# Percentage change formula involving negative numbers

Discussion in 'General Math' started by Petre Huile, Sep 4, 2004.

1. ### Petre HuileGuest

I need to calculate the percentage change (or increase) for a series
of number, such as 100, 200, 500, 300. The percentage change/increase
from 100 to 200 will be 100%, from 200 to 500 will be 150%, from 500
to 300 will be -40%.

However, when negative numbers are involved, I am not sure how to do
this correctly. For example, I have a series of numbers:

-15000, -3000, 10000

What is the percentage change/increase from -15000 to -3000? Should it
be 80%, since you have to increase -15000 by +12000 to get to -3000?
And from -3000 to 10000 is 433.33%? These percentages doesn't seem to
look right since you are going from a negative to a negative. Take the
first example, going from -15000 to +12000 in order to make -3000
seems to indicate that the percentage change/increase should be more
than 80%.

Can someone help me out on this? What is the proper formula for
calculating this percentage when negative numbers are involved?

Thanks,
Petre

Petre Huile, Sep 4, 2004

2. ### Mike TerryGuest

Start with the simpler question: given two numbers A and B, what percentage
is B of A? The answer is (B/A) * 100%.

E.g. what percentage is 15 of 20? Answer, (15/20) * 100% = 75%.
Or what % is -3000 of -15000? Answer, (-3000/-15000)) * 100% = 20%.

Now, if the question is what is the percentage increase from A to B, just
ask "what percentage is B of A?" then subtract 100%, since we only want the
increase bit.

E.g. what is percentage increase from -15000 to -3000? Well, -3000 is 20%
of -15000, so the percentage increase here is 20%-100% = -80%.

Regards,
Mike.

Mike Terry, Sep 4, 2004

3. ### Peter WebbGuest

The formula is (a-b)/b * 100.

However, mathematicians seldom use percentages; the main practical use of
percentages seems to be to obscure the truth.

Consider the following "percentage changes":

From 1 to 5,000 is 499,900%.
From 5,000 to 1 its 99.98%
From 5,000 to 0 its 100%
From 0 to 5,000 its undefined (infinite if you like)
From 2 to 5,000 is 249,000%.
From -1 to 5,000 its 500,100%.
From .01 to 5,000 its 49,990,000%
From 5,000 to 500,000 its 9,900%.

These figures are basically meaningless. If percentages mean anything, its
for comparing two similar numbers. As soon as one number is more than (say)
double the other number, percentages become increasingly useless.

Maybe if you posted what you are actually trying to do with percentages, we
could suggest a more meaningful way of comparing numbers.

Peter Webb, Sep 4, 2004
4. ### Petre HuileGuest

For two numbers a, then b in a series, I used the formula (b-a)/a *
100 (initially) since I want a formula that can compare the
change/increase from one number in a series to the next. The use of
the comparison is, for example, in a series of dollar figures to
measure sales. What is the percentage increase from \$10 to \$20? Which
will be 100%.

However, the formula doesn't work as well (as least as I see it with
my limited math knowledge) when there is a negative number involved.
If the series goes like this:

-\$50, -\$20, \$50

Applying the formula, from -\$50 to -\$20, you will get -60%, which
doesn't look right since you need to earn \$30 in order to get to -\$20,
so the percentage should be a positive number to reflect the positive
change. Also, from -\$20 to \$50, the formula will give -350% which
again looks wrong since you have to *add* +70 onto -20 in order to get
to 50. So the increase, whether it is a dollar figure or not, should
be a positive figure.

Is there anything I am missing here?? That's what I want to know, and
the above example illustrate pretty much what I want to do.

Thanks,
Petre

Petre Huile, Sep 4, 2004
5. ### Mike TerryGuest

Percentages are a way of representing *ratios*, and the ratio of a positive
number to a negative number is negative, so -350% is correct.
I don't think you're missing anything, although maybe the percentage figures
you're generating aren't particularly useful? In the end you have to ask
what you want to use these figures for...

Regards,
Mike.

Mike Terry, Sep 4, 2004
6. ### BobGuest

The point, which others have made, that you need to think about the
application, is important.

You say these are sales numbers. How can you have negative sales? (Ok,
I can imagine, but I doubt you meant that.) So the question does not
make sense.

Now, let's say they are profit/loss. If loss changed from 50 to 25, it
(the loss) decreased by 50%. If a loss of 50 is followed by a profit
of 25, that is good news. But I don’t see how you can meaningfully
speak of the change as a percentage.

(From memory... tables in newspaper showing financial results for
companies do not show the % change when one number involved is
negative.)

I think your own intuition is telling you that the formula is not
serving you well here. Indeed. Don't be a slave to it. I might even
suggest that the formula you give is restricted to a and b both >0. If
they are not, think carefully about what you intend. Note that the
restriction I suggest is not a mathematical restriction, but a logic
or relevance restriction.

bob

Bob, Sep 5, 2004
7. ### John O'FlahertyGuest

Even when a and b are >0, percentage loses meaning around zero. Imagine
a company that had \$.01/share earnings- barely profitable. Next year
they turn the corner, and have \$1/share earnings. This could easily
happen, and it would be an increase of 10,000%. Not a very useful figure.

John O'Flaherty, Sep 14, 2004