Measuring: Greatest Possible Error

Date: 04/18/98 at 14:54:39
From: Ann
Subject: Greatest possible error for measurement

Hi! I don't understand what the greatest possible error for 
measurement is or how to find it. Can you help me, please?

An example problem would be: 

The length of a room is given as 7 meters. What is the greatest 
possible error for this measurement?  

Date: 04/20/98 at 11:27:24
From: Doctor Derrel
Subject: Re: Greatest possible error for measurement

Hi Ann,

You have found the place where pure mathematics and the real world 
collide. I'm not sure that my answer will satisfy you, but I will try 
to give you a little insight.

The answer is: It depends.

I know that isn't much of answer, but I'll explain. Whenever you make 
a measurement, you have some error. How much the error is depends on 
the equipment you are using to make the measurement and how you use 
the equipment. If you are measuring the water in the beaker by looking 
at the volume markings, the accuracy of your measurement of the volume 
of water depends first on how you look at the water. That is, are you 
measuring at the top or the bottom of the meniscus? (The meniscus is 
the curved shape of the water's surface caused by its surface tension 
and interaction with it's container. If you aren't sure what this 
means, go get yourself an ordinary glass of water and look at how the 
water curves up where its surface touches the glass.) The second is 
how accurately measuring lines are painted on the beaker. The third is 
how thick the measuring lines are: i.e., is it the top, middle, or 
bottom of the measuring line that is the correct measurement?

For measurements you make in numerous steps that you add together, 
each additional step introduces more error. Let's look at the water in 
the beaker again. If you have a 500ml (milliliter) beaker, but need 10 
liters of water in a bucket, you will have to fill the beaker 20 times 
and empty into a bucket. If you consistently put in only 499ml, you 
would end up 20ml short of water in your bucket. 

Now, let's look at measuring distances. If you measure your room with 
a professional measuring tape, you will be more accurate than if you 
try to measure it by laying meter sticks end-to-end. However, if you 
are measuring a real room, you also have to decide where you are going 
to measure it. Most people might measure it on the floor along one 
wall. But is the opposite wall exactly the same length? If you measure 
it at the ceiling, is that exactly that same length? Is the wall 
curved? (Put your head against a wall and look down a hallway - you 
will probably see waves and curves in the wall.)

In short, the term "greatest possible error" has no meaning unless the 
measurement method is specified, the error of the measurement method 
when done by properly trained people is known, and the people who are 
doing the measurement have been trained properly.

My discussion has been based on the assumption that the room is 
really seven meters long. But what if you are going to build a room 
designed to be seven meters long, then measure it. The room is built 
base on a lot of measurements, so it may or may not be seven meters 
long before you measure it. You now have more sources of error.

My discussion is also based on the the assumption that you were 
talking about measurement error. On thing you have to know 
about dimensions (e.g., length, diameter, radius, angle, etc.) when 
you design something is that the dimension always has a tolerance 
stated or implied. That is, if you get a blueprint that says a 
dimension on a machine part is 37.2 millimeters (mm), some place on 
that blueprint it will say something like "Tolerance for xx.x is 
+/- 0.03." (The "+/-" is read as "plus or minus.") That means that for 
a dimension like 37.2 millimeters, you can actually build it so that 
it is anywhere between 36.9 and 37.5 millimeters. Alternately, it will 
be stated with the dimension as 37.2 +/- 0.03. This means, once again, 
it can have the dimension between 36.9 and 37.5 millimeters.

If the question really has to do with tolerance, it may depend on how 
your seven meters is written (i.e, 7m, 7.0m, 7.00m, or even as 700cm). 
In these cases, you you might reasonably assume that the tolerance is 
one-half the last digit. That is, for 7.0 meters, it is +/- 0.05m, for 
7.00 meters, it is +/- 0.005m, for 700cm it is +/- 0.5cm. This 
reasoning implies that 7 meters would have a tolerance of +/- 0.5m, 
but you have to ask yourself if a tolerance of half a meter in seven 
meters (about 7 centimeters in a meter) is reasonable.  

I hope I've helped. If you are still confused, write again.

-Doctor Derrel,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/