Untangling Complex Systems, page 101
tion of the traveled distance and the time spent; if we want to know the density of a material we need
to measure its mass and its volume.
524
Appendix D
40
3 )
V = L 3
V(cm 30
( Lbest + dL)3
20
( Lbest)3
( Lbest − dL)3 10
00.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
L (cm)
Lbest − dL Lbest Lbest + dL
FIGURE D.8 The propagation of the L uncertainty in the determination of the volume.
best
How do the errors made in the direct measurements propagate in the estimation of the
derived variables? Imagine having measured the length L of a cube’s side directly and wanting
to determine its volume. If Lbest = (2.6 ± .
0 )
1 cm, then the best estimate of the volume will be
V
3
3
best = ( Lbest ) = 17.576 cm . How much is the uncertainty in the volume? (Figure D.8) If the uncertainty in L is small, the values of V for L = Lbest + dL and L = Lbest − dL are pretty close to V . The function V( L), around V , can be approximated by a straight line. In other words, best
best
V( L) can be described by a truncated Taylor’s series expansion having V as its accumulation point: best
dV
V( Lbest + dL) = V( Lbest) +
dL V
= best + dV [D.6]
dL Lbest
dV
V( Lbest − dL) = V( Lbest) −
dL V
= best − dV [D.7]
dL Lbest
It derives that the uncertainty in the volume is dV = ( dV dL)
dL = (3 L 2 ) dL = 2 028 3
.
cm . The
L
best
best
final result is V
3
best = (17.6 ± .
2 0) cm . This analysis of the error’s propagation can be generalized. For
any function z of the variable x that is measured directly with an uncertainty dx, the error will be: dz
dz =
dx
[D.8]
dx xbest
Note that in equation [D.8], we have the absolute value of the first derivative because sometimes the
first derivative may be negative, but the error never decreases when it propagates. If z is a function
of two or more variables, x , , …., , the uncertainty will be:
1 x 2
xn
dz
dz
dz
dz =
dx 1
dx
+
2 +… +
dx [D.9]
dx
n
1
dx 2
dx
x
1,
x
n
best
2, best
xn, best
Appendix D
525
The uncertainty in z determined through either the [D.8] or [D.9] formula is called the maximum
a priori absolute error. In the absence of miscalibration errors, the z value will be surely within
true
the range z
best + dz, zbest − dz. When the uncertainties in the variables x
that have been
1 , x 2 , …, xn
measured directly, are independent and random, we may use formula [D.10] for estimating the
uncertainty in z.
2
2
2
dz
dz
dz
dz =
dx
+…+
dx
1
+
dx
[D.10]
dx
2
dx
n
n
1
dx
x
2
1, best
x 2, best
xn, best
Alternatively, we may calculate as many values of z as are those collected for the independent
variables x
. Let us assume to have repeated
1 , x 2 , …, x
m measurements for each variable. The best
n
estimation of z and its uncertainty is the average ( z
m
= (∑ z / and the standard deviation of the
=1
) )
j
j
m
mean ( S
m
1
2
z =
(
) /
, respectively.
1 ∑
z − z
m
( − =1
) )
m
j
j
d.6 sTudenT’s T-disTribuTion
We have already learned that when we repeat the measurement of a value of the variable x, n times,
with n → ∞, and the uncertainty is mainly random,3 the data are well-described by a Gaussian function. The Gaussian function will be centered on the mean value x , and its width will be the standard
deviation of the mean, σ x. By exploiting the Gaussian distribution, we can calculate the probability
that a further determination will be within the range x ± gσ x. Such probability is a function of the parameter g (see the values listed in Table D.1). For instance, the probability that a measurement is within the range x +
σ x, x −σ x , with g = 1, amounts to 68.27%.
TABLE D.1
Percent Probability (Prob) That a
Measurement is within the Range
x + gσ
x, x − gσ x
as a Function of g and
Calculated under the Hypothesis That the
Measurements are Described by a
Gaussian distribution
g
Prob (%)
0.00
0.00
0.50
38.29
0.68
50.35
1.00
68.27
1.04
70.17
1.44
85.01
1.50
86.64
1.65
90.11
1.96
95.00
2.00
95.45
2.50
98.76
3.00
99.73
3 In other words, ε random >> ε sys.
526
Appendix D
Most often, we cannot repeat the same measurement an infinite number of times (i.e., many
times), but just a few. In such cases, our data are not properly described by a Gaussian distribution
but a Student’s t- distribution. The Student’s t- distribution is defined for the continuous random variable t = ( x − xGauss) / Sx where xGauss is the expectation value of the corresponding Gaussian distribution and Sx is the experimental standard deviation of the mean x . The probability density function
of the t variable is
ν +1
ν
Γ
+1
1
2
2 − 2
t
f ( t,ν ) =
1+
[D.11]
πν
ν
ν
Γ
2
In equation [D.11], ν is the number of the degrees of freedom that is equal to m − 1, if m is the
finite number of times a measurement has been repeated; Γ is the gamma function. The function
f ( t,ν ) is symmetric and bell-shaped (like the Gaussian distribution), but it has heavier tails than the Gaussian distribution; its expectation value is zero, and its variance is ν / ν
( − 2) for ν > 2. If ν → ∞,
the t-distribution approaches a Gaussian distribution centered at 0 and with a standard deviation
σ = 1. The probability that the ( m + 1) - th measurement is within the interval x + tS
x , x − tSx
is a
function of t and ν; some values are reported in Table D.2. The last row of Table D.2 (with ν = ∞) corresponds to the values defined for the Gaussian distribution.
We may use the Student’s t- distribution for comparing the mean values of two independent sets
of measurements, xA and xB, and ask us if they belong to the same distribution. This test is suitable to ascertain if the experimental conditions have not changed in the two series of measurements or if
the measurements of the same physical quantity carried out in different laboratories can be merged
to give a single best estimate. First, we calculate the t-value through the following formula
xA x
t
B
=
−
[D.12]
S 2
2
A
SB
+
nA nB
TABLE D.2
Values of the t Variable Expressed as a Function of the Percentage Probability (Prob) That
a Further Measurement is within the Range x + tS
x, x − tSx
. The t-Value also Depends
on the Degrees of Freedom ν = m − 1
Prob (%)
ν
50
60
70
80
90
95
99
1
1.000
1.376
1.963
3.078
6.314
12.706
63.657
2
0.816
1.061
1.386
1.886
2.920
4.303
9.925
3
0.765
0.978
1.250
1.638
2.353
3.182
5.841
4
0.741
0.941
1.190
1.533
2.132
2.776
4.604
5
0.727
0.920
1.156
1.476
2.015
2.571
4.032
6
0.718
0.906
1.134
1.440
1.943
2.447
3.707
7
0.711
0.896
1.119
1.415
1.895
2.365
3.499
8
0.706
0.889
1.108
1.397
1.860
2.306
3.355
9
0.703
0.883
1.100
1.383
1.833
2.262
3.250
10
0.700
0.879
1.093
1.372
1.812
2.228
3.169
20
0.687
0.860
1.064
1.325
1.725
2.086
2.845
30
0.683
0.854
1.055
1.310
1.697
2.042
2.750
∞
0.674
0.842
1.036
1.282
1.645
1.960
2.576
Appendix D
527
In [D.12], S 2
2
A and SB are the variances (i.e., the standard deviations S and S , both to the power of A
B
two) of the two series of measurements, A and B, whose numbers of elements are n and n , respec-
A
B
tively. If the estimated t-value is larger than the expected t- value for a confidence level of the 95%, we may assert that the experimental conditions have been changed. Otherwise, the two series may
be considered as belonging to the same distribution and may be used to determine a better estimate
through the calculation of the weighted average:
xA xB
+
2
2
x
SA SB
best =
[D.13]
1 + 1
S 2
2
A
SB
The formula [D.13] can be generalized to more than two separate sets of measurements that are
consistent among them. If the number of independent sets of measurements is k, x 1, x 2,…, xk are the corresponding mean values, and S , , …, are their standard deviations, the best estimate is
1 S 2
Sk
k wixi
∑
x
i 1
best =
=
[D.14]
k wi
∑ i=1
The terms w
2
i = 1 / Si are called weights. The larger the standard deviation, the smaller the weight.
The sets of measurements that are much less precise than the others contribute very much less to
xbest. For example, if one set of measurements is three times less precise than the others, its weight is
nine times smaller than the other weights. The uncertainty in the weighted average xbest is
k
−1/2
S
x =
wi
∑
[D.15]
i=1
d.7 leasT-sQuares fiTTing
Sometimes the final goal of an experiment is the determination of the mathematical equation relat-
ing two variables. For instance, a chemist may be interested in determining the order of a chemical
reaction with respect to a reagent X. In other words, the chemist may be interested in determining
the exponent that appears in the equation v k X nX
= [ ] . The chemist measures the reaction rates ( v ,
1
v , …, ) at different concentrations of
, [
, …, [
) maintaining constant all the other
2
v
X ([ X]
X]
X]
n
1
2
n
experimental conditions. It is convenient to work with the logarithmic form of the kinetic equa-
tion ( v = k[ X ] nX ), i.e., log( v) = log( k) + nX log[ X ]. In fact, by plotting log( v ), log( ), …, log( ) vs.
1
v 2
vn
log[ X] , log[
, …, log[
, the experimental data can be described by a straight line, y = A + Bx.
1
X]2
X] n
The slope is B = nX, and the intercept is A = log( k). The best estimates of the coefficients A and B
are obtained by the linear regression, also called the least-squares method.
More in general, imagine having collected pairs of data ( x
), (
), …, (
). The uncer-
1 , y 1
x 2 , y 2
x , y
n
n
tainties on the x values are negligible with respect to those on the y values, and the uncertainties on i
i
the y values are of the random type. Each y value belongs to a Gaussian distribution centered on its i
i
true value that is ( A + Bx ). Therefore, the probability of collecting y is:
i
i
( y
2
i − A− Bxi )
−
1
2
σ
P y
yi
( i) ∝
e
[D.16]
σ yi
528
Appendix D
The probability of obtaining the complete set of independent measurements, y , , …, , is the
1 y 2
yn
product of the probabilities of obtaining every single value:
n ( y
2
i − A− Bx )
n
i
1 −∑
2
σ
P( y , y , ,
… y
i 1
yi
) ∝
1
2
n
e
∏
=
[D.17]
σ
i
yi
1
=
According to the principle of maximum likelihood, the best estimates of the A and B parameters are
those that maximize the total probability, P( y
)
1, y 2,
,
… yn , or minimize the exponent4
n
2
( iy − A− Bxi)
χ2 = ∑
[D.18]
σ 2
i 1
=
yi
For the determination of these values, we differentiate χ 2 with respect to A and B and set the derivatives equal to zero:
∂χ2
n
= −
wi yi A Bx
A
∑
2
( − − i ) = 0
∂
i=1
[D.19]
∂χ2
n
= −
wixi yi A Bx
