ASTR 3130, Majewski [FALL 2012]. Lecture Notes
ASTR 3130 (Majewski) Lecture Notes
ERROR ANALYSIS, PART I: SCIENTIFIC NOTATION AND SIGNIFICANT DIGITS
 No measurement is infinitely precise.
 For a variety of reasons that we will explore a little bit later, each measurement will have
certain levels of associated "random and systematic errors".
 This means that when we cite a value for a measurement, it cannot be stated with exactness, but
is acknowledged to have an associated level of uncertainty.
 Science has developed certain conventions for conveying a rough sense of the
the level of precision in a cited numerical value via the number of cited digits in the number.
 One of the most common mistakes made by students of science is to violate these conventions and
thereby convey an improper level of implied precision in a cited number.
 As a common example, students will often use their calculator to compute a number, and
then record every digit put out by that calculator, even when some of those digits are spurious
and well beyond where they have actual meaning compared to the true level of precision in the
number.
 Mastering the conventions of stating measurements/values with the correct number of significant
digits is important, and the subject of this webpage.
WEB REFERENCES:
www.chem.sc.edu/faculty/morgan/resources/sigfigs/index.html.
http://hrsbstaff.ednet.ns.ca/benoitn/chem11/units/measurement/sig_fig.htm.
Scientific Notation
Before proceeding with the conventions of siginficant digits, it is important to review the conventions
used for writing very large or very small numbers using scientific notation. In scientific
notation we write all numbers in the form:
a x 10^{b}
where:
 the exponent, b, of the base (in this case 10) is an integer,
 the mantissa (also called the significand), a, is a real number between 1 and 10.
(Note that if b = 0, we do not usually write " x 10^{0} ", which is just unity anyway.)
So we have these examples (shamelessly lifted from Wikipedia):
Standard decimal notation 
Normalized scientific notation 
2 
2x10^{0}, or just 2 
300 
3x10^{2} 
4,321.768 
4.321768x10^{3} 
53,000 
5.3x10^{4} 
6,720,000,000 
6.72x10^{9} 
0.2 
2x10^{1} 
0.000 000 007 51 
7.51x10^{9} 
Note that the base and exponent are often expressed in other ways. So, for example, the number
1.23 x 10^{6} can be written in these alternative forms:
 1.23e+6. This is called "e notation", often used in, e.g., computer programs
because superscripts cannot always be displayed
conveniently. Here the "e" stands for exponent;
however, this format can be confusing with the "e" potentially confused with the
mathematical constant e. This potential confusion can be avoided by using...
 1.23E+6, the "E notation".
 1.23 x 10^6 .
Significant Digits (Also Called "Significant Figures")
 Significant Digits (Significant Figures) in a cited number are those that can be claimed to be
known with some degree of reliability.
 It is presumed that as you go from left to righthand digits in a cited number,
the degree of reliability in the digits decreases.
As a measurement to derive a number is made increasingly more reliably (e.g., with new experiments
or measurements of higher precision, or by performing more experiments
of the same precision and avaeraging the results), the number of digits/figures in the
number that can be stated with confidence generally increases.
 For example, a number like 123.4 is said to have 4 significant figures, while a number
like 123.45 is said to have 5 significant figures.
 An interesting historical example:
 If I say that the height of Mt. Everest is 29,029 feet, the implication is
that I know the mountain is precisely 29,029 feet to "about" 1 foot accuracy,
or at least better than 10 foot accuracy.
That is, I can assume with great confidence that the mountain is not
likely to be 29,039 feet or 29,019 feet, but very close to (within a
few feet of), if not exactly, 29,029 feet.
 But if I say that Mt. Everest is 29,000 feet high, the implication could be
that I only know the height of the mountain to about 1,000 foot accuracy, because
I am not giving anything but zeros for the last three digits.
 This particular example is interesting, because in 1852, an Indian mathematician
and surveyor, Radhanath Sikdar,
discovered that Mt. Everest was the highest peak in the world. He did this
using trigonometric surveying
measurements made with giant (precise) theodolites carried by teams of men led by
James Nicholson (who did the hard measuring work, but came down with malaria
and was forced to return home) as part of the "Great Trigonometric Survey of India" started
in 1802.
Later, in 1856, Andrew Waugh, the British Surveyor General of India, using Nicholson's data,
determined that the height of Mt. Everest (then called "Peak XV") was exactly 29,000 feet
high.
However, Waugh was fearful that if he reported this number it would give the impression that
it was nothing more than a rounded off estimate. So he arbitrarily added 2 feet to the height
of Peak XV and announced publicly that the mountain was 29,002 feet high.
I think most people can appreciate that "29,002 feet" sounds more exact than
"29,000 feet", but in this case, it most definitely wasn't!
 Waugh would have avoided this problem had he reported that the mountain was "29,000.0 feet
high", but he didn't actually want to claim that he knew things to the level of 0.1 foot
accuracy.
Following the current norms for citing significant figures (given below), if Waugh had stated that
Everest was 2.9000 x 10^{4} feet or
29,000 feet,
it would be clear that his measurement was intended
to be good to the foot.
As one can see, it is somewhat intuitive how people interpret numbers based on how they are reported.
The conventions now followed for stating numbers with significant digits relies on, and codifies,
this intuition via a set of rules.
Rules for the Number of Significant Figures to Quote in a Measured Quantity
These rules given here and in the next sections
are adapted from those listed in the tutorial by Stephen L. Morgan at
http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/sigfigs3.html and
at
http://hrsbstaff.ednet.ns.ca/benoitn/chem11/units/measurement/sig_fig.htm.
 All nonzero digits are significant:
 1.234 g has 4 significant figures,
 1.2 g has 2 significant figures.
 Zeroes between nonzero digits are significant:
 1002 kg has 4 significant figures,
 3.07 mL has 3 significant figures.
 Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point:
 0.001 C has only 1 significant figure,
 0.012 g has 2 significant figures.
 Trailing zeroes that are also to the right of a decimal point in a number are significant:
 0.0230 mL has 3 significant figures,
 0.20 g has 2 significant figures.
 When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant:
 190 miles may be 2 or 3 significant figures,
 50,600 calories may be 3, 4, or 5 significant figures.
 As in the Everest example above, 29,000 feet could be 2, 3, 4, or 5 significant figures.
The potential ambiguity in this rule can be avoided by the use of "scientific," notation, because
this format places the significant 0's to the right of a decimal point.
For example, depending on whether the number of significant figures is 2, 3, 4, or 5, we would write
29,000 feet as:
 2.9 x 10^{4} (2 significant figures),
 2.90 x 10^{4} (3 significant figures),
 2.900 x 10^{4} (4 significant figures), or
 2.9000 x 10^{4} (5 significant figures),
By writing a number in scientific notation, the number of significant figures is clearly
indicated by the number of digits in the decimal, as shown by these examples.
An alternative method sometimes employed is using a bar over the figures intended to be significant.
For example:
 29,000 (2 significant figures),
 29,000 (3 significant figures),
 29,000 (4 significant figures),
 29,000 (5 significant figures),
Exact Numbers
Some numbers are "exact" because they are known with complete certainty and are assumed
to have an infinite number of significant digits.
Two types of exact numbers are common:
 Numbers obtained by counting things that can be done with certainty and that
are not divisible. For example:
 For example, there are 13 people taking ASTR3130 (one would not say there are
13.000 people in the class, or 13.078!)
 Or, there are 6 types of quark (not 6.00).
 Numbers derived by definition. For example:
 1 m = 100 cm.
 12 objects are in a dozen.
 12 inches are in a foot.
Because these numbers have an infinite number of significant digits, we ignore these as a limiting
factor in determining the number of significant figures in the result of a calculation (see below).
Most (but not all) exact numbers are integers, and it is common in the scientific literature to
actually spell out such exact numbers when they are less than a certain value (commonly, 20 = "twenty"). For
example:
 "We undertook three separate tests...", not "We undertook 3 separate tests...".
 "The measurement was made by eleven students.", not "The measurement was made by 11 students."
Rules for Retaining Sigificant Figures in Mathematical Calculations
DON'T SIMPLY COPY YOUR CALCULATOR OUTPUT!
When carrying out calculations, the general rule is that the accuracy of a calculated result is
limited by the least accurate measurement involved in the calculation.
This then makes very clear rules for how many digits to report after the calculation:
 Addition and Subtraction: Add or subtract in the normal fashion.
Then round the answer to the LEAST number of decimal places as offered in any number in the problem.
For example:
 100 (assume 3 significant digits) + 23.643
(5 significant digits) = 123.643, but
this should be rounded to 124 (3 significant digits).
 20.63 + 6.6 + 3.786 =
31.016, but this should be stated as (rounded to) 31.0.
 145.67  98.432 = 47.238, but
this should be stated as (rounded to) 46.24.
 Multiplication and Division: The LEAST number of significant figures in any
number in the calculation determines the number of significant figures in the answer. For example:
 3.0 (2 significant figures) x 12.60 (4 significant figures) = 37.8000, which should
be rounded to 38 (2 significant figures).
 30.2 m x 2.3 m = 69.46 m^{2} = 69 m^{2}
 124 cm / 41 s = 3.0 cm/s
Rules for Rounding Off Numbers
 If the digit to be dropped is greater than 5, the last retained digit is increased by one.
For example,
 12.6 is rounded to 13.
 559 is rounded to 560.
 If the digit to be dropped is less than 5, the last remaining digit is left as it is.
For example,
 12.4 is rounded to 12.
 562 is rounded to 560.
 If the digit to be dropped is 5, and if any digit following it is not zero,
the last remaining digit is increased by one. For example,
 12.51 is rounded to 13.
 12.500001 is rounded to 13.
 If the digit to be dropped is 5 and is followed only by zeroes, the last remaining digit is
increased by one if it is odd, but left as it is if even. For example,
 11.5 is rounded to 12,
 but 12.5 is also rounded to 12.
Essentially this rule means that if the digit to be dropped is 5 followed only by zeroes,
the result is always rounded to the even digit.
The rationale for this is to avoid biases by how we round, because half of the time we will be rounding up,
and half the time we will be rounding down.
 If you adopted a rule of always "rounding up", you will be always biasing your results
to higher numbers.
Using Calculators and When to Round
Despite my warnings above not to simply report what appears in your calculator output, it is important
to be clear about when in the course of doing a calculation it is appropriate to apply the
rules given above for significant digits and rounding.
Some helpful tips:
 It is a good practice when using a calculator to also write down what is happening at each step
in the calculation. if you work the entirety of a long calculation without writing down
any intermediate results, you may not be able to tell if an error is made. On the other hand,
even if you realize that an error has occurred in your calculation, you may not be able to tell
at what step that error was made.
 In a long calculation involving a mix of mathematical operations, carry as many digits as
practical throughout the entire set of calculations and then only round the final result
according to the appropriate rules.
For example,:
(5.00 / 1.235) + 3.000 + (6.35 / 4.0) =
4.04858... + 3.000 + 1.5875 =
8.630829... =
8.6
The following rules apply to this calculation:
 The division in the first term
should result in 3 significant figures (shown in red).
 The second term is clearly declared to be 4 significant terms.
 The division in the last term should result in 2 significant figures.
 The three numbers added together should result in a number that is rounded off to the last
significant digit occurring furthest to the right common to all three terms, which is in the third term.
 Thus, the final result should be rounded with 1 digit after the decimal because the
accuracy in the result has been limited by the accuracy of the division in the third term.
 Fortunately, many calculators allow you to carry through the results of intermediate calculations
in the active display and memory when performing a complex series of calculations. This means that
you can automatically carry through the results of each individual step without rerentering it, and
this allows you to retain many digits without truncation/rounding errors introduced at intermediate
stages of the calculation.
 On the other hand, if you prematurely round off an intermediate result while carrying out a
long calculation, you
risk doing so at a digit that is too far to the left to retain the proper level of accuracy at the
digit you need to keep significant by the end of the calculation.
For this reason, it is good practice to carry all digits through to the final result before
rounding, which should happen at the end of the calculation.
 Note, however, if in the narrative of your lab report you are describing an intermediate result
of a calculation,
you should not in your text cite that number with more digits than is appropriate at that
point in the calculation (i.e., too many significant digits).
A rule of thumb is to always retain as many digits as possible at all stages of a calculation,
BUT if one of the intermediate results of the calculation is being discussed in a
narrative, apply the rounding to that number when you are citing it, even if more digits are
carried through to the final calculation.
Now Test Yourself
NOW THAT YOU KNOW THE RULES FOR SIGNIFICANT FIGURES, PLEASE TAKE THE SELFCORRECTING QUIZ
LOCATED AT THIS WEBSITE.
Also, try the quiz
here.
All material
copyright © 2012 Steven R.
Majewski. All rights reserved. These notes are intended for the private,
noncommercial use of students enrolled in Astronomy 3130 at the
University of Virginia.
