Lee, if we're going to talk technically, then "percentage difference" is technically not a valid measure of distance or "closeness", because distance is something that happens in a metric space, and a metric has to be symmetrical.

Also, I think you'd find subjects saying that "102 is approximately 100" sounds more natural than "100 is approximately 102".

I think in certain contexts it makes sense to think about the closeness of two quantities in terms of percentage difference. For example, let's say we're not just talking about the numbers 98 and 100, but the rates 98 mph and 100 mph. When we talk about speed, what we're actually interested in is usually not the speed itself but rather the amount of time it takes to cover a certain distance when traveling that speed.

So in this context, it makes sense to say that 98 mph is about 100 mph to the same degree that 980 mph is about 1000 mph--because they have the same marginal relation in the time required to cover a certain distance at those speeds.

Comment author:Kindly
15 June 2012 11:31:49AM
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But the relation you're describing is itself percentage-based! If you go from the rates to, say, the time it takes to cover a distance of 100 miles, then you get (roughly) 102 and 100 hours in the first case, and 10.2 and 10 hours in the second case. These only have the same relation if we use percentage differences or ratios to think about how close two times are.

Comment author:Kindly
15 June 2012 05:47:01PM
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I thought that iemnitable was trying to justify the use of ratios when comparing speeds (as an example), and I pointed out that this requires us first to justify the use of ratios when comparing times.

I was thinking of it more like: if there's a certain place I can get to in (roughly) 102 hours going 98 mph, and I want to get there in 100 hours, I need to speed up to 100 mph. Similarly, if there's a another place that I can get to in roughly 102 hours going 980 mph, and I want to get to that place in 100 hours, I need to speed up to 1000 mph.

I kind of wanted to clarify that in the original post but I hadn't really thought of a good way to express it at the time.

Furthermore, I think that your interpretation of the example even makes it more clear that it makes sense to think of it in terms of a ratio. In the first case, you've sped up by 2 mph and gotten a gain of about 2 hours, straightforward enough. But in the second case, you've sped up by 20 mph, and only gotten a gain of about 0.2 hours. Here's where I think most people's intuition is probably screaming "whaaaaaaat!?"

But if we think of it in terms of the ratios, then everything fits together nicely again and the screaming intuition voice shuts up. Plus the math we have to do to get to the right answer is a lot easier.

Comment author:Kindly
15 June 2012 10:40:13PM
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Oh, I see what you mean now.

(Incidentally, Eliezer's original objection can be resolved by taking logs. Suddenly although the ratios 102/100 and 100/102 are not symmetrical, log(102/100) and log(100/102) are.)

## Comments (66)

OldLee, if we're going to talk

technically, then "percentage difference" istechnicallynot a valid measure of distance or "closeness", because distance is something that happens in a metric space, and a metric has to be symmetrical.Also, I think you'd find subjects saying that "102 is approximately 100" sounds more natural than "100 is approximately 102".

I think in certain contexts it makes sense to think about the closeness of two quantities in terms of percentage difference. For example, let's say we're not just talking about the numbers 98 and 100, but the rates 98 mph and 100 mph. When we talk about speed, what we're actually interested in is usually not the speed itself but rather the amount of time it takes to cover a certain distance when traveling that speed.

So in this context, it makes sense to say that 98 mph is about 100 mph to the same degree that 980 mph is about 1000 mph--because they have the same marginal relation in the time required to cover a certain distance at those speeds.

But the relation you're describing is itself percentage-based! If you go from the rates to, say, the time it takes to cover a distance of 100 miles, then you get (roughly) 102 and 100 hours in the first case, and 10.2 and 10 hours in the second case. These only have the same relation if we use percentage differences or ratios to think about how close two times are.

It looks to me like that was iemnitable's goal.

I thought that iemnitable was trying to justify the use of ratios when comparing speeds (as an example), and I pointed out that this requires us first to justify the use of ratios when comparing times.

Ah; I got a different impression from the great-grandparent. I agree with your point in the parent.

I was thinking of it more like: if there's a certain place I can get to in (roughly) 102 hours going 98 mph, and I want to get there in 100 hours, I need to speed up to 100 mph. Similarly, if there's a another place that I can get to in roughly 102 hours going 980 mph, and I want to get to that place in 100 hours, I need to speed up to 1000 mph.

I kind of wanted to clarify that in the original post but I hadn't really thought of a good way to express it at the time.

Furthermore, I think that your interpretation of the example even makes it more clear that it makes sense to think of it in terms of a ratio. In the first case, you've sped up by 2 mph and gotten a gain of about 2 hours, straightforward enough. But in the second case, you've sped up by 20 mph, and only gotten a gain of about 0.2 hours. Here's where I think most people's intuition is probably screaming "whaaaaaaat!?"

But if we think of it in terms of the ratios, then everything fits together nicely again and the screaming intuition voice shuts up. Plus the math we have to do to get to the right answer is a lot easier.

Oh, I see what you mean now.

(Incidentally, Eliezer's original objection can be resolved by taking logs. Suddenly although the ratios 102/100 and 100/102 are not symmetrical, log(102/100) and log(100/102) are.)