Friday, January 17, 2014

Significant figures

This post is about math.

(Now that should have made 99% of readers go away).

In a recent comment one of my readers remarked that I had quoted some number, that he wasn't sure about the precision of that number and that "numbers are pedantic". And I was under the impression that not everybody understands the precision of numbers in mathematical terms. Precision is very important in math and sciences, so there are well-established conventions which result in a number not only having a value, but also containing some information about the precision of that value.

The way this precision is expressed is by giving more or less significant figures. By convention, trailing zeroes are placeholders that indicate the scale of the number, and are not significant figures. So if I say that game X has 291,938 players, that is a pedantic number. It only has significant figures, and that basically claims that the number is exact. There are not 291,937 or 291,939 players in that game, there are EXACTLY 291,938.

But if I use trailing zeroes, and say that game X has 300,000 players, only the 3 is a significant figure, the five trailing zeroes aren't. Saying that a game has 300,000 players does not exclude the possibility of the game having some more or less players than that. Any mathematician would read that "300,000" as meaning "between 250,000 and 349,999". Because using all those trailing zeroes means that any number which would be rounded up or down to 300,000 is included. So "300,000" is not a very pedantic number, it is a number with a 16.7% error margin. Likewise if you express the number of players of a game in millions, you include the possibility of an error margin of plus or minus half a million.

These mathematical conventions about significant figures allow us to talk about numbers which are impossible to know exactly. Nobody knows EXACTLY how many inhabitants the United States of America has, and obviously the number changes every day through births, deaths, arrivals and departures. But mathematically it is perfectly okay to claim that the USA has 300 million inhabitants, because the implied error margin of that number is bigger than the probable error margin of a U.S. census. Saying that the 300 million number is wrong would mean saying that it is wrong by over 50 million.

Note that some player numbers expressed on blog posts are "wrong" or at least "disputable" by more than their error margin. To give the most quoted example, the number of World of Warcraft players, even if you express it in millions, varies by more than half a million depending on whether you count or don't count Chinese players in that number. The numbers of EVE Online vary by more than the implied error margin of 50,000 depending on whether you count Chinese players or not, and whether you count "accounts" or "players". And nobody agrees what should be counted for a Free2Play game, as the number of people who ever made an account is rather obviously inflated and not significant at all.

(Now that should have made 99% of readers go away).

In a recent comment one of my readers remarked that I had quoted some number, that he wasn't sure about the precision of that number and that "numbers are pedantic". And I was under the impression that not everybody understands the precision of numbers in mathematical terms. Precision is very important in math and sciences, so there are well-established conventions which result in a number not only having a value, but also containing some information about the precision of that value.

The way this precision is expressed is by giving more or less significant figures. By convention, trailing zeroes are placeholders that indicate the scale of the number, and are not significant figures. So if I say that game X has 291,938 players, that is a pedantic number. It only has significant figures, and that basically claims that the number is exact. There are not 291,937 or 291,939 players in that game, there are EXACTLY 291,938.

But if I use trailing zeroes, and say that game X has 300,000 players, only the 3 is a significant figure, the five trailing zeroes aren't. Saying that a game has 300,000 players does not exclude the possibility of the game having some more or less players than that. Any mathematician would read that "300,000" as meaning "between 250,000 and 349,999". Because using all those trailing zeroes means that any number which would be rounded up or down to 300,000 is included. So "300,000" is not a very pedantic number, it is a number with a 16.7% error margin. Likewise if you express the number of players of a game in millions, you include the possibility of an error margin of plus or minus half a million.

These mathematical conventions about significant figures allow us to talk about numbers which are impossible to know exactly. Nobody knows EXACTLY how many inhabitants the United States of America has, and obviously the number changes every day through births, deaths, arrivals and departures. But mathematically it is perfectly okay to claim that the USA has 300 million inhabitants, because the implied error margin of that number is bigger than the probable error margin of a U.S. census. Saying that the 300 million number is wrong would mean saying that it is wrong by over 50 million.

Note that some player numbers expressed on blog posts are "wrong" or at least "disputable" by more than their error margin. To give the most quoted example, the number of World of Warcraft players, even if you express it in millions, varies by more than half a million depending on whether you count or don't count Chinese players in that number. The numbers of EVE Online vary by more than the implied error margin of 50,000 depending on whether you count Chinese players or not, and whether you count "accounts" or "players". And nobody agrees what should be counted for a Free2Play game, as the number of people who ever made an account is rather obviously inflated and not significant at all.

Comments:

<< Home

Newer› ‹Older
Since when have Europeans spoken about 'Math'? Isn't it Maths? Don't tell me American cultural imperialism has claimed another victim?

Ironically enough, my oldest learned the same thing this fall in her Chemistry class. The things you're exposed to in an MMO blog.... ;-)

In french it is Math. The real word is MathÃ©matiques, but when we do abreviation, we never put the s. In French, we write : 1 TV, 2 TV

About the precision and the convention : It can be misleading. For exemple if you know there is 300 000 +/- 5 000, you still write 300 000 ;-) But mainly I agree !

About the precision and the convention : It can be misleading. For exemple if you know there is 300 000 +/- 5 000, you still write 300 000 ;-) But mainly I agree !

Your basic point is quite true. I find that I always use qualifiers e.g. say almost all; never say always so the no-value-added troll can't object to the 0.1% exception.

I would tend to disagree - I think many mathematicians would read 300,000 as an exact integer; any scientist would read it as a real number.

It has been ages so I'm sure you are correct but at first glance I would assume 300,000 is +-1 but 3e5 would be +-100,000 and 300 million or 300e6 would be +- million.

I think you are being quite conservative with your varies examples.

My readings are that the WoW Chinese numbers are perhaps half and certainly above half a million. ( 3.5 +-2 million seems defensible; if there are 700,00 losses that the majority occurred in Asia, and the next quarter the losses were about half, you have to start with a number a bit above a half million IMO. Also a survey of realms shows more than in E & NA.)

However much I am disappointed by what EVE is versus what it could be, I admire the honesty of CCP e.g. that you can see the concurrent user count so can have http://eve-offline.net/?server=tranquility

So if there X people playing in E/NA EVE then there 1.2 X playing worldwide and 2.5 X accounts playing worldwide assuming the 2+ accounts/person. So a subtle wording difference can make a 250% difference in the correct answer.

I would tend to disagree - I think many mathematicians would read 300,000 as an exact integer; any scientist would read it as a real number.

It has been ages so I'm sure you are correct but at first glance I would assume 300,000 is +-1 but 3e5 would be +-100,000 and 300 million or 300e6 would be +- million.

I think you are being quite conservative with your varies examples.

My readings are that the WoW Chinese numbers are perhaps half and certainly above half a million. ( 3.5 +-2 million seems defensible; if there are 700,00 losses that the majority occurred in Asia, and the next quarter the losses were about half, you have to start with a number a bit above a half million IMO. Also a survey of realms shows more than in E & NA.)

However much I am disappointed by what EVE is versus what it could be, I admire the honesty of CCP e.g. that you can see the concurrent user count so can have http://eve-offline.net/?server=tranquility

So if there X people playing in E/NA EVE then there 1.2 X playing worldwide and 2.5 X accounts playing worldwide assuming the 2+ accounts/person. So a subtle wording difference can make a 250% difference in the correct answer.

If you are writing for an audience of mathematicians then you could reasonably expect the conventions you outline to to be understood. If, on the other hand, you are writing for an audience that you yourself believe to be so inimical to the mere mention of math that "99%" will stop reading then, as a writer and communicator rather than a scientist, you would be best advised to use more qualifiers and clarifications.

@Bhagpuss, if the people who needed the qualifiers stopped reading at the first line, I can write for the remaining 1% without having to talk down to children.

Post a Comment
#### Links to this post:

<< Home