Introduction to Engineering
Camera Lab #2
An Introduction to Experimental Error
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 large the error
in our measurement is so that we can use the measurement with some
degree of confidence. Another reason for estimating our “normal error” is that
when a mistake does occur, we can spot it much easier, because it will lie
outside the range of errors we expect to see.

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 mean and a standard
deviation based on that sample.
For the population, for the mean, and  for the standard deviation.
The mean, represented by the Greek letter , or mu, is our “best guess” of
the true value of the quantity we’re measuring. It’s just the arithmetic
average of all the measurements:

WHHS Intro to Exp Error
08/06/02 mjh
1
N
N
x
i 1
i
where N is the total number of measurements.
The sample standard deviation, represented by the Greek letter , 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:
x 
1
 x i  x 2

N 1
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, , or , 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 Stanford University:
http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html
WHHS Intro to Exp Error
08/06/02 mjh