# Quality Progress - February 2018 - 55

```FIGURE 2

Histogram of the sample without replacement
results showing a bell-shaped curve
20

Extreme example

17
Frequency

Suppose we have a simple distribution of data in which each number
is equally likely and the data are the
first 10 numbers: 1, 2, 3, 4, 5, 6, 7, 8,
9 and 10. Note that the probability
of taking a sample of size one from
this group is 0.1, or 10%.
This is a small sample, but we
can illustrate the power of the CLT
by looking at the distribution of
the means from all possible samples of size three. Table 1 shows
the initial eight samples and the
last seven samples for samples
without replacement. Note that
the means of each group of three
are not necessarily close, varying
from a mean of 2-9.
Sampling without replacement,
there are 120 unique combinations of three numbers from the
numbers 1-10. When all the combinations are counted together in
a histogram, the average value of
a sample of three approaches is a
normal distribution (Figure 2) with
an overall mean of 5.5. The overall
mean of 5.5 is identical to the
mean of the original distribution.
While the original population contains 10 possible values, because
our sampled population is more
than 30, the CLT is applicable.
Sampling with replacement
would increase the combinations
to 220 and add both lower means
(1 from the sample 1, 1, 1) and
higher means (10 from the sample
10, 10, 10). The increased sample
size shows a smoother normal

20
17

12

12

7
1

3

2

2.7

7
4

3.4

4.1

4.8

5.5

6.2

6.9

7.6

8.3

More

FIGURE 3

Histogram of the sample with replacement
results showing a smoother bell-shaped
curve
80

70

60

50

Group 1

greater than the lower limit of 30
samples. In these cases, the CLT
may be valid, but other issues
do arise when looking for the
statistical signal in all the data that
are not accounted for when the
theorem and many parametric
tests were developed.

40

30

20

10

0

Editor's note: The overall mean of the 1,000 means is 5.5, identical to the mean of the original
distribution and the sample without replacement results.

qualityprogress.com ❘ February 2018

QP 55

```
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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
Quality Progress - February 2018 - 17
Quality Progress - February 2018 - 18
Quality Progress - February 2018 - 19
Quality Progress - February 2018 - 20
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Quality Progress - February 2018 - The Crown Jewels of Design
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Quality Progress - February 2018 - Open Lines
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Quality Progress - February 2018 - Less Is More
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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
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Quality Progress - February 2018 - Six Sigma Solutions
Quality Progress - February 2018 - 51
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Quality Progress - February 2018 - Statistics Spotlight
Quality Progress - February 2018 - 54
Quality Progress - February 2018 - 55
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Quality Progress - February 2018 - 57
Quality Progress - February 2018 - Marketplace
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Quality Progress - February 2018 - Footnotes
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Quality Progress - February 2018 - Try This Today
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