Chapter 11
Errors
Abstract All measurements, no matter how carefully undertaken, have errors. These
errors need to be quantified so that different experiments can be compared. Results
may have precision, self-consistency, and/or accuracy close to the real result. Results
may have random errors and/or systemic errors to be addressed. This chapter explains
the techniques for calculating and minimising the errors within an experiment as well
as methods to propagate the errors when performing mathematical operations on the
results.
11.1 Introduction
You will often see scientific results quoted as x ± y, where y indicates the range
of possible error. Errors, also called perhaps more accurately uncertainties, are an
expression of how much confidence can be put in a result. It does not mean that a
mistake has been made or that the scientists think that their result is wrong in some
way. In fact, all measurements are to some degree affected by uncertainness, because
a level of randomness is a fundamental part of the universe as expressed by both
quantum mechanics and chaos theory.
However, understanding how much of your result consists of random events is
an important skill, and you will be expected to express uncertainties for all your
experimental results. Not only do the uncertainties express a level of confidence,
they also allow comparison between different experiments. Say, for example, that
one person does an experiment that gives a result of 2.5 ± 0.1, while another person
employs a different method to make the same measurement and gets 2.6 ± 0.2.
Although the quoted results differ in value, they are experimentally consistent due
to their uncertainties.
The terms accuracy and precision are much more constrained in science than
in everyday life, where they are used interchangeably. The term accuracy reflects
how close a measurement is to its real value, while precision refers to how selfconsistent the results are. It is possible to be precise without being accurate and
accurate without being precise. For example, later on in this book, you will be asked
© Springer Nature Switzerland AG 2020
M. Gallaway, An Introduction to Observational Astrophysics,
Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-43551-6_11
161
Errors
Abstract All measurements, no matter how carefully undertaken, have errors. These
errors need to be quantified so that different experiments can be compared. Results
may have precision, self-consistency, and/or accuracy close to the real result. Results
may have random errors and/or systemic errors to be addressed. This chapter explains
the techniques for calculating and minimising the errors within an experiment as well
as methods to propagate the errors when performing mathematical operations on the
results.
11.1 Introduction
You will often see scientific results quoted as x ± y, where y indicates the range
of possible error. Errors, also called perhaps more accurately uncertainties, are an
expression of how much confidence can be put in a result. It does not mean that a
mistake has been made or that the scientists think that their result is wrong in some
way. In fact, all measurements are to some degree affected by uncertainness, because
a level of randomness is a fundamental part of the universe as expressed by both
quantum mechanics and chaos theory.
However, understanding how much of your result consists of random events is
an important skill, and you will be expected to express uncertainties for all your
experimental results. Not only do the uncertainties express a level of confidence,
they also allow comparison between different experiments. Say, for example, that
one person does an experiment that gives a result of 2.5 ± 0.1, while another person
employs a different method to make the same measurement and gets 2.6 ± 0.2.
Although the quoted results differ in value, they are experimentally consistent due
to their uncertainties.
The terms accuracy and precision are much more constrained in science than
in everyday life, where they are used interchangeably. The term accuracy reflects
how close a measurement is to its real value, while precision refers to how selfconsistent the results are. It is possible to be precise without being accurate and
accurate without being precise. For example, later on in this book, you will be asked
© Springer Nature Switzerland AG 2020
M. Gallaway, An Introduction to Observational Astrophysics,
Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-43551-6_11
161