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11.2.3 Statistical Analysis
For the final value of S to carry meaning, it needs to be supplemented with an
estimate of its standard deviation σ S . To come up with a meaningfull estimate
of this standard deviation turns out to be somewhat involved. A simple-minded
approach is to break up an experiment consisting of N runs into n sections of N n
runs each. For each section i, an S-value S i can be calculated and the resulting
sample set can be statistically analyzed to obtain an estimator for the sample
mean S, the sample standard deviation σ Si , and the resulting standard error of
the sample mean σ S :
S = 1 n
σ Si = √ 1
n − 1
n∑
S i (11.1)
i=1
n∑ (
Si − S ) 2
i=1
(11.2)
σ S = σ S
√ i
(11.3)
n
The problem with this analysis is that it assumes the errors of S i to be uncorrelated,
an assumption that does not hold in the presence of systematic errors in
the experiment like drift and 1/f noise. This leads to different estimates of σ S
for different choices for the number of sections n. In reality, the overall standard
error in S as a function of sampling-time is the result of two competing effects:
• S is calculated from a linear combination of 16 probabilities (P | 00 〉 (ab), . . . ,
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