Quality Progress - February 2018 - 53

Solving quality quandaries through statistics

Statistics Spotlight
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PROBABILITY

Staying Relevant
The central limit theorem remains a
key concept in probability theory
by Julia Seaman, I. Elaine Allen and Samuel Zetumer

In the beginning, there was the central limit theorem (CLT). We
use it every day as the basis for our statistical analyses. We base the
reproducibility of our analyses on what the CLT tells us about our data,
and we make inferences from the results of our hypothesis tests based
on the CLT.
This theorem forms the basis for the majority of hypothesis testing
and prediction models in statistics. In simplified terms, the CLT states
that the sum of many different independent results tends toward a
normal distribution and gives us a method to estimate
sampling error.
But is this theorem-first postulated in 1733-still relevant and appropriate in the age of big data? Specifically,
was the theorem really proven to encourage
us to perform parametric analyses with every
sufficiently large data set? In its first formulation, it was not.

Early origins

First and foremost, the CLT is a theorem
proven in probability theory and not
(initially) proposed to be used as
a normal theory to approximate
data from other distributions for

statistical analysis. It was first
proposed by Abraham de Moivre in
1733 when he discovered he could
approximate binomial distribution
probabilities from an integral of
exp(-x2),1 although he did not name
this integral.
It wasn't until 1920 that George
Polya gave the name to the theorem that we are now familiar with.
There were several developments
during the almost 200-year span
from de Moivre to Polya to supplement the theorem to the results
we use today. Over the centuries,
the proof of the theorem included
contributions by Pierre-Simon
Laplace, Siméon Denis Poisson,
Peter Gustav Lejeune Dirichlet, Bessel, Augustin Louis Cauchy, Pafnuty
Chebyshev, Aleksandr Liapounov,
Jarl Waldemar Lindeberg, Paul
Lévy and W. Feller.2 Throughout
all these contributions, the CLT
was formulated as a theoretical probability theorem
for the development
of density functions of
distributions.3 This is
not how statisticians apply this
theorem.

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QP 53


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Table of Contents for the Digital Edition of Quality Progress - February 2018

Seen and Heard
Expert Answers
Progress Report
Mr. Pareto Head
Career Coach
Office Efficiency
The Crown Jewels of Design
Open Lines
Less Is More
ASQ 2018 Six Sigma Resource Guide
Standard Issues
Six Sigma Solutions
Statistics Spotlight
Marketplace
Footnotes
Try This Today
Quality Progress - February 2018 - intro
Quality Progress - February 2018 - cover1
Quality Progress - February 2018 - cover2
Quality Progress - February 2018 - 1
Quality Progress - February 2018 - 2
Quality Progress - February 2018 - 3
Quality Progress - February 2018 - 4
Quality Progress - February 2018 - 5
Quality Progress - February 2018 - Seen and Heard
Quality Progress - February 2018 - 7
Quality Progress - February 2018 - Expert Answers
Quality Progress - February 2018 - 9
Quality Progress - February 2018 - Progress Report
Quality Progress - February 2018 - 11
Quality Progress - February 2018 - Mr. Pareto Head
Quality Progress - February 2018 - 13
Quality Progress - February 2018 - Career Coach
Quality Progress - February 2018 - 15
Quality Progress - February 2018 - Office Efficiency
Quality Progress - February 2018 - 17
Quality Progress - February 2018 - 18
Quality Progress - February 2018 - 19
Quality Progress - February 2018 - 20
Quality Progress - February 2018 - 21
Quality Progress - February 2018 - The Crown Jewels of Design
Quality Progress - February 2018 - 23
Quality Progress - February 2018 - 24
Quality Progress - February 2018 - 25
Quality Progress - February 2018 - 26
Quality Progress - February 2018 - 27
Quality Progress - February 2018 - 28
Quality Progress - February 2018 - 29
Quality Progress - February 2018 - Open Lines
Quality Progress - February 2018 - 31
Quality Progress - February 2018 - 32
Quality Progress - February 2018 - 33
Quality Progress - February 2018 - 34
Quality Progress - February 2018 - 35
Quality Progress - February 2018 - 36
Quality Progress - February 2018 - 37
Quality Progress - February 2018 - Less Is More
Quality Progress - February 2018 - 39
Quality Progress - February 2018 - 40
Quality Progress - February 2018 - 41
Quality Progress - February 2018 - 42
Quality Progress - February 2018 - 43
Quality Progress - February 2018 - ASQ 2018 Six Sigma Resource Guide
Quality Progress - February 2018 - 45
Quality Progress - February 2018 - Standard Issues
Quality Progress - February 2018 - 47
Quality Progress - February 2018 - 48
Quality Progress - February 2018 - 49
Quality Progress - February 2018 - Six Sigma Solutions
Quality Progress - February 2018 - 51
Quality Progress - February 2018 - 52
Quality Progress - February 2018 - Statistics Spotlight
Quality Progress - February 2018 - 54
Quality Progress - February 2018 - 55
Quality Progress - February 2018 - 56
Quality Progress - February 2018 - 57
Quality Progress - February 2018 - Marketplace
Quality Progress - February 2018 - 59
Quality Progress - February 2018 - Footnotes
Quality Progress - February 2018 - 61
Quality Progress - February 2018 - 62
Quality Progress - February 2018 - 63
Quality Progress - February 2018 - Try This Today
Quality Progress - February 2018 - cover3
Quality Progress - February 2018 - cover4
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