IEEE Solid-State Circuits Magazine - Spring 2014 - 54
! 3v bounds is valid for normal distributions, the values that should be
presented for unknown and potentially asymmetric distributions are
as follows:
■■ instead of n ! v, the two percentiles P 15.87 and P 84.13
■■ instead of n ! 2v,
P 2.275 and
P 97.725
■■ instead of n ! 3v,
P 0.135 and
P 99.865 .
Also, we would then not give the
mean value n, but the median
M = P 50.0, below which 50% of the
samples are found.
This is annotated in Figure 1. As
mentioned above, inside the P 15.87 and
P 84.13 lines, our data set looks normally
distributed, but outside these lines,
we cannot tell much from such a small
number of samples. At this point, the
only way to get further with statistics
based on the mean and standard deviation is to assume that the underlying
distribution is a normal distribution.
n and v lie for any confidence level.
Since ! 3s implies an error probability
of 0.27%, as mentioned above, it might
be a good idea to ask for an interval
within which the true values lie with
a probability of 95%: c = 0.95 and
a = 1 - c = 0.05. It is well known that
the confidence interval for n is then [4]
P ' m - t (1 -a/2, N -1) · s # n # m
N
s
(4)
+ t (1 -a/2, N -1) ·
1 = c,
N
where t (1 -a/2, N -1) is the inverse
cumulative Student-t distribution,
the solution of the integral equation
t (1 -a/2, N -1)
#-3
(N - 1) r ·C c N - 1 m
2
# c1 +
-N
2
t
m
N -1
2
(a/2, N -1)
#-3|
N -1 -1
2
N -1
2
c
x
2
x
·e - 2 dx = a .
2 (7)
-1 m
N
·C
2
Solving these integrals numerically (in ScientificPython, e.g., implemented as stdtrit and chdtri) or
using a table [4], [5] for the data set
x, N = 24, gives estimated ranges
for the true n and v:
n x =-3.3966 g 2.5791,
v x = 5.4994 g 9.9256,
and for the data set y, N = 8,
Cc N m
2
2
where | 2(a/2, N -1) is the inverse cumulative Chi-square distribution, the
solution of the integral equation
n y = 1.8648 g 11.7264,
v y = 5.3744 g 16.5438.
Therefore, what we should plot are
ranges rather than points, as shown in
(5)
Figure 2(b). These ranges are quite large,
for
the
integration
boundary
because both n and v are uncertain,
t (1 -a/2, N -1) . Similarly for v [5],
so, e.g., the confidence range for the
location of the n + 3v point on the x
Assuming a Normal Distribution
axis extends from min (n) + 3 min (v)
(N - 1) s 2
(N - 1) s 2
So let us now look at the two data
to max (n) + 3 max (v) . The figure
3
P)
#v #
2
| (a/2, N -1)
| (21 -a/2, N -1)
sets x i and y i . Equations (1) and
already makes it clear how little we
= c,
(2) give m x =-0.4088, s x = 7.0758,
actually know with N = 24 and 8 sam
(6)
m y = 4.9308, and s y = 8.1285. Not
ples, respectively.
surprisingly, the results from
Now imagine that someone
the data sets x i and y i look
assumes that the estimated
99
s x = v x . That person would
quite different. A graphical
98 P97.725 or ″µ + 2σ″
then believe that the probrepresentation of m x ! 3s x
and m y ! 3s y is shown in Figability that a sample lies out95
ure 2(a).
side the range m x ! 3s x is only
90
0.27%. If the true n and v both
What should alarm readP84.13 or ″µ + σ″
are the maxima of their respecers and authors alike is that
80
tive confidence intervals, then
all ! 3s points now lie out70
side the range of which we
the probability that a sample
60
do indeed have measurement
lies outside the range m x ! 3s x
P50 or ″µ″
50
is actually 6.05%. This means
data: by drawing these points,
40
that the error probability
we have implied information
30
was underestimated by a factor
about a possible value range
20
P15.87 or ″µ - σ″
of 22.4.
of x and y for which we have
no empirical evidence. Also,
All this must be shocking
10
the points drawn are just estienough for authors, reviewers,
5
mates of the real values of n
and readers alike who have
P2.275 or ″µ - 2σ″
and ! 3v, and nothing is said
simply plotted and requested
2
3v bounds up to date. It gets
about confidence yet.
1
-10
-5
0
5
10
15
20
even
worse: Remember that
If and only if we are certain
x
even this is valid only if we
that the underlying distribuhave prior knowledge that
tion is a normal distribution, Figure 1: The data series x plotted to look like a cumulative
i
the data we look at is exactly
then we can actually calculate distribution. The vertical axis is scaled such that a perfect nornormally distributed. And this
the ranges in which the true mal distribution would be represented as a straight line [1].
Cumulative Distribution (%)
54
s p r i n g 2 0 14
IEEE SOLID-STATE CIRCUITS MAGAZINE
dt = 1 - a/2
Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Spring 2014
IEEE Solid-State Circuits Magazine - Spring 2014 - Cover1
IEEE Solid-State Circuits Magazine - Spring 2014 - Cover2
IEEE Solid-State Circuits Magazine - Spring 2014 - 1
IEEE Solid-State Circuits Magazine - Spring 2014 - 2
IEEE Solid-State Circuits Magazine - Spring 2014 - 3
IEEE Solid-State Circuits Magazine - Spring 2014 - 4
IEEE Solid-State Circuits Magazine - Spring 2014 - 5
IEEE Solid-State Circuits Magazine - Spring 2014 - 6
IEEE Solid-State Circuits Magazine - Spring 2014 - 7
IEEE Solid-State Circuits Magazine - Spring 2014 - 8
IEEE Solid-State Circuits Magazine - Spring 2014 - 9
IEEE Solid-State Circuits Magazine - Spring 2014 - 10
IEEE Solid-State Circuits Magazine - Spring 2014 - 11
IEEE Solid-State Circuits Magazine - Spring 2014 - 12
IEEE Solid-State Circuits Magazine - Spring 2014 - 13
IEEE Solid-State Circuits Magazine - Spring 2014 - 14
IEEE Solid-State Circuits Magazine - Spring 2014 - 15
IEEE Solid-State Circuits Magazine - Spring 2014 - 16
IEEE Solid-State Circuits Magazine - Spring 2014 - 17
IEEE Solid-State Circuits Magazine - Spring 2014 - 18
IEEE Solid-State Circuits Magazine - Spring 2014 - 19
IEEE Solid-State Circuits Magazine - Spring 2014 - 20
IEEE Solid-State Circuits Magazine - Spring 2014 - 21
IEEE Solid-State Circuits Magazine - Spring 2014 - 22
IEEE Solid-State Circuits Magazine - Spring 2014 - 23
IEEE Solid-State Circuits Magazine - Spring 2014 - 24
IEEE Solid-State Circuits Magazine - Spring 2014 - 25
IEEE Solid-State Circuits Magazine - Spring 2014 - 26
IEEE Solid-State Circuits Magazine - Spring 2014 - 27
IEEE Solid-State Circuits Magazine - Spring 2014 - 28
IEEE Solid-State Circuits Magazine - Spring 2014 - 29
IEEE Solid-State Circuits Magazine - Spring 2014 - 30
IEEE Solid-State Circuits Magazine - Spring 2014 - 31
IEEE Solid-State Circuits Magazine - Spring 2014 - 32
IEEE Solid-State Circuits Magazine - Spring 2014 - 33
IEEE Solid-State Circuits Magazine - Spring 2014 - 34
IEEE Solid-State Circuits Magazine - Spring 2014 - 35
IEEE Solid-State Circuits Magazine - Spring 2014 - 36
IEEE Solid-State Circuits Magazine - Spring 2014 - 37
IEEE Solid-State Circuits Magazine - Spring 2014 - 38
IEEE Solid-State Circuits Magazine - Spring 2014 - 39
IEEE Solid-State Circuits Magazine - Spring 2014 - 40
IEEE Solid-State Circuits Magazine - Spring 2014 - 41
IEEE Solid-State Circuits Magazine - Spring 2014 - 42
IEEE Solid-State Circuits Magazine - Spring 2014 - 43
IEEE Solid-State Circuits Magazine - Spring 2014 - 44
IEEE Solid-State Circuits Magazine - Spring 2014 - 45
IEEE Solid-State Circuits Magazine - Spring 2014 - 46
IEEE Solid-State Circuits Magazine - Spring 2014 - 47
IEEE Solid-State Circuits Magazine - Spring 2014 - 48
IEEE Solid-State Circuits Magazine - Spring 2014 - 49
IEEE Solid-State Circuits Magazine - Spring 2014 - 50
IEEE Solid-State Circuits Magazine - Spring 2014 - 51
IEEE Solid-State Circuits Magazine - Spring 2014 - 52
IEEE Solid-State Circuits Magazine - Spring 2014 - 53
IEEE Solid-State Circuits Magazine - Spring 2014 - 54
IEEE Solid-State Circuits Magazine - Spring 2014 - 55
IEEE Solid-State Circuits Magazine - Spring 2014 - 56
IEEE Solid-State Circuits Magazine - Spring 2014 - 57
IEEE Solid-State Circuits Magazine - Spring 2014 - 58
IEEE Solid-State Circuits Magazine - Spring 2014 - 59
IEEE Solid-State Circuits Magazine - Spring 2014 - 60
IEEE Solid-State Circuits Magazine - Spring 2014 - 61
IEEE Solid-State Circuits Magazine - Spring 2014 - 62
IEEE Solid-State Circuits Magazine - Spring 2014 - 63
IEEE Solid-State Circuits Magazine - Spring 2014 - 64
IEEE Solid-State Circuits Magazine - Spring 2014 - 65
IEEE Solid-State Circuits Magazine - Spring 2014 - 66
IEEE Solid-State Circuits Magazine - Spring 2014 - 67
IEEE Solid-State Circuits Magazine - Spring 2014 - 68
IEEE Solid-State Circuits Magazine - Spring 2014 - 69
IEEE Solid-State Circuits Magazine - Spring 2014 - 70
IEEE Solid-State Circuits Magazine - Spring 2014 - 71
IEEE Solid-State Circuits Magazine - Spring 2014 - 72
IEEE Solid-State Circuits Magazine - Spring 2014 - 73
IEEE Solid-State Circuits Magazine - Spring 2014 - 74
IEEE Solid-State Circuits Magazine - Spring 2014 - 75
IEEE Solid-State Circuits Magazine - Spring 2014 - 76
IEEE Solid-State Circuits Magazine - Spring 2014 - 77
IEEE Solid-State Circuits Magazine - Spring 2014 - 78
IEEE Solid-State Circuits Magazine - Spring 2014 - 79
IEEE Solid-State Circuits Magazine - Spring 2014 - 80
IEEE Solid-State Circuits Magazine - Spring 2014 - 81
IEEE Solid-State Circuits Magazine - Spring 2014 - 82
IEEE Solid-State Circuits Magazine - Spring 2014 - 83
IEEE Solid-State Circuits Magazine - Spring 2014 - 84
IEEE Solid-State Circuits Magazine - Spring 2014 - 85
IEEE Solid-State Circuits Magazine - Spring 2014 - 86
IEEE Solid-State Circuits Magazine - Spring 2014 - 87
IEEE Solid-State Circuits Magazine - Spring 2014 - 88
IEEE Solid-State Circuits Magazine - Spring 2014 - 89
IEEE Solid-State Circuits Magazine - Spring 2014 - 90
IEEE Solid-State Circuits Magazine - Spring 2014 - 91
IEEE Solid-State Circuits Magazine - Spring 2014 - 92
IEEE Solid-State Circuits Magazine - Spring 2014 - 93
IEEE Solid-State Circuits Magazine - Spring 2014 - 94
IEEE Solid-State Circuits Magazine - Spring 2014 - 95
IEEE Solid-State Circuits Magazine - Spring 2014 - 96
IEEE Solid-State Circuits Magazine - Spring 2014 - 97
IEEE Solid-State Circuits Magazine - Spring 2014 - 98
IEEE Solid-State Circuits Magazine - Spring 2014 - 99
IEEE Solid-State Circuits Magazine - Spring 2014 - 100
IEEE Solid-State Circuits Magazine - Spring 2014 - 101
IEEE Solid-State Circuits Magazine - Spring 2014 - 102
IEEE Solid-State Circuits Magazine - Spring 2014 - 103
IEEE Solid-State Circuits Magazine - Spring 2014 - 104
IEEE Solid-State Circuits Magazine - Spring 2014 - Cover3
IEEE Solid-State Circuits Magazine - Spring 2014 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/mssc_fall2023
https://www.nxtbook.com/nxtbooks/ieee/mssc_summer2023
https://www.nxtbook.com/nxtbooks/ieee/mssc_spring2023
https://www.nxtbook.com/nxtbooks/ieee/mssc_winter2023
https://www.nxtbook.com/nxtbooks/ieee/mssc_fall2022
https://www.nxtbook.com/nxtbooks/ieee/mssc_summer2022
https://www.nxtbook.com/nxtbooks/ieee/mssc_spring2022
https://www.nxtbook.com/nxtbooks/ieee/mssc_winter2022
https://www.nxtbook.com/nxtbooks/ieee/mssc_fall2021
https://www.nxtbook.com/nxtbooks/ieee/mssc_summer2021
https://www.nxtbook.com/nxtbooks/ieee/mssc_spring2021
https://www.nxtbook.com/nxtbooks/ieee/mssc_winter2021
https://www.nxtbook.com/nxtbooks/ieee/mssc_fall2020
https://www.nxtbook.com/nxtbooks/ieee/mssc_summer2020
https://www.nxtbook.com/nxtbooks/ieee/mssc_spring2020
https://www.nxtbook.com/nxtbooks/ieee/mssc_winter2020
https://www.nxtbook.com/nxtbooks/ieee/mssc_fall2019
https://www.nxtbook.com/nxtbooks/ieee/mssc_summer2019
https://www.nxtbook.com/nxtbooks/ieee/mssc_2019summer
https://www.nxtbook.com/nxtbooks/ieee/mssc_2019winter
https://www.nxtbook.com/nxtbooks/ieee/mssc_2018fall
https://www.nxtbook.com/nxtbooks/ieee/mssc_2018summer
https://www.nxtbook.com/nxtbooks/ieee/mssc_2018spring
https://www.nxtbook.com/nxtbooks/ieee/mssc_2018winter
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_winter2017
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_fall2017
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_summer2017
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_spring2017
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_winter2016
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_fall2016
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_summer2016
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_spring2016
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_winter2015
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_fall2015
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_summer2015
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_spring2015
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_winter2014
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_fall2014
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_summer2014
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_spring2014
https://www.nxtbookmedia.com