When Percentage Calculations are Inappropriate
Date: 02/11/2003 at 09:58:57 From: David Subject: Negative Goals How do I best calculate the % of goal when at least one of the numbers is a negative number? With positive numbers, this is easy… Goal = 100 Actual = 98 % to Goal = 98 / 100 = .98 = 98% of goal but it appears more difficult when your goal is negative. Here is my example: A company that provides a monthly service brings on new customers every day, but unfortunately also has customers deactivate from their service each month as well. In these tough economic times, they have a quota to reach: –1000 customers ("negative" 1000 customers) this year (meaning that they will have 1000 more customers deactivate than those that activate). If they lose fewer than 1000 customers, they will have surpassed their goal. For this example, let's say they blow away their goal and bring on 100 MORE customers than they deactivate. What % did they beat their goal? If we use the calculation above, it would be … Goal = -1000 Actual = 100 % to Goal = 100 / -1000 = -.10 = -10% of goal?? Example 2: Goal = -1000 Actual = -500 % to Goal = -500 / -1000 = .50 = 50% of goal?? It seems simple. What am I doing wrong? Thank you.
Date: 02/11/2003 at 12:12:51 From: Doctor Peterson Subject: Re: Negative Goals Hi, David. This is not an uncommon problem; I've discussed something like it here in our archives: Calculating Percent Change when the Base is a Negative Number http://mathforum.org/library/drmath/view/55720.html The problem is that percentage is not the right way to measure achievement of this sort of goal. It's really inappropriate in any case, but it becomes obvious when the goal is zero or negative. What's wrong is that the goal is not proportional to the effort expended, so the percentage of the goal attained does not measure anything meaningful. In particular, attaining a negative goal does not require negative effort! Your goal is to keep the loss from getting too great. If you lose half as many customers as you hoped, you have done very well, not half as well as you intended. (That's what your 50% example shows.) How can we adequately measure such performance? The best approach, I think, is to ignore percentages entirely, and just say that you retained 200 more customers than you had hoped, rather than that you lost only 800 of the 1000 you had expected. If you need a percentage, you have to decide what to compare this with. You might just use the absolute numbers, rather than the loss: you had, say, 15000 customers, and your goal was to have no less than 14000 at the end of the year. If you have 14,200, then you accomplished 14,200/14,000 * 100% = 101.4% of your goal of customer retention. But although the numbers taken this way can't go negative, they still are not necessarily proportional to effort; if you had 15,000,000 customers to start with, and lost 800 rather than 1000, the percentage would look a lot less, namely 100.001%. That doesn't seem appropriate (but perhaps it is). The problem is that you are running up a down escalator: as hard as you run, you are being carried backward by a force you have no control over. You want to measure how well you are running by how slowly you move backward, rather than by how fast you move forward. If you knew how fast the escalator was moving (how many customers would have left if you had made no effort), you could subtract that before measuring your progress. But you don't know that number. When the escalator was moving up, making it look as if you were making a big effort to gain customers even if you did nothing, you wouldn't have thought to subtract the amount of gain that could be attributed to "good times"; but your numbers then were just as meaningless as they are now! It's just harder to hide it now. Again, I recommend just reporting the number of extra customers retained, and not trying to compare it with any arbitrary base. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 02/12/2003 at 09:40:46 From: David Subject: Thank you (Negative Goals) What a stumper! I have floundered with this issue for years and couldn't find anyone who could answer my question. I read through the Wall Street Journal article that you linked and I can see that I'm not the only one who was stumped. Thank you for answering this difficult question for me. David
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