Greatest Possible Error

Date: 02/11/99 at 23:06:58
From: Brenda 
Subject: Greatest possible error

I need an elementary level answer to what is the greatest possible 
error. I went into your archives and noticed that you had an answer, 
but I need an answer that students in fourth grade can understand, with 
an example. If you could help, that would be great.

Thanks,
Brenda Brouwer

Date: 02/12/99 at 12:14:42
From: Doctor Peterson
Subject: Re: Greatest possible error

Hi, Brenda. I'll see if I can help.

It would be helpful if you could give me an example to show the context 
of your question - what kind of error you have in mind. I assume you 
are referring to this answer, which covers several different aspects of 
the question:

   http://mathforum.org/dr.math/problems/ann4.18.98.html   

Suppose you are given a measurement a friend has made - say that the 
height of his room's ceiling is 7 feet. In one sense the greatest 
possible error is "infinite" - your friend might simply be lying! But 
we'll assume that we can trust him, and the height really is 7 feet. 
The question is, how far off might he be and still be considered 
correct? That depends on how he stated his measurement. He might say, 
"It's exactly seven feet, according to precise surveying instruments", 
or he might have said "It's about seven feet, as close as I can 
estimate without using a ruler." But let's again suppose that he 
actually made a measurement, but all he said was "7 feet." What might 
the actual height be?

Any number that would round off to 7 feet would be reasonable. If he 
measured 7 feet 3 inches, but didn't trust his measurement much, he 
might round it off to 7 feet. But if he measured 9 feet, he certainly 
couldn't honestly tell you 7. Even if he measured 7 feet 9 inches, he 
would have rounded it off to 8 feet. So any number between 6 1/2 and 
7 1/2 feet is possible for the actual height. So we would say that the 
"maximum error" in the measurement as he expressed it is 1/2 foot.

The hard part is that we don't know for sure how much accuracy he was 
trying for. If he told us 80 inches, we wouldn't know whether he 
rounded it off to the nearest inch (up to a 1/2-inch error) or to the 
nearest 10 inches (up to a 5-inch error). So when accuracy matters, we 
try to be more specific about it, and say something like "7 feet to 
the nearest inch," or "7 feet plus-or-minus half an inch." That's a lot 
more useful than trying to read something into the number we were 
given.

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