Error Representation and Curvefitting
(
http://www.owlnet.rice.edu/%7Elabgroup/pdf/errorsandfits.pdf
)
This article is a follow-up to the article titled "Error analysis and significant figures,"
which introduces important terms and concepts. The present article covers the rationale
behind the reporting of random (experimental) error, how to represent random error in
text, tables, and in figures, and considerations for fitting curves to experimental data.
WHEN TO REPORT RANDOM ERROR
Random error, known also as experimental error, contributes uncertainty to any
experiment or observation that involves measurements. One must take such error into
account when making critical decisions. When you present data that are based on
uncertain quantities, people who see your results should have the opportunity to take
random error into account when deciding whether or not to agree with your conclusions.
Without an estimate of error, the implication is that the data are perfect. Random error
plays such an important role in decision making, it is necessary to represent such error
appropriately in text, tables, and in figures.
When we study well defined relationships such as those of Newtonian mechanics, we
may not require replicate sampling. We simply select enough intervals at which to
collect data so that we are confident in the relationship. Connecting the data points is
then sufficient, although it may be desirable to use error bars to represent the accuracy
of the measurements. When random error is unpredictable enough and/or large enough
in magnitude to obscure the relationship, then it may be appropriate to carry out
replicate sampling and represent error in the figure.
REPRESENTING EXPERIMENTAL ERROR
The definitions of mean, standard deviation, and standard deviation of the mean were
made in the previous article. You may also encounter the terms standard error or
standard error of the mean, both of which usually denote the standard deviation of the
mean. The first set of terms are unequivocal, and their use is preferred. However, in
the biological sciences one most often encounters the term standard error of the mean
(SEM) rather than standard deviation of the mean.
Assumptions
The methods described here assume that you have an unbiased sample that is subject
to random deviations. Furthermore, it is assumed that the deviations yield a valid
sample mean with individual data points scattered above and below the mean in a
distribution that is symmetrical, at least theoretically. We call such a distribution the
normal distribution, but you may know it better as a "bell curve." Most of us assume that
we have a normal distribution, and sometimes that assumption is not correct.
Some data distributions are skewed (i.e., shifted to the right or left) or multi-modal