The concept of error is
inherent to any experimental process. In
fact, it is a part of nature. This brief
introduction will present the concept of experimental error, explain why it is
important to us as engineers, and illustrate some of the ways that we deal with
error as we encounter it.

·
*So what is error?*

Most of us have grown up with
the idea that the word “error” invariably means that someone made a mistake,
got a wrong answer on an exam, etc. When scientists, engineers, and
statisticians speak of experimental error, however, they are not referring to a
mistake of any kind. Errors in
scientific measurements are simply *unavoidable*** uncertainties**. As technically sophisticated people, we
recognize that perfect measurements are simply not attainable, and we find
methods of dealing with this fact.

There can be many sources of
error. For example, if you use a
stopwatch to time a worker performing a task, you introduce an error in the
time measurement because it takes a fraction of a second for you to hit the
start and stop buttons. There is no way
to ** exactly**
account for that small deviation from the “true” elapsed time, and so an error
is present in the reading. The same applies for the experiments we are doing in
the labs.

·
*If error is inherent to all measurements, why do
we care about it?*

It’s true that the presence of
error is a given. What becomes
important, then, is to try to ** quantify** that error. That is, we need to know

·
*How can we get a handle on error?*

Simply measuring a quantity
once tells us little about the error associated with that measurement. The best way to get a handle on error is to
take a series of measurements (a ** sample**)and then
calculate a

For the
population, for the mean, and *s* for
the standard deviation.

The ** mean, **represented by the
Greek letter

where **N** is the total number of
measurements.

The sample standard deviation,
represented by the Greek letter s, or ** sigma**, is the actual
estimate of the error. It measures the
average difference between our observed values and our best guess of the
mean. The standard deviation is
calculated from the sample using the formula:

It can be shown
that, in most cases, measurements subject to random errors have a *normal
distribution.*

·
*What is
the normal distribution?*

The normal, or
Gaussian, distribution is represented by a bell-shaped curve as shown
below. The value of the quantity of
interest is shown on the x-axis, and the probability of observing that value in
a measurement situation is shown on the y-axis.
Note that the highest probability corresponds to the mean and that the
probabilities are symmetric about this mean value.

The normal
distribution occurs very often in nature – in fact, your grades in this class
will likely be normally distributed! It also has some interesting
characteristics. Regardless of the value of N, m, or s,
about 68% of your observations will fall within one standard deviation of the
mean, as shown on the graph below.
Ninety-five percent will fall within two standard deviations, and 99.7%
will fall within 3 standard deviations.

We only have time
to examine the normal distribution very briefly here. To learn more, check out
the following website, courtesy of

http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html