Untangling Complex Systems, page 100
lyzed inside a laboratory where it is easier to control variables and parameters. When we face Complex
Systems, this is not always feasible. Whether the environment is the laboratory or the real world, it
often plays a relevant role. In fact, the environment can influence the phenomenon under study (espe-
cially if it obeys nonlinear laws), and it can also affect the performances of the observer, and the instru-
ment. The use of an instrument is mandatory for the observer; the device allows making quantitative,
objective, and reproducible determinations. Any measurement, however accurate and performed care-
fully, is tainted by errors, and the results have inevitable uncertainties. There are two principal sources
of errors and uncertainties: one is the instrument that we use, and the other is the operator.1 There are two types of sources of uncertainty: they are the systematic and the random errors.
D.2 SYSTEMATIC ERRORS
Any instrument determines the value of a variable with a definite degree of accuracy. When we
choose an instrument to collect data, we must be aware of the error we make by using it. Almost
all direct measurements require reading either a marked scale (e.g., the marked scale of a ruler, a
chronometer, or a pipette) or a digital display (e.g., the digital display of a spectrophotometer).
When we read a marked scale, the main source of error is the interpolation between the scale
markings. For instance, let us measure the length of the clip shown in Figure D.2 with a ruler
graduated in millimeters. We might reasonably decide that the clip’s length is undoubtedly closer
to 3.2 cm than it is to 3.1 or 3.3 cm, but no more precise reading is possible. We conclude that our
best estimate is 3.20 cm, and the uncertainty of our determination is in the range 3.15 and 3.25 cm.
1 The contribution of the environment is included in those of the observer and instrument.
517
518
Appendix D
Environment
Observer
Instrument
Object or
phenomenon
FIGURE D.1 Protagonists of a measurement.
FIGURE D.2 Determination of the length of a clip by a marked ruler.
The correct way to present the result of the measurement will be lbest = ( .
3 20 ± 0. )
05 cm. More in
general, when we measure a quantity x , we collect our best estimate x and we specify the range
true
best
(∆ x) within which we are confident x lies:
true
xbest ± x
∆ [D.1]
The term ∆ x is the absolute error of our measurement, and it is usually stated with one significant
figure. At most, with two significant figures, but no more. Moreover, the last significant figure in
x must be of the same order of magnitude of the uncertainty, that is in the same decimal position.
best
If we want to increase the accuracy of our determination and reduce ∆ x , we need a finer instrument.
For instance, in the case of the length of the clip, we may use a Vernier caliper, like that shown in
Figure D.3. First, we close the jaws lightly on the clip. Then, we measure its length. In the Vernier caliper, there are two marked scales: a fixed one and a sliding one. The fixed scale is identical to the scale
of the ruler of Figure D.2. The separation between two consecutive marks is equal to 1 mm. The sliding scale has 20 marks. The left-most tick mark on the sliding scale will let us read from the fixed scale
the number of whole millimeters that the jaws are opened. In Figure D.3, it is between 3.2 and 3.3 cm.
For the estimation of the clip’s length at the level of one-tenths of mm, we must determine the number
of the first mark of the sliding scale that is aligned with a mark of the fixed scale. In Figure D.3, it is the seventh mark that corresponds to 0.30 mm. The uncertainty, as indicated onto the caliper, amounts
to 0.05 mm. Therefore, the final best estimate of the clip’s length is lbest = ( .
3 230 ± .
0
)
005 cm. Note
that the use of the caliper has increased the accuracy of our determination of one order of magnitude.
The quality of a collected datum is expressed by the ratio of ∆ x over x , which is called the
best
relative uncertainty (ε rel):
x
ε rel = ∆ [D.2]
xbest
Appendix D
519
FIGURE D.3 Determination of the length of a clip by a Vernier caliper.
FIGURE D.4 Reading of a scale whose marks are rather far apart. The distance between two consecutive
marks is (1.85 ± 0.05) cm.
If we multiply ε rel by 100, we obtain the percentage relative error. The percentage relative error
is ε rel = 1 5
. 6% with the ruler, whereas it is ε rel = 0 155
.
% with the caliper. We confirm that the use of
the caliper reduces the uncertainty of one order of magnitude.
Usually, the uncertainty in reading a scale when the marks are fairly close together is half the
spacing. If the scale markings are rather far apart, like those of the pipette shown in Figure D.4, we might reasonably assume that we can read to one-fifth of a division. The volume of the liquid taken
with the pipette of Figure D.4 is reasonably estimated to be (0.24 ± 0.02) mL.
The error we make when we read a value on a digital display is usually of one unit on the last
significant figure, unless indicated differently by the specifications of the instrument we are using.
The errors described so far generated by the inherent inaccuracy of the instrument we use, are
said systematic because they are always present, and their absolute extent (∆ )
x is constant and
known, even before making the measurement.
An instrument may be a source of another type of systematic error that cannot be known unless
we measure standard quantities, or we repeat our measurement by using different devices. Possible
miscalibration of our equipment generates such a second type of systematic instrumental error.
520
Appendix D
A
B
FIGURE D.5 Parallax error in reading a pipette’s scale. Case A represents the correct viewing angle,
whereas case B represents the incorrect one.
Moreover, the operator may introduce systematic errors in the measurements, as well. For
instance, the right way of reading a volume on a scale of a pipette is to put our eyes at the height of
the liquid’s meniscus, like in situation A of Figure D.5. If we read the volume systematically by viewing the meniscus from above (see situation B of Figure D.5), we determine the volume constantly
by defect. In situation B of Figure D.5, we will read (1.40 ± 0.05) mL rather than (1.50 ± 0.05) mL.
The introduction of such uncertainty, called parallax error, might be even worse depending on the
angle of viewing and the type of scale.
D.3 RANDOM UNCERTAINTIES
It may also occur that sometimes we read a scale assuming the right position, but in some others
not. In this case, not all the measurements are affected by the same parallax error, but only some of
them. The uncertainty introduced in such a way is not systematic but random.
The instrument itself may become a source of random errors. For instance, a chronometer may
run faster or slower accidentally; a photomultiplier may change abruptly and temporarily its sensi-
tivity to photons. Often, it is the environment that can contribute to the occurrence of random errors.
In fact, fluctuations in the surrounding conditions may influence the performances of the instrument
and the observer.
The only way for estimating the contribution of the random error is to repeat the same measure-
ment many times. For example, imagine determining the lifetime of a long-lived phosphor by using
a chronometer. A source of error is our reaction time in starting and stopping our clock: if we delay
in starting or we anticipate in stopping, we underestimate the lifetime; if we delay in stopping or
we anticipate in starting, we overestimate its lifetime. Since all these possibilities are equally likely,
Appendix D
521
our measurements will spread statistically. We will have a set of results: τ , , …, whose limiting
1 τ 2
τn
distribution is Gaussian. The best estimate of τ true is the center of the Gaussian distribution (read
Box 1 of this Appendix), which is the average when n → ∞:
n τ i
∑
τ
i 1
best = τ =
=
[D.3]
n
The uncertainty on the average2 is the standard deviation of the mean:
1
n τ( i −τ 2)
n 1∑ i=
σ
1
τ =
−
[D.4]
n
BOX 1 THE GAUSSIAN DISTRIBUTION
The form of the Gaussian function describing how the measurements of the τ value are
true
(τ τ
−
2
true)
distributed over the variable τ is f (τ ) 1/σ π
2
e
σ
= (
) − 22 .
Why is the average the best estimate of τ ?
true
If we collect n values ( τ , , …, ), the probability of getting is proportional to
1 τ 2
τn
τ 1
(τ
2
1 τ
− true)
1
−
f (τ ) ∝
e
σ
2 2
1
σ
π
2
The probability of obtaining τ is proportional to
2
(τ
2
2 τ
− true)
1
−
f (τ ) ∝
e
σ
2 2
2
, and so on.
σ
π
2
(τ
2
n τ
− true)
The probability of getting τ is proportional to f (τ
σ
1
2 ) −
2
2
n ) ∝ ( /σ
π e
.
n
The probability of observing the entire set of n values ( τ , , …, ) is just the product of the
1 τ 2
τn
probabilities of observing each of them when all the n measurements are independent events.
n
n
(τ τ
∑
−
i
i
1
1
− =
ttrue)2
prob(τ
2
2σ
1,τ 2 ,
,
… τ n) ∝ f (τ1) f (τ2)… f (τ n) =
e
σ
π
2
According to the principle of the Maximum Likelihood, the best estimate of τ is that value
true
for which prob(τ
…
τ
…
1 τ 2
τ
1,τ 2,
,τ n) is maximized. Obviously,
(
prob , , , n) is maximum if the
exponent n
∑ (τ
2
2
2 is minimum. If we differentiate the exponent with respect to
=1
i −τ true ) / σ
i
τ and we set the derivative equal to zero, we achieve that τ
n
best = τ = ∑
τ n
/ .
true
=1 i
i
( Continued)
2 The uncertainty on the average value τ is the standard deviation of the mean, whereas the uncertainty on the single 2
determination τ is the standard deviation σ =
1
n (
) .
n 1 ∑
τ i −τ
i
−
i=1
522
Appendix D
BOX 1 (Continued) THE GAUSSIAN DISTRIBUTION
By using the principle of the Maximum Likelihood, it is possible to also find the best esti-
mate for σ, which is the width of the limiting distribution. If we differentiate prob(τ
…
1,τ 2,
,τ n)
with respect to σ, and we set the derivative equal to zero, we confirm that σ is the standard
best
deviation of the mean. In the figure within this box, there are two Gaussian distributions, both
centered at the same τ , but having two different widths, i.e., two different
true
σ values.
0.9
( τ − τtrue)2
−
f ( τ) = 1
2 σ 2
e
σ√2 π
0.6
σ = 1
f ( τ)
0.3
σ = 3
0.0 5
10
15
τ
d.4 ToTal uncerTainTy in direcT measuremenTs
In synthesis, all the direct measurements are subjected to both random and systematic errors (see
Figure D.6). Some random errors are generated by the observer, whereas some others by the equip-
ment. The unique strategy to cope with them is to repeat and repeat, as much as possible, the
experimental determination. Also, the systematic errors may derive from both the observer and the
instrument. The remedy consists in changing the operator or the experimental device. Whatever is
the accuracy of a device, we cannot completely wipe out the systematic instrumental error due to
the inherent uncertainty of any apparatus.
This means that our measurements of x can never be represented as dots along the x-axis
true
(see Figure D.7) but as circles, whose area shrinks by upgrading our experimental apparatus.
Figure D.7 shows four possible situations that may arise when we determine through direct mea-
surements the value x of a variable. Case (1) occurs when the random error (
) is small and
true
εrandom
when the only type of systematic error ( ε ) is that generated by the inherent uncertainty of the
sys
instrument. In fact, the measurements are represented by small circles distributed symmetrically
with respect to x and pretty close to it. Such determinations are said to be accurate and precise.
true
Case (2) comes about when the random error is small, the inherent uncertainty is tiny, but our
measurements are affected by miscalibration. In case (2) the determinations are still precise ( ε
random
is small like in case [1]) but they are inaccurate (the overall ε is large). Case (3) refers to a situa-
sys
tion where our measurements suffer from scarce precision, although they are accurate because the
instrument has small inherent inaccuracy and there are no problems with miscalibrations. Case (4)
represents the worse situation among all because the distribution and size of the circles along x-axis
reveal that our determinations are imprecise (they are spread in a broad range of x values) and inac-
curate (they are big circles, all located at values smaller than x ).
true
Appendix D
523
Change
Change
instrument
Miscalibration
operator
Observer’s
systematic
errors
Inherent
instrumental
inaccuracy
PHYSICAL
QUANTITY
Operator’s
Instrumental
random
random
errors
Repeat
Repeat
errors
FIGURE D.6 Types of uncertainties in a direct measurement and their remedies.
(1)
xtrue
(2)
xtrue
(3)
xtrue
(4)
xtrue
FIGURE D.7 Four possible situations we may encounter when we measure x more than once.
true
When we know both the random ( ε
) and the systematic uncertainty (
), and they are com-
random
εsys
parable, the reasonable estimation of the total uncertainty will be the sum of the two contributions:
xbest + x
∆ = x + ε random + ε sys = x +σ x + ε sys [D.5]
From equations [D.5] and [D.4], it is evident that if we repeat many times our measurement, i.e.,
then n → ∞, ε
can become very tiny and negligible with respect to
. However, even if we
random
εsys
repeat an infinite number of times our measurement, the total uncertainty in x cannot be smaller
best
than ε that is the inherent inaccuracy of our experimental device.
sys
d.5 ProPagaTion of The uncerTainTies
Many variables are measured indirectly. They need to be calculated from the values of variables
that have been measured directly. For instance, the speed of a vehicle requires the direct determina-
