Percentage change formula involving negative numbers

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

  1. Petre Huile

    Petre Huile Guest

    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?

    Petre Huile, Sep 4, 2004
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  2. Petre Huile

    Mike Terry Guest

    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%.

    Mike Terry, Sep 4, 2004
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  3. Petre Huile

    Peter Webb Guest

    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 Huile

    Petre Huile Guest

    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.

    Petre Huile, Sep 4, 2004
  5. Petre Huile

    Mike Terry Guest

    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...

    Mike Terry, Sep 4, 2004
  6. Petre Huile

    Bob Guest

    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

    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, Sep 5, 2004
  7. 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
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