# Introduction to Data Analysis Data Reduction Example by gregoria

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```									                                                                                     Data Reduction

• Translate measured data into one or more physical
variables of interest
Introduction to Data Analysis                   • Obtain
– best estimate for physical variable
– estimate for precision and accuracy of measurement
c.f. Bevington                            (systematic and statistical uncertainty)
Chapters 1-3                       • Example (from my experiment):

Sep 13 2004                                                  Sep 13 2004

Example: Determination of particle
Histograms
momentum distribution

z
Beam

By
R = pzx/(q*By)             Binned representation of data in 1, 2 or 3 dimensional
2                                           variable space

1

x
10 cm

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1

Statistical and Systematic Error                                                   Distributions
Mean

Sample
• Systematic error                                                 Distribution
– inherent to measurement, apparatus, methods                                                           Variance
– estimate magnitude by comparing different approaches
– limits accuracy
• Statistical error
– Measurements jitter around truth
– Average many measurements to improve estimate (if they
are independent)
– limits precision, but lim(N-> inf) = truth                                Mean
for N -> inf :
Parent distribution
Variance
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Binominal Distribution                                                   Poisson Distribution

Mean                                                                  Mean

Std. Dev                                                              Std. Dev

P(x:n,p): Probability to get ‘yes’ x times out of n tries,             Derives from binomial distribution for p << 1 with n*p = µ
if probability of ‘yes’ for single try is p                            Important for counting experiments with low count rate

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2

Gaussian Distribution                                                  Statistical Error on Mean

Mean

Std. Dev                                                              • Repeated measurements increase precision
– but only as sqrt(N)
– ultimate limit may come from systematic uncertainty

Derives from Poisson distribution for n*p >> 1

Seen everywhere b/c of Central Limit Theorem

Sep 13 2004                                                                Sep 13 2004

Error propagation

Interested in error on ‘x’, but measure ‘u’, with x = f(u)

What if x = f(u,v):