9%	59.6%	35.8%
5	AUT	1	4	0	5	8.5	11.8%	47.3%	0.0%	59.1%	35.5%
6	CAN	4	4	2	10	34.9	11.5%	11.5%	5.7%	28.7%	18.6%
7	SVK	1	0	0	1	5.4	18.5%	0.0%	0.0%	18.5%	18.5%
8	SWE	0	3	1	4	9.5	0.0%	31.5%	10.5%	42.0%	18.4%
9	LAT	0	0	1	1	2.0	0.0%	0.0%	49.4%	49.4%	12.3%
10	CZE	0	2	1	3	10.5	0.0%	19.0%	9.5%	28.5%	11.9%
11	BLR	1	0	0	1	9.5	10.6%	0.0%	0.0%	10.6%	10.6%
12	FIN	0	1	0	1	5.4	0.0%	18.5%	0.0%	18.5%	9.2%
13	GER	6	1	1	8	81.9	7.3%	1.2%	1.2%	9.8%	8.2%
14	RUS	2	4	3	9	143.5	1.4%	2.8%	2.1%	6.3%	3.3%
15	POL	1	0	0	1	38.5	2.6%	0.0%	0.0%	2.6%	2.6%
16	FRA	1	0	2	3	65.7	1.5%	0.0%	3.0%	4.6%	2.3%
17	AUS	0	1	0	1	22.7	0.0%	4.4%	0.0%	4.4%	2.2%
18	KOR	1	0	0	1	50.0	2.0%	0.0%	0.0%	2.0%	2.0%
19	USA	3	1	5	9	313.9	1.0%	0.3%	1.6%	2.9%	1.5%
20	ITA	0	1	1	2	60.9	0.0%	1.6%	1.6%	3.3%	1.2%
21	JPN	0	2	1	3	127.6	0.0%	1.6%	0.8%	2.4%	1.0%
22	UKR	0	0	1	1	45.6	0.0%	0.0%	2.2%	2.2%	0.5%
23	GBR	0	0	1	1	63.2	0.0%	0.0%	1.6%	1.6%	0.4%
24	CHN	0	1	0	1	1,351.4	0.0%	0.1%	0.0%	0.1%	0.0%

Norway remains #1 with a weighted medal total of 1.345 per million citizens.

Switzerland surges to #2 with .406 weighted medals per million citizens.

Tiny Slovakia takes third with .370 weighted medals per million.

Canada moves up to #6 (.186) from #8 in a non-weighted system, but down from #2 in a crude medal count unadjusted for population.

The USA (.015) drops to #19 from #18, and way down from #5 in the non-population-adjusted crude medal rankings.  

China (.00037) and Great Britain (.004) remain the last-ranking of the medal-scoring countries.  However, since only 24 of the 96 competing countries (technically, countries don't compete - their national olympic committees do) have earned medals, even China finds itself at the 75th percentile of all competing countries.  

Appendix

Population and Predicted Medals, a More Detailed Mathematical Approach

A final mathematical demonstration (skip this paragraph if you hate math):  let's assume that athletic ability is roughly normally distributed in the population (a fair assumption).  In other words, if most people run a given distance after a certain amount a training, let's say that most will run it in 10 seconds with a standard deviation of 1 second.   This means that 15.9% of any large population will run it in 9 seconds or faster (10 - 1), and less than 3% will run it in 8 seconds or faster (10 - 1 x 2).  But in the Olympics, only three people can win a medal, so it does you no good if most of your team is very, very fast.  All that matters is how fast your top 3 racers are compared to the top 3 racers of another team.   If we know that the current world champion runs it in 6 seconds, we can calculate that we would need a pool of 31,575 runner simply to find one that can run that fast.   To find 3 of either gender, we would need a pool of 94,725 trained runners with the time, energy, and will to compete in the Olympics.   But this pool cannot come from the general population, since the very old and very young cannot compete.  Since most athletes come from an age range from 15 to 29, and about 20% of the American population was in this range in 2010 (obviously this varies some from country to country), we would therefore need a population of at least 473,625 people (94,725 / .2) to find 3 medal-likely runners.  But it gets worse, since the Olympics is segregated by gender.   Just to send 3 Olympic caliber women to the Olympics, we would need a population of twice 473,625 (assuming half are women) or 947,250.  Of course, most of the population is not athletically active.  According to WebMD, 49.6% of Americans report exercising for at least 30 minutes for 3 times a week.  Not all of these run, of course, and not all are still in their competitive years, and few exercise the fanatical amount required of an Olympic athlete, or have the means or desire to drop whatever other responsibilities they have to join an Olympic team, but let's make a very liberal assumption that 1 in 10 do exercise this much AND can drop everything to compete internationally.  This means that we would need a population of 9,472,500 (947,250 x 10) to find 3 Olympic caliber athletes.  Now, this by no means guarantees that they would win or remain uninjured or be able to compete in all events.   Returning to the winter Olympics, the United States team numbers over 2,000 athletes.  To find 2,000 athletes all at Olympic caliber requires a huge population.  It is no surprise then that the countries with the largest populations tend to dominate the medal count.  

Yes, there are diminishing returns (the difference in time between the 100 millionth and the millionth fasters runners is no doubt far less than the difference in time between the millionth and the thousandth or the thousandth and the hundredth, and of course there are differences in cultures (how much a society values, encourages, and supports athletics and what type will influence how large the competing talent pool as well), access to appropriate venues (the Bahamas, despite the Disney movie to the contrary, in general is at a severe disadvantage in the winter games), and ethnicity (certain ethnic groups dominate sprinting and marathon running regardless of what country members of that group happen to be citizens of).  But some correction for population size seems fair, and ignoring massive differences in population size seems absurd.