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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 

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 

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.

Associated Topics:
Middle School Fractions
Middle School Number Sense/About Numbers
Middle School Statistics

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