Download Significant Figures, Errors and Error Propagation

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Topic 8
Significant Figures, Errors and
Error Propagation
Contents
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
8.2 Significant Figures and Rounding . . . . . . . . . . .
8.2.1 Significant Figure Examples . . . . . . . . . .
8.2.2 Significant Figure Rules . . . . . . . . . . . .
8.2.3 Significant Figure Summary . . . . . . . . . .
8.2.4 Rounding . . . . . . . . . . . . . . . . . . . .
8.3 Significant Arithmetic . . . . . . . . . . . . . . . . . .
8.3.1 Multiplication . . . . . . . . . . . . . . . . . .
8.3.2 Division . . . . . . . . . . . . . . . . . . . . .
8.3.3 Addition . . . . . . . . . . . . . . . . . . . . .
8.3.4 Subtraction . . . . . . . . . . . . . . . . . . .
8.4 Experimental Errors . . . . . . . . . . . . . . . . . .
8.5 Propagation of Experimental Error . . . . . . . . . .
8.5.1 Addition and Subtraction of Measurements .
8.5.2 Multiplication and Division of Measurements
8.6 Tutorial Topic 8 . . . . . . . . . . . . . . . . . . . . .
8.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . .
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Prerequisite knowledge
• Familiarity with scientific notation.
• Understanding of how to round numbers.
• Ability to manipulate algebraic expressions.
• Some background in experimental error assessment.
Learning Objectives
By the end of this topic, you should be able to:
• Identify significant figures.
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TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
• Distinguish between precision and accuracy.
• Apply rounding and work in scientific notation.
• Apply significant figures to algebraic operations - addition and subtraction
• Apply significant figures to algebraic operations - multiplication and division.
• Assess errors in experimental results.
• Identify and quantify how errors propagate.
©H ERIOT-WATT U NIVERSITY
TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
8.1
Introduction
The first requirement is to distinguish between two terms that are used interchangeably
in common language - however, for scientists and engineers these terms have very
different meanings:
• Accuracy
• Precision
Say the density of a liquid is repeatedly measured using a density bottle, together
with the suppliers Standard Operating Procedure (SOP) - assume that experimental
conditions are kept the same.
Given a sufficient number of measurements the results may be plotted as a frequency
distribution - such as the one shown below:
Mean
value
True
value
Number of
density
measurements
falling into a
specific band
- The true value is known
because a standard is
available.
Accuracy
- There will be a spread of results
depending on the equipment,
the operator and the procedure.
Precision
Density
bands
The precision is the spread of results obtained when all efforts are made to keep the
experimental conditions the same. The precision can also be called the repeatability or
the reproducibility - it is basically the consistency of the measurement.
The accuracy is how close the mean value of the measured results is to the true value,
as can be seen accuracy and precision are different.
Standard chemical engineering textbooks usually offer little support in this area,
however, [Felder and Rousseau, 2008] have a useful section on significant figures.
©H ERIOT-WATT U NIVERSITY
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TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
8.2
Significant Figures and Rounding
The more significant figures there are associated with a measurement then the more
precision there is in that measurement. However, it is important not to give the
impression of a false precision.
For instance when making up a reagent the weight may have been measured by a
technician as 23.5 g - now, consider that this is written down as follows:
• Case (1), Weight = 23.5 g; this has three significant figures and informs anyone
who might pick up the result, that its value could actually lie anywhere in the range
23.45g - 23.55 g - the precision is 23.5 ± 0.05 g.
• Case (2), Weight = 23.500 g; five significant figures are now being claimed. This
means that the result could now lie anywhere in the range 23.4995 - 23.5005,
notice that a false precision of 23.5 ± 0.0005 g is now being claimed.
Therefore, writing down the number in the wrong way, in this case with too many
decimal points (although same applies to numbers without decimal points), creates a
false precision.
At this point do not confuse precision with accuracy. A laboratory balance may be very
precise, that is give highly reproducible answers, but at the same time, it may be out of
calibration and therefore quite inaccurate.
8.2.1
Significant Figure Examples
To establish the correct precision within a data set, and then maintain a consistent
precision throughout a calculation using this data, requires a clear understanding of
the rules governing significant figures.
First consider a sequence of examples and then it should be possible to summarise a
set of clear rules:
1. All non-zero numbers are significant - for instance the average molar mass of a
mixture is 28.6 kg kmol-1 . The numbers "2, 8 and 6" are non-zero and "significant",
therefore, the molar mass is known to three significant figures.
2. Zeros within a string of non-zero numbers also count as significant figures, for
instance standard atmospheric pressure is taken as 101.325 kN m-2 . The numbers
"1, 0, 1, 3, 2, 5" are all "significant" - there are six significant figures.
3. Leading zeros do not count as significant figures - for instance the triple point
pressure of water is 0.006112 bar. The three leading zeros are insignificant, only
"6, 1, 1, 2" are "significant", so that there are four significant figures.
4. If a number has a decimal point then trailing zeros are significant - for instance
atmospheric pressure could be written as 0.611200 kN m-2 thus "6, 1, 1, 2, 0, 0"
are all "significant" - there are now six significant figures.
a) To maintain the same precision, but now working in N m-2 instead of kN m-2 ,
atmospheric pressure would have to be written as 611.200 N m-2 thus, there
are still six significant figures.
©H ERIOT-WATT U NIVERSITY
TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
b) The weight of a sample is 120.00 g - there are five significant figures, because
whenever there is a decimal point trailing zeros count.
5. If a number has no decimal point then trailing zeros are ambiguous. The weight of
a large van may be quoted as 3,500 kg - this is ambiguous:
a) The weight may be exactly 3,500 kg.
b) The weight may have been rounded to the nearest hundred kg.
c) If it is exactly 3,500 kg, then the ambiguity may be resolved in two ways; first,
write 3,500. kg (note the decimal point); second, write 3.500 tonnes; in both
cases there are four significant figures.
8.2.2 Significant Figure Rules
Having considered a number of examples, the rules about significant figures may be
listed as follows:
1. All non-zero digits in a number are significant.
2. Zero digits within a string of non-zero digits count as significant.
3. Leading zero digits are not significant.
4. When a number has a decimal point trailing zero digits are significant.
5. When a number has no decimal point trailing zero digits are ambiguous, either
insert a decimal point, or change the unit prefix or use scientific notation (there are
other conventions but they are rarely used).
8.2.3 Significant Figure Summary
All of the above rules can be summarised in a very condensed way as follows:
"For any measured quantity the number of significant figures is the number of digits
starting from the first non-zero digit on the left and ending either, if there is a decimal
point, with last digit (whether zero or non-zero) or, if there is no decimal point, the
last non-zero digit".
Working in scientific notation can simplify the question of significance:
Measurement
Scientific Notation
9600
9600.0
96050
0.0096
0.00605
9.6 x 103
9.6000 x 103
9.605 x 104
9.6 x 10-3
6.05 x 10-3
©H ERIOT-WATT U NIVERSITY
Number of Significant
Figures
2
5
4
2
3
5
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TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
8.2.4
Rounding
When using a calculator numbers must be rounded - there are two ways of doing this:
• Rounding to some arbitrary number of decimal places. In financial work rounding
to two decimal places is usual.
• Rounding to significant figures. When the number of significant figures is known
the result is rounded to the known number of significant figures - this is the
recommended approach.
The first approach of rounding to a certain number specified decimal places is not
recommended: too many decimal places lead to false precision; too few decimal places
lead to loss of information.
When rounding a calculated result the approach should be as follows:
• The number of significant figures will be known from the values entered into the
calculator and the operation (add, subtract, multiply, divide) - see next section on
how to do this.
• Take the calculated result and apply the number of significant figures to the value
displayed on the calculator.
• Retain all digits that are "significant" - the most significant to the left the least
significant to the right and then inspect the digits immediately to the right of the
least significant digit.
• These digits are not significant and will be discarded. If they are over "5" then
round the least significant figure up, if less than "5" then round it down. Now apply
these rules to the number 16.72532:
Significant Figures
5
4
3
2
1
Final Result
16.725
16.73
16.7
17
2 x 101
• Now take the same number 16.72532 and round it using same number of decimal
places:
Decimal Places
5
4
3
2
1
Final Result
16.72532
16.7253
16.725
16.73
16.7
©H ERIOT-WATT U NIVERSITY
TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
Take the case where two significant figures apply - the correct answer should be written
as 17, meaning that the result could actually lie anywhere in the range 16.5 - 17.5,
hence, the precision is ±0.5.
However, using two decimal places provides a result 16.73 - clearly the result, using this
incorrect approach, could be anywhere in the range 16.725 - 16.735 which implies a
false precision of ±0.005.
A problem arises when rounding 16.725 to four significant figures - applying the rules as
before. Only four digits are significant and these are "1, 6, 7, 2" the rest are insignificant
and the least significant digit is "2".
In this case there is only one digit to the right of the least significant digit and this is "5",
clearly it is difficult to know whether to round the least significant digit up or down. There
are two different ways of doing this:
1. Expand the previous rule from "greater than five - move up" to "five or more - move
up", the 16.725 would then appear as 16.73.
2. Apply the rule "round to nearest even number" - that is if the least significant figure
is an even number then leave it, if the least significant number is odd then round
to the nearest even; using this rule 16.725 becomes 16.72.
It is a matter of choice which rule to apply - rule 1 has the effect of always rounding up,
which can shift the result of many calculations in this direction. Rule 2 has the advantage
of randomising the rounding process - it is a superior approach.
©H ERIOT-WATT U NIVERSITY
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TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
8.3
Significant Arithmetic
In the previous section the number of significant figures was given; in fact the number of
significant figures depends on the precision of the starting data and the operation that
is being carried out (multiplication, division, addition or subtraction). This is known as
significant arithmetic.
8.3.1
Multiplication
When multiplying or dividing two values together the number of significant figures that
should be applied to the result can be simply stated as follows: "The starting value
with the least number of significant figures by itself determines the number of significant
figure of the final result".
In the case of significant multiplication, the precision of the final result cannot be more
precise than the least precise starting value - now apply this rule given to the following
examples (involving multiplication only):
Starting Data Multiplication
Number of Significant
Figures
Final Rounded Result
9 x 9 = 81
9 x 9.0 = 81
9.0 x 9.0 = 81
9.2 x 9.54 = 87.758
1501 x 271 = 406,771
3.146 x 16.0 = 50.3360
3.146 x 16 = 50.336
1
1
2
2
3
3
2
80
80
81
88
407,000
50.3
50
At this point some confusion may arise in the mind of a student; it is well-known that
9x9 =81, but the result is rounded to 80 which seems wrong - however, the confusion is
easily clarified:
• It is important to realise that the "9" in question is not an integer it is a measured
value, say 9 cm. Thus the student could be working out the area of a square 9
cm×9 cm, which might appear to be 81 cm2 .
• However, there is only 1 significant figure, which means that the "9 cm" can actually
lie anywhere in the range 8.5 - 9.5 cm. With such a low measurement precision
the calculated area must be stated as 80 cm2 .
The above approach is strictly correct when a single multiplication is being carried
out. Sometimes a sequence of calculations is needed and then it is wise to hold all
intermediate answers to an extra significant figure and round at the end.
This avoids "rounding errors" that gradually enter calculations if each intermediate result
is rounded, as a general loss of information builds up over the calculation as a whole.
Consider 1200x1.610 = 1932 - the factor "1.610" has four significant figures but there is
ambiguity about how many significant figures apply to "1200":
©H ERIOT-WATT U NIVERSITY
TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
• If there is no decimal place "1200" has only two significant figures and the answer
is must be written as 1900.
• If "1200" just happens to be exactly "1200" then there are four significant figures
and the answer is 1932 - there are three ways getting round this ambiguity:
1. place a decimal point after the final "0"
2. modify the unit prefix - if it is 1200 g write this as 1.200 kg
3. use scientific notation, i.e. 1.200×103 g
One final point, if there are 1200 samples each with average mass 1.61327 g calculate
the mass of the entire batch = 1200×1.61327 g = 1935.924 g. In this case there are 6
significant figures and the answer must be rounded to 1935.92 g
When a value has been arrived at by counting (not by measurement) there are an
infinite number of significant figures - basically this number can in no way limit the overall
number of significant figures of the final result.
By taking an infinite number of significant figures what is being inferred is that the
number is known precisely.
8.3.2 Division
The same significant figure rule that applies to significant multiplication also applies to
significant division. That is the final answer can have no more precision than the factor
with the least precision - consider the following examples:
Starting Data Division
Number of Significant
Figures
Final Rounded Result
9 / 4 = 2.25
9.0 / 4 = 2.25
9.0 / 4.0 = 2.25
9.00 / 4.00 = 2.25
9.2 / 9.54 = 0.964361
1501 / 271 = 5.5387454
3.146 / 16.0 = 0.196625
3.1461 / 16.11 = 0.195289
1
1
2
3
2
3
3
4
2
2
2.3
2.25
0.96
5.54
0.197
0.1953
The above approach is strictly correct when a single division is being carried out.
Sometimes a sequence of calculations is needed and again it is wise to hold all
intermediate answers to an extra significant figure and then round at the end.
Consider the following division 1200/1.610 = 745.34161 - the factor "1.610" has four
significant figures but there is ambiguity about how many significant figures apply to
"1200". The rules are the same as before, but just to recap:
• If there is no decimal place, "1200" has only two significant figures and the answer
is must be written as "750".
• If "1200" just happens to be exactly "1200" then there are four significant figures
©H ERIOT-WATT U NIVERSITY
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TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
and the answer is "745.3" - once again there are three ways getting round this:
1. place a decimal point after the final "0"
2. modify the unit prefix - if it is 1200 g write this as 1.200 kg
3. use scientific notation, i.e. 1.200×103 g
If there are 1200 samples with a total mass 1935.924 g the average mass of each
sample = 1935.924 g/1200 = 1.613270 g. There are seven significant figures.
8.3.3
Addition
In the case of significant addition, the number of significant figures that should be
applied to the result can be stated as follows:
• When two or more values are added the position of the least significant digit in the
starting values should be noted.
• The position of the least significant figure in the final result should correspond to
the furthest left of the least significant digit in the starting values.
• Notice that it not the number of significant figures that matter, it is the position of
the least significant figure that is important.
Apply this rule given to the following addition examples:
Starting Data Addition
Position of Least Significant
Final Rounded Result
Figure From Left
4 + 5.1 = 9.1
One’s place
9
4.0 + 5.1 = 9.1
Tenth’s place
9.1
22.1 + 130 = 352.1
Tens place
350
22.1 + 130. = 352.1
One’s place
352
22.1 + 130.0 = 352.1
Tenth’s place
352.1
8.3.4
Subtraction
For significant subtraction, the result is once again rounded to the position of the least
significant figure in the more imprecise of the two starting values - for example:
Starting Data Subtraction
Position of Least Significant
Final Rounded Result
Figure From Left
15 - 7.9 = 7.1
One’s place
7
15.0 - 7.9 = 7.1
Tenth’s place
130 - 22.1 = 107.9
Tens place
7.1
110
130. - 22.1 = 107.9
One’s place
108
130.0 - 22.1 = 107.9
Tenth’s place
107.9
©H ERIOT-WATT U NIVERSITY
TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
8.4
Experimental Errors
There are two general categories of error in any measurement:
• Random error is an unpredictable variation in, either the measured reading of an
instrument from its true value, or a random variation of an observer’s interpretation
of the measurement from its true value.
• Systematic error is a predictable offset of the measured value from its true value,
caused either by the instrument or the observer’s technique or environmental
interference.
Random errors are evident when the measured values vary randomly around the true
value - all measurements contain random errors.
Systematic errors are evident when the measured value varies randomly but is always
offset, either on one side or the other, from the true value. Systematic errors are
generally caused by the following - either singly or in combination:
• Instrument calibration problems or zero errors - these may be corrected.
• Problems with experimental technique - these may be eliminated.
• Interference in the measurement caused by outside factors - these may be
eliminated.
Systematic errors should be eliminated but there will always be a random error
associated with any measurement.
As far as random errors are concerned: "The experimental error of a measurement is
likely to be about half of the smallest division on that instrument".
If x is the measured value and ∆x the estimated error then the result should be quoted
as follows:
x ± ∆x .................(8.1)
For instance if an electronic timer is accurate to 0.5 s and the measured time is 35 s
then the result should be quoted as follows
35 ± 0.5 s
©H ERIOT-WATT U NIVERSITY
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TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
Sometimes two readings are required - for instance, when measuring length using a
ruler a measurement is made first at one end and then at the other.
• If the smallest division is 1 mm then the estimated error is 0.5 mm.
• But, there are two measurements one at either end thus, the estimated error will
be 1 mm. If the measurement is 46 mm it should be quoted as follows:
46 ± 1 mm
It is important to distinguish between the absolute error and the relative error as follows:
ε = |v − ve | .................(8.2)
η=
|v − ve |
.................(8.3)
v
Where,
v = True value of the quantity being measured.
ve = Measured or estimate value of the quantity being measured.
ε = Absolute error between the true and measured values.
η = The relative error is absolute error divided by the true value.
The relative error may also be expressed as a percentage - simply multiply the relative
error, or fractional error, by one hundred; relative error is good where there are big
differences in size between different experimental values.
©H ERIOT-WATT U NIVERSITY
TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
8.5
Propagation of Experimental Error
The usual textbook used in this course [Cutnell and Johnson, 2012] has little or no
information on any of the issues raised in this topic. However, [Preston and Dietz, 1999]
covers this material well.
8.5.1 Addition and Subtraction of Measurements
Suppose that it is necessary to calculate some value using the formula shown below, in
this formula individual measurements are added or subtracted:
W = Ax + By + Cz .................(8.4)
Where,
A = is a constant, likewise B and C.
x = is a measured quantity, likewise y and z.
W = is the calculated quantity.
The key factor is how the errors in the individual measurements will propagate through
to the final calculated value - a common way of finding this propagation error is given
below:
∆W =
q
(A ∆x)2 + (B ∆y)2 + (C ∆z)2 .................(8.5)
Where,
∆x = is the known error in quantity x, likewise ∆y and ∆z.
∆W = is estimated error in the calculated quantity.
Equation (8.4) may be used to find the calculated value of W from measured values of
x, y and z. Equation (8.5) is then used to estimate the likely error ∆W associated with
the calculated value.
©H ERIOT-WATT U NIVERSITY
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TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
Example : 8.5.1
Problem:
Take the three measured lengths shown below L1 , L2 , L3 and L4 together with the
formula shown below and then determine the following:
a) Calculate the length L4 according to the formula - see below.
b) Use equation (8.5) to calculate the estimated error ∆L4 associated with L4 .
Solution:
The individual measured lengths and the formula for finding the calculated length L4 are
supplied below:
L1 = 5.0 ± 0.1 cm
L2 = 3.4 ± 0.2 cm
L3 = 9.5 ± 0.1 cm
L4 = L1 + 3L2 − 2L3
a) Students are asked to calculate L4 = _________________________
b) Students are asked to calculate ∆L4 = ________________________
c) Students are asked to quote the value of L4 = __________________
..........................................
©H ERIOT-WATT U NIVERSITY
TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
8.5.2 Multiplication and Division of Measurements
Suppose the starting point is now a function similar to the one shown below:
W = k xa y b z c .................(8.6)
Where,
k = is a constant, likewise a, b and c.
x = is a measured quantity, likewise y and z.
W = is the calculated quantity.
A different formula will now have to be used to estimate the likely propagation error in
the calculated value as follows:
∆W
=
W
s
a∆x
x
2
+
b∆y
y
2
+
c∆z
z
2
.................(8.7)
Where,
∆x = is the known error in the measured quantity x, likewise ∆y and ∆z.
∆W = is the estimated error in the calculated quantity.
Equation (8.6) may be used to find the calculated value of W from measured values of
x, y and z. Equation (8.7) may then be used to estimate the likely error ∆W associated
with the calculated value.
The above formula is derived using the sum of partial derivatives found from
differentiating equation (8.6), partially, with respect to each measured quantity.
©H ERIOT-WATT U NIVERSITY
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TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
Example : 8.5.2
Problem:
Take the four measured quantities shown below m1 , m2 , r and G, together with the
formula shown (gravitational attraction formula) and then determine the following:
a) Calculate the gravitational attraction F according to the formula supplied - see
below.
b) Use equation (8.7) to calculate the estimated error ∆F associated with F .
The individual measured quantities and the formula for finding the calculated quantity F
are supplied below:
m1 = 19.7 ± 0.2 kg
m2 = 9.4 ± 0.2 kg
r = 0.641 ± 0.009 m
G = 6.67 × 10−11 N m2 kg−2
F =
G m1 m2
r2
Solution:
a) The gravitational attraction between these two masses can be found from the
formula supplied in the question
F =
G m1 m2
6.67 × 10−11 × 19.7 × 9.4
=
r2
0.6412
∴ F = 3 × 10−8 N
b) To find the error in the calculated value use equation (8.7) - modified below to for
the gravitational attraction expression
∆W
=
W
s
a∆x
x
2
+
b∆y
y
2
+
c∆z
z
2
©H ERIOT-WATT U NIVERSITY
TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
∆F
=
F
s
∆m1
m1
∆F
=
F
s
0.2
19.7
∴
2
2
+
+
∆m2
m2
2
+
2
+
0.2
9.4
−2∆r
r
2
−2 × 0.009
0.641
2
∆F
= 0.0367
F
The above is the relative error, now convert this to an absolute error
∆F = 0.0367 × F = 0.0367 × 3 × 10−8 N
∆F = 1.10 × 10−9 N
The calculated force of attraction with its estimated error is given by
F = (3 ± 0.1) × 10−8 N
..........................................
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TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
8.6
Tutorial Topic 8
1. Determine the number of significant figures assuming that the following quantities
have been measured in the laboratory:
a) The mass of a sample is 128.63 g.
b) The absolute temperature of a gas is 307 K.
c) The absolute pressure of a gas is 0.00712 bar.
d) The absolute temperature of a gas is 307.00 K.
e) The mass of a sample is 100 g.
2. A sample weight is stated as 1,200 g, however, if it weighs exactly 1,200 g resolve
the ambiguity as to the number of significant figures as follows:
a) Use the decimal point method and state number of significant figures.
b) Change the unit prefix and state number of significant figures.
c) Use scientific notation and state number of significant figures.
3. A sample weight is stated as 1,200 g, however, if it weighs 1,200 g to one
hundredth of a gram then resolve the ambiguity as to the number of significant
figures as follows:
1. Use the decimal point method and state number of significant figures.
2. Change the unit prefix and state number of significant figures.
3. Use scientific notation and state number of significant figures.
4. Consider the following measurements 1300 (precise to hundred units), 0.0013,
0.00105, 13040 (precise to ten units),1300.0 and determine the following:
a) The number of significant figures in each case.
b) Write the numbers in scientific notation.
5. Consider the following measured values and round them to four significant figures:
a) 18.843 g.
b) 18.845 g apply the rule "five or more round up".
c) 1.845 g apply the rule "round to even".
d) 1.855 g apply the rule "five or more round up".
e) 1.855 g apply the rule "round to even".
6. The following measured values have to be multiplied and divided, see below, apply
the rules of significant arithmetic to round the final result (the result that would
appear on a calculator is also indicated below):
a) 7×7 = 49
b) 7.0×7 = 49
c) 7.0×7.0 = 49.
d) 1605×152 = 243,960
e) 10.0/4 = 3.25
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TOPIC 8. SIGNIFICANT FIGURES, ERRORS AND ERROR PROPAGATION
f) 10.0/4.0 = 3.25
g) 10.0/4.00 = 3.25
h) 5.782/3.200 = 1.806875
7. The following measured values have to be added and subtracted, see below, apply
the rules of significant arithmetic to round the final result (the result that would
appear on a calculator is also indicated below):
a) 4 + 3.1 = 7.1
b) 4.0 + 3.1 = 7.1
c) 15.2 + 160 = 175.2
d) 15.2 + 160. = 175.2
e) 15.2 + 160.0 = 175.2
f) 17 - 5.1 = 11.9
g) 17.0 - 5.1 = 11.9
8. An A4 sheet of paper has a measured length (L = 29.7 ±0.1) cm and measured
width (W = 21 ±0.1) cm, use the formula below to calculate the perimeter "P" (cm)
of the A4 sheet. Calculate how the uncertainty of the individual measurements
propagate through an uncertainty in the final calculated result and quote the final
result to an appropriate number of significant figures.
P = 2L + 2W
9. An A4 sheet of paper has a measured length (L = 29.7 ±0.1) cm and measured
width (W = 21 ±0.1) cm, use the formula below to calculate the area "A" (cm2 )
of the A4 sheet. Calculate how the uncertainty of the individual measurements
propagate through an uncertainty in the final calculated result and quote the final
result to an appropriate number of significant figures.
A = LW
8.7
Bibliography
1. Felder, Richard M. and Rousseau, Ronald W. 2008. Elementary Principles of
Chemical Processes. 3rd ed. New Jersey: Wiley
2. Preston, Daryl W. and Dietz, Eric R. 1999. The Art of Experimental Physics. New
Jersey: Wiley
©H ERIOT-WATT U NIVERSITY
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GLOSSARY
Glossary
Accuracy
When an experiment is repeated a sufficient number of times under identical
conditions the results may be plotted as a distribution. The difference between
the average of this distribution and the true value is the accuracy. Thus accuracy
is different from precision; see "Precision" in this glossary.
Precision
When an experiment is repeated a sufficient number of times under identical
conditions the results may be plotted as a distribution. The spread or scatter
of the experimental values is the precision. It is also called the reproducibility
or repeatability. Thus precision is different from accuracy; see "Accuracy" in this
glossary.
Propagation Error
Propagation error arises because experimental measured values are entered into
a function to find some result and, as a result of the nature of the function, errors
propagate through to the final result. However, knowing the functional relationship
between the starting values mathematics may be used to predict the propagation
error in the final result.
Random Error
Random error is an unpredictable variation in either the measured instrument
reading from its true value, or a random variation of an observer’s interpretation
of the measurement from its true value. Random errors are evident when the
measured values vary randomly around the true value; all measurements contain
random errors.
Rounded
A calculator often produces an answer to many decimal places. Such a result
should be rounded to retain only the number of significant digits in accordance
with the rules of significant figures and the rules of significant arithmetic, see both
"Significant Figures" and "Significant Arithmetic" in this glossary. It is good practice
in a chain of calculations to hold intermediate answers to a higher precision and
then round once the final result has been found.
Significant Addition
The rule of significant addition is that if two experimental quantities are added
together, the result of the calculation should be rounded to the position of the least
significant digit of the more imprecisely known of the two starting values. This is
the same rule that applies to "Significant Subtraction", see this glossary. What is
important is the position of the least significant figure not the number.
Significant Arithmetic
Significant arithmetic refers to the multiplication, division, addition and subtraction
of experimentally determined values. There are rules to determine the number
of significant figures that should be applied to the result. However, the basic
requirement is that the result can be no more precise than the least precisely
known of the starting value.
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GLOSSARY
Significant Division
The rule of significant division is that if two experimental quantities are divided
one into the other, the result of the calculation should be rounded to the number
of significant figures that applies to the least precisely known of the two starting
values. This is the same rule that applies to "Significant Multiplication", see this
glossary. What is important is the smaller number of significant figures of the two
starting values.
Significant Figures
Significant figures and significant digits are used interchangeably. The number
of significant figures is the number if digits in an experimental value that are
significant in terms of the precision of the measurement.
Significant Multiplication
The rule of significant multiplication is that if two experimental quantities are
multiplied together, the result of the calculation should be rounded to the number
of significant figures that applies to the least precisely known of the two starting
values. This is the same rule that applies to "Significant Division", see this
glossary. What is important is the smaller number of significant figures of the
two starting values.
Significant Subtraction
The rule of significant subtraction is that if two experimental quantities are
subtracted one from the other, the result of the calculation should be rounded
to the position of the least significant digit of the more imprecisely known of the
two starting values. This is the same rule that applies to "Significant Addition", see
this glossary. What is important is the position of the least significant figure not the
number.
Systematic Error
A systematic error is identified as an offset of the measured value from its true
value and is caused either by the instrument or the observer’s technique or
environmental interference. Systematic errors are evident when the measured
value varies randomly, but is always offset either on one side or the other from the
true value.
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