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Measuring: Greatest Possible Error

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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?
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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.

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
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/
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Associated Topics:
Middle School Measurement

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