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I'm trying to quantify the relative change between two items. First item goes from $15$ to $37$ in $2$ minutes and second item goes from $15$ to $10$ in the same time. The rate of change for first item is $37 - 15 = 22$ in $2$ minutes and for the second item its $10 - 15 = -5$. Now, i want to know how much times is the first item better than second item.

For example if the change in first item would have been $20 (35 - 15)$ and second item is $2 ( 17 - 15)$ then we would have said first item is $10$ times better than second item. but here due to negative number I'm not sure.

Diwakar Jha
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    "For example if the change in first item would have been 20 (35-15) and the second item is 2 (17-15) then we would have said first item is 10 times better than second item" Talking about "times better" or "percent more" and similar related comparisons are often very ambiguous and it is unclear without using more words what the number is in reference to. Now... if we were to assume you really do intend what you say in your second paragraph to be correct... then your notion of "times better" is simply looking at the ratio of the differences between later values compared to initial ones... – JMoravitz Jun 26 '20 at 16:46
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    In such a case... then you would have $\frac{22}{-5}$. If you don't like having something being "negative times better than" then that just means that your definition of "times better than" that you are using is flawed and you need to rethink your second paragraph. – JMoravitz Jun 26 '20 at 16:47

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In a sense the answer is clearly $-\frac{22}{5}$, but you're right to feel dis-satasfied with that answer. But the problem is with the question, not the answer.

You have two things that are changing over time. We could assume that they're changing at constant rates, or we could assume we're always talking about the average rate-of-change during a particular 2-minute interval, whatever. $$ f_1(t) = 15 + 11 x $$ $$ f_2(t) = 15 - \frac{5}{2} x $$ The slope of $f_1$ is 11; the slope of $f_2$ is $-\frac{5}{2}$. $11 = -\frac{22}{5} \times -\frac{5}{2}$; so "$-4.4\times$"; no problem.

But your post consistently asks about "better than", not "times", and you want to quantify this. That's weird because it implies a system of goodness or value (a utility function), context that isn't included in your post.

We could be a little less worried about linguistic ambiguity if we instead talked about "faster than". These are values changing over time, so it makes total sense to ask

How much faster is $f_1$ changing than $f_2$?

Pay attention to how weird it sounds to say

$f_1$ is negative 4.4 times faster than $f_2$.

That's because an expression of speed does not on its own contain directional (sign) information.
We would instead say

$f_1$ is [increasing] 4.4 times faster than $f_2$ [is decreasing].

So what about "better than"?
If you really do have a quantifiable notion "good" and "bad" in which the goodness of $f_n$ is just $f_n$'s rate of increase, then you could say

$f_1$ is 4.4 times better than $f_2$ is bad.

But you probably don't.

ShapeOfMatter
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  • Thanks for the detailed answers, much appreciate. I'm sorry for the confusion regarding "better than" and "times". I think "times" is what i'm more interested in. Looks like, mathematically "f1 is 4.4 times better than f2 is bad" seems right but it is still not as intuitive as F1 is 10 times faster than f2. Just wondering is there a better way to put it (again, quantitatively)! – Diwakar Jha Jun 28 '20 at 00:01