IEEE Solid-State Circuits Magazine - Spring 2014 - 55

is knowledge that we never, ever have
at all. Our example data comes from
a trimmed temperature sensor, in
which case we are even certain that
the distribution is not normal. In that
case, the estimation errors we make
can be arbitrarily much higher.
The sobering conclusion of this
section is therefore not "apply this
theory correctly instead of incorrectly." The conclusion is "do not even
use this theory." We should never just
assume a normal distribution without
having a valid reason to do so.

What if the Underlying
Distribution Is Not Known?
So what can we do if we have no
knowledge about the shape of the
underlying distribution? The interesting answer is that there is a distribution-independent method to
obtain confidence intervals for percentiles that is even simpler, mathematically, than the standard method
described in the previous section.
The percentiles of a distribution
have a very nice property than can be
explained with a simple thought experiment: What do we know about the percentile Pp or the median M = P 50.0 if
we have just one single measurement?
The question sounds absurd, but being
what they are, we know that having
one single measurement value x 1,
then the percentile P p lies below that
value with a probability of 1 - p and
above it with a probability p.
So it is for every measurement
value, independent of all the others. Therefore, for a sorted list of
samples x i, i = 1f N, the probability that a percentile lies somewhere
between two measured values follows the binomial distribution [6]

	

+3sx

mx

x, y
−3sy

−10

0

my +10

+20

+3sy

(a)
″µx - 3σx″

µx

″µx + 3σx″
x, y

″µy - 3σy″

″µy + 3σy″

µy
(b)

Figure 2:  (a) A standard way of drawing `n ! 3v_ (actually m ! 3s) in papers. (b) The
intervals labeled `n ! 3v_ are the intervals where the values n ! 3v can lie for all possible n and v in their respective 95% confidence intervals. The intervals labelled n x and
n y are simply the respective 95% confidence intervals.

Figure 3 shows this both for the
median and for the 15.78 percentile.
Adding the probabilities on the intervals, we see that the probability that
the true median is within the range
of measured values is 87.5%, but
the probability that P 15.87 is within
is only 49.9%. This means that four
measurements are only sufficient to
estimate the median of the true distribution with a confidence of 87.5%.
This can now easily be generalized using (8) for any number N of
measurements, any percentile p,
and any integer 1 # m 1 N/2:
P " x m # M # x N -m +1 ,
= 1 -2

N

/ e k o 21N , 

m -1

(9)

k =0

P " x m # Pp , = P " x N -m +1 $ P1 -p ,
m -1 N
= 1 - / e o p k (1 - p) N -k,
k =0 k

(10)

where p = (1/2) is inserted into (8) to
obtain (9). This lets us, as described
in [7] for the median, decide which
values x i we should use as bounds
for different percentiles and different confidence levels, as shown in
Table 1.
For example, if you have ten
samples and need a 75% confidence
interval for M, Table 1 says three,
meaning that the interval x 3 gx 8 is
a 75% (or better) confidence interval. The most extreme data, x 1, 2, 9, 10,
are simply dropped. So we have a
statistical method where ignoring
outliers is not an ad-hoc strategy
but a proven part of the procedure.
The more measurements we have,
the more outliers we can ignore, as
Table 1 readily shows.
We will show examples in the
following section, but let us stress,
right here, a very important point:
if we choose actually measured values as interval bounds, then it is

p
0.501

P " x k # Pp # x k +1 ,

0.378


N
= c m p k (1 - p) N -k, k = 0gN, (8)
k

where it is assumed that x 0 " -3 and
x N +1 " 3. This is valid independent of
the underlying probability distribution,
as long as that distribution is sufficiently well behaved (its being continuous is already more than sufficient).

	

−3sx

0.375

0.25

0.25
0.107

0.0625

0.0625

0.0135
x1

x2

x3

x4

0.0005

x

Figure 3:  The probability that the true median (P 50%, black) and the percentile P 15.87 % (red,
dashed) of a process generating four measurements x i , i = 1f 4, lie in the intervals defined
by the measured data x i .

	 IEEE SOLID-STATE CIRCUITS MAGAZINE	

s p r i n g 2 0 14	

55



Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Spring 2014

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