Very simple maths question to determine my fate

[J3N0VA]

When they work out your kill to death ratio on the game Call of Duty Black Ops, if they calculate your total kills divided by your total deaths, is it the same as finding your average kill to death ratio?

Basically, is

sum(k/d)/g = sum(k)/sum(d)
where all sums are from 1 to g

true?

If I have the wrong answer I think I will change degree. Please check that the text in blue match before attempting to answer the question.

 

[LCKurtz]

When they work out your kill to death ratio on the game Call of Duty Black Ops, if they calculate your total kills divided by your total deaths, is it the same as finding your average kill to death ratio?

Basically, is

sum(k/d)/g = sum(k)/sum(d)
where all sums are from 1 to g

true?

If I have the wrong answer I think I will change degree. Please check that the text in blue match before attempting to answer the question.

It's hard to figure out what you are really asking. I am not familiar with that game. I have several questions:

What is g?
What is the index of summation?
Does each play of the game result in a single kill or death or several kills and deaths?

For example, in your first game might you have k₁ kills and d₁ deaths? If so is k₁/d₁ what you call the kill-to-death ratio for game 1? Is that what you are trying to average over several games? If not that, then what?

More details please.

 

[J3N0VA]

It's hard to figure out what you are really asking. I am not familiar with that game. I have several questions:

What is g?
What is the index of summation?
Does each play of the game result in a single kill or death or several kills and deaths?

For example, in your first game might you have k₁ kills and d₁ deaths? If so is k₁/d₁ what you call the kill-to-death ratio for game 1? Is that what you are trying to average over several games? If not that, then what?

More details please.

Yes, what you have said is what I meant. And g stands for the total number of games played.

 

[LCKurtz]

If we let k_i and d_i be the numbers of kills and deaths in the i'th game and we play n games, then what I think you are asking is whether

\frac{\sum_{i=1}^n\frac{k_i}{d_i}}{n}=\frac{\sum_{i=1}^nk_i}{\sum_{i=1}^nd_i}

If that is what you are asking, the answer is no. I might also observe that you would have no hope of answering the question given the difficulty you have stating it in the first place.

 

[J3N0VA]

If we let k_i and d_i be the numbers of kills and deaths in the i'th game and we play n games, then what I think you are asking is whether

\frac{\sum_{i=1}^n\frac{k_i}{d_i}}{n}=\frac{\sum_{i=1}^nk_i}{\sum_{i=1}^nd_i}

If that is what you are asking, the answer is no. I might also observe that you would have no hope of answering the question given the difficulty you have stating it in the first place.

Well I wrote that equation and proved it wrong by contradiction.

Don't you think it's interesting that it isn't true?

 

[disregardthat]

Even more interesting, the two numbers have drastically different interpretations. See

http://en.wikipedia.org/wiki/Simpson's_paradox

So basically if you compare your average kill to death ratio with another player and find that it is better, he can still have a better total win/loss ratio.

 

[LCKurtz]

Well I wrote that equation and proved it wrong by contradiction.

Easy enough, once the problem is formulated correctly.

Don't you think it's interesting that it isn't true?

It didn't strike me as "interesting that it isn't true". My reaction was more like "why would anyone expect those to be equal?".

Even more interesting, the two numbers have drastically different interpretations. See

http://en.wikipedia.org/wiki/Simpson's_paradox

So basically if you compare your average kill to death ratio with another player and find that it is better, he can still have a better total win/loss ratio.

Thanks for the link. I had never heard of Simpson's paradox.

 

[J3N0VA]

It didn't strike me as "interesting that it isn't true". My reaction was more like "why would anyone expect those to be equal?".

Perhaps you have a stronger mathematical intuition than me.

 

[Kindayr]

The best way I look at them is that the left side is the average K/D per game. The second number is your total kills over your total deaths.

They can vary a lot, depending on your own skill. if you consistently get the same K/D per game, then they will be the same, or at least closer to one another. But if your skill is inconsistent, then these numbers will differ much more substantially. For example, if you get a game where you get 0 kills and 200 deaths, your K/D for that game is 0, so it affects the left side very small. However, it adds 200 deaths to the right side, affecting your total more drastically.

I hope that opens some intuition.

 

[J3N0VA]

Even more interesting, the two numbers have drastically different interpretations. See

http://en.wikipedia.org/wiki/Simpson's_paradox

So basically if you compare your average kill to death ratio with another player and find that it is better, he can still have a better total win/loss ratio.

Can you explain this article in mathematical terms? I find it confusing.