Percentage Error

Date: 10/24/2002 at 20:47:35
From: Matthew Dean
Subject: Percentage error

Dr. Math,

Please help me find the percentage error for the following numbers.

   5.7 estimated findings
   5.8 actual findings

Date: 10/25/2002 at 16:17:00
From: Doctor Achilles
Subject: Re: Percentage error

Hi Matthew,

Thanks for writing to Dr. Math.

Percentage error is just how much your guess was off from the actual 
value. The formula is:

  |estimate - actual|/actual * 100%

[That is: the absolute value of (the estimate minus the actual) all 
divided by the actual, all multiplied by 100%.]

Let's think about why we use this formula. If you want to know how 
close your estimate is, the first thing to do is just to ask how much 
you missed by, that is the absolute value of the difference between 
the two numbers (the absolute value is used because you're only 
concerned with how much you missed by, not whether you were too high 
or too low). So in this case, you missed by 0.1.

0.1 is a small number, so it sounds like your guess is pretty good.

So why do we bother with this business of dividing by the actual 
amount?

Let's take a couple of other examples:

Example 1:

  100.0 (estimated)
  105.3 (actual)

Example 2:

  10.0 (estimated)
  15.3 (actual)

Example 3:

  1.0 (estimated)
  6.3 (actual)

Example 4:

  0.1 (estimated)
  5.4 (actual)

In all cases, you missed by the same amount (5.3). But in the first 
example, it seems as if your guess was a lot better. Even though it's 
off by just as much as the last example, missing by 5.3 out of 105.3 
isn't too bad, but missing by 5.3 out of 5.4 seems pretty darn bad.

So we're concerned here not just with how much you missed by, but with 
what percent of the actual value you missed by.

So in example 1, you missed by 5.3 out of a total of 105.3:

  5.3/105.3 * 100%

equals

  0.0503 * 100%

equals

  5.03%

And in example 4, you missed by 5.3 out of total of 5.4:

  5.3/5.4 * 100%

equals

  .9815 * 100%

equals

  98.15%

This is a good way to represent the intuition that missing by 5.3 out 
of 105.3 is a pretty good guess, but missing by 5.3 out of 5.4 is a 
pretty rotten guess. In one case, your percent error is only about 5% 
(small error), while in the other case, your percent error is a big 
98% (huge error).

Can you apply the formula to find the percentage error for your 
numbers now?

I hope this helps. If you have other questions or you'd like to talk 
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/