# Quality Progress - February 2018 - 54

```Statistics Spotlight
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124345464755497541275299949514654237204317
91545154134541213231315447133415341545721157945495241262
124345464755497541275299949514654237204317
91545154134541213231315447133415341545721157945495241262
124345464755497541275299949514654237204317
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Statistical application

FIGURE 1

The theorem is derived under the
assumption of an infinite population with observations that are
independent, and identically distributed with constant mean and
variance. Using basic calculus, it is
not difficult to prove and is often
included in high school curricula.
Further, for this infinite population with mean, μ, and standard
deviation, σ, there is the added
assumption that for our sample to
be normally distributed we must
take sufficiently large random
samples from the population with
replacement. What de Moivre
showed was that this will hold true

Uniform distribution of the numbers 1-10
and mean 5.5
0.12
0.10
0.08
0.06
0.04
0.02
0

TA B L E   1

Means of 3 from a
uniform distribution of
the numbers 1-10
Take samples of size 3
1

2

3

2.0

1

2

4

2.3

1

2

5

2.7

1

2

6

3.0

1

2

7

3.3

1

2

8

3.7

1

2

9

4.0

1

2

10

4.3

...

54 QP

Average

...

...

...

6

8

9

7.7

6

8

10

8.0

6

9

10

8.3

7

8

9

8.0

7

8

10

8.3

7

9

10

8.7

8

9

10

9.0

February 2018 ❘ qualityprogress.com

1

2

3

4

5

6

7

8

9

regardless of the distribution of the source population.
In its most familiar form, this theorem does not apply to sampling
from a finite population-for example, the number of factories an
organization owns or the number of transit subway riders per day.4 Two
important modifications of the CLT were necessary before statisticians
could apply the results to finite populations and sampling without
replacement. Andrey Markov showed that the theorem can be relaxed
for use with dependent sampling (without replacement) and Lévy
showed that the same properties of the CLT with theoretical distribution can be applied to empirical distributions (that is, real data). 5
In general, statisticians assume that whether the underlying distribution is normal or skewed, provided the sample size is sufficiently
large (usually n > 30), the sample will be normal. If the population is
already normal, the theorem holds true even for samples smaller than
30. In practice, this means we can use the normal probability model to
quantify uncertainty when making inferences about a population mean
based on the sample mean.
However, the essential component of the CLT is that it is referring to
the distribution of our sample means approaching the normal distribution, and the mean of our sample means will be the same as the
population mean, not a specific mean from one specific sample-as
how the CLT is used today.
We are now analyzing large data sets from nonrandomized and
from samples without replacement. The CLT, while very generalizable,
was developed before the advent of computers and age of big data.
Now, it's too easy to have too much data and therefore be magnitudes

10

```
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Seen and Heard
Progress Report
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
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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
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Quality Progress - February 2018 - 26
Quality Progress - February 2018 - 27
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Quality Progress - February 2018 - Open Lines
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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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