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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
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

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
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
```
Associated Topics:
Middle School Fractions