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Joe Sterling
science forum beginner

Joined: 21 Jan 2006
Posts: 2

Posted: Sat Apr 22, 2006 11:55 pm    Post subject: Percent change between negative and positive numbers

Is there any authoritative source on computing the percent change
between integers of opposite sign?

I have a debate going with someone on this topic. My feeling is the
formula

(A1-B1)/ABS(B1)*100 yields the percent change from B1 to A1, whatever
the sign of the numbers.

But a colleague pointed out that the formula applied to

B1 = -10
A1 = +10

shows a 200% increase, whereas if

B1 = -1
A1 = +10

we get a 1100% increase. And how can an increase from a smaller number
(-10) to 10 be a lesser percentage than an increase from a larger
number (-1) to 10?

It sounds like a good counter argument to the formula, but intuitively
it seems to that it depends on how percent increase is formally
defined. It does seem to me that to get to 10 from -10 you need to add
10 twice so that you do have a 200% increase.

I would be tempted to say that the reason for the apparent difficulty
is that in cases like this the percent increase is not related to the
difference in magnitudes of the numbers, whereas it is if the formula
is applied to positive integers. The apparent assymetry is analogus to
the fact that one would expect that doubling a number (multiplying by
2) always increases it, but in the case of negative numbers it does
not.

Any opinions on this?

Thanks,

Joe Sterling
Carlos Moreno

Joined: 11 May 2005
Posts: 80

Posted: Sun Apr 23, 2006 12:48 am    Post subject: Re: Percent change between negative and positive numbers

Joe Sterling wrote:
 Quote: Is there any authoritative source on computing the percent change between integers of opposite sign? I have a debate going with someone on this topic. My feeling is the formula (A1-B1)/ABS(B1)*100 yields the percent change from B1 to A1, whatever the sign of the numbers.

I don't know if there is an authoritative answer to this (well, I'm
sure there has to be), but this definition does not make sense to
me -- not in the least.

Why the mysterious absolute value?

Percentage implies a proportional measure. If the exact same
multiplicative operation that was applied to your number were to
be applied to 100.

So, if from -10 you increase 20, that means that you added -2 times
your original number -- an increase of -2 times your original number;
an increase of -200%

 Quote: It sounds like a good counter argument to the formula, but intuitively it seems to that it depends on how percent increase is formally defined. It does seem to me that to get to 10 from -10 you need to add 10 twice so that you do have a 200% increase.

Wait -- the percentage of increase is measured with respect to
the initial number, and not the final number (look at your
formula). So no, it's not that you're adding 10 twice. It's
that you're -2 times adding -10 (I reversed the order of the
phrasing to avoid the confusing adding -10 -2 times)

Carlos
--
Chip Eastham
science forum Guru

Joined: 01 May 2005
Posts: 412

Posted: Sun Apr 23, 2006 12:55 am    Post subject: Re: Percent change between negative and positive numbers

Joe Sterling wrote:
 Quote: Is there any authoritative source on computing the percent change between integers of opposite sign? I have a debate going with someone on this topic. My feeling is the formula (A1-B1)/ABS(B1)*100 yields the percent change from B1 to A1, whatever the sign of the numbers. But a colleague pointed out that the formula applied to B1 = -10 A1 = +10 shows a 200% increase, whereas if B1 = -1 A1 = +10 we get a 1100% increase. And how can an increase from a smaller number (-10) to 10 be a lesser percentage than an increase from a larger number (-1) to 10? It sounds like a good counter argument to the formula, but intuitively it seems to that it depends on how percent increase is formally defined. It does seem to me that to get to 10 from -10 you need to add 10 twice so that you do have a 200% increase. I would be tempted to say that the reason for the apparent difficulty is that in cases like this the percent increase is not related to the difference in magnitudes of the numbers, whereas it is if the formula is applied to positive integers. The apparent assymetry is analogus to the fact that one would expect that doubling a number (multiplying by 2) always increases it, but in the case of negative numbers it does not. Any opinions on this? Thanks, Joe Sterling

While I applaud your attempt to apply rigor to this situation,
the use of "percentage increase" (or decrease) in comparing
negative and positive numbers is likely to mislead, whatever
formula is chosen.

So the practical answer is not to use this terminology except
when comparing "before" and "after" values that are both
positive (or at least of the same sign which should otherwise

It is well-established usage that a "before" figure of (say)
3 and 1/3 changing to an "after" figure of 10 constitutes
a 200% increase.

To describe a change from -10 to +10 in the same terms
cannot help but mislead. Certainly the latter is an increase
but one more impressive than merely going from 3 and 1/3
to 10, is it not?

regards, chip
Klueless
science forum beginner

Joined: 30 May 2005
Posts: 19

Posted: Sun Apr 23, 2006 1:42 am    Post subject: Re: Percent change between negative and positive numbers

"Joe Sterling" <bgmpsl@hotmail.com> wrote in message news:1145750134.988878.188690@g10g2000cwb.googlegroups.com...
 Quote: (A1-B1)/ABS(B1)*100 yields the percent change from B1 to A1, whatever the sign of the numbers.

I would use this formula without the absolute value. The reason is
that if F=100*(A-B)/B then the equation can be rewritten as an equation
linear in A and B which allows easy solution for A and B as functions
of the other two quantities. The ABS would unnecessarily complicate
this elegance.

An increase from B=1 to A=10 is a 900% increase and from B=1 to A=100
is a 9900% increase seems perfectly reasonable to me. Therefore, having
no strongly preconceived idea what percentage increases and decreases
amongst the negatives ought to be, I let the formula tell me that from B=-1 to
A=10 is a -1100% increase -- i.e. a 1100% decrease in the value of B. This
seems puzzling at first. Then, consider that the formula says from B=-1 to
A=-1/2 is a 50% decrease in B -- we see what is happening. Movement
towards 0 is "decreasing" the negative B. Reaching 0 is a 100% decrease
of the negative B. So going beyond 0 into the positive numbers would have
to be an even bigger decrease, something bigger than 100%.
David W. Cantrell
science forum Guru

Joined: 02 May 2005
Posts: 352

Posted: Sun Apr 23, 2006 1:58 am    Post subject: Re: Percent change between negative and positive numbers

"Chip Eastham" <hardmath@gmail.com> wrote:
[snip]
 Quote: While I applaud your attempt to apply rigor to this situation, the use of "percentage increase" (or decrease) in comparing negative and positive numbers is likely to mislead, whatever formula is chosen. So the practical answer is not to use this terminology except when comparing "before" and "after" values that are both positive (or at least of the same sign which should otherwise be known to the reader or hearer of your comments). It is well-established usage that a "before" figure of (say) 3 and 1/3 changing to an "after" figure of 10 constitutes a 200% increase. To describe a change from -10 to +10 in the same terms cannot help but mislead.

I can see it now: The dawning of an important new technique
in "How to Lie with Statistics"!

Cheers,
David

 Quote: Certainly the latter is an increase but one more impressive than merely going from 3 and 1/3 to 10, is it not? regards, chip

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