Quality Progress - February 2018 - 57

distribution curve than the previous histogram, as expected from the CLT
(Figure 3, p. 55).
This extreme example may be unrealistic. Because one of the authors
is a consultant, clients often share their data with the caveat that they
know it is not normally distributed so parametric statistics based on the
normal distribution should not be used. A good example of non-normally
distributed data is housing prices. See Figure 4, which shows few homes
are close to zero in cost and there are some expensive properties creating
a highly skewed distribution (usually following a Weibull distribution).6
As discussed in a New York Times article, "...even when raw data does
not fit a normal distribution, there is often a normal distribution lurking
within it."7,8
It is not the raw data that form the basis for our use of statistics based
on the normal distribution, but our reliance on the theorem to assure to us
that the distribution of our sample mean will be normally distributed.
For samples of size five and size 50 from this distribution, the histograms in Figure 5 show much less skewness. Therefore, some parametric
tests may be appropriate with this data set. When the sample is small
and the data are skewed, for example, a test based on the median rather
than the mean may be appropriate. The median always will represent the
center of the distribution while the mean will be influenced (pulled in the
direction) by the extreme skewness of the data.

Caveats and conclusions

If our sample size exceeds 30, can we always assume the CLT and use our
parametric statistical results? Probably, but not always.
First, we should look at the data. If our key variable shows a strong
bimodal distribution, using the normal distribution will mask the real
differences in the two peaks in our data.
Next, if we are fitting models, look at the distributions of the independent and dependent variables and how they are related. If we are
predicting ordinal or categorical variables, the normal distribution may

Julia E. Seaman is research
director of the Quahog
Research Group and a
statistical consultant for the
Babson Survey Research Group
at Babson College in Wellesley,
MA. She earned her doctorate
in pharmaceutical chemistry
and pharmacogenomics from
the University of California, San
Francisco.

I. Elaine Allen is professor of
biostatistics at the University
of California, San Francisco,
and emeritus professor of
statistics at Babson College.
She is also director of the
Babson Survey Research
Group. She earned a doctorate
in statistics from Cornell
University in Ithaca, NY. Allen
is a member of ASQ.

not be appropriate regardless of the
sample size.
Finally, with small samples, use
parametric (based on the normal distribution) and nonparametric methods.
Do they agree? Great! If not, look at
your data more closely to understand
these differences and which method
assumptions are most accurate for the
data.
REFERENCES
1. William J. Adams, The Life and Times of
the Central Limit Theorem, second edition.
American Math Society, 2009.
2. Max Mether, "A History of the Central
Limit Theorem," Specialized Applied
Mathematics, 2003, https://tinyurl.com/
mether-clt-history.
3. Hans Fischer, A History of the Central
Limit Theorem: From Classical to Modern
Probability Theory, Springer, 2011.
4. D.R. Bellhouse, "The Central Limit Theorem
Under Simple Random Sampling,"
American Statistician, Vol. 55, No. 4, 2001.
5. L. Le Cam, "The Central Limit Theorem
Around 1935," Statistical Science, Vol. 1, No.
1, 1986, pp. 78-96.
6. Wikipedia, "ISSS608 2016-17 T1
Franky Eddy," https://tinyurl.com/
wiki-franky-eddy.
7. Casey Dunn, "As 'Normal' as Rabbits'
Weights and Dragons' Wings," New York
Times, Sept. 23, 2013, https://tinyurl.com/
nyt-casey-dunn.
8. Michelle Paret, "Explaining the Central
Limit Theorem With Bunnies and Dragons,"
Minitab Blog, Oct. 15, 2013, https://tinyurl.
com/minitab-blog-clt.

Samuel Zetumer is a medical
degree candidate at the
University of California,
San Francisco, and a tutor
and teaching assistant
in biostatistics in the
department of epidemiology
and biostatistics. He earned
a bachelor's degree in
mathematics from Princeton
University in New Jersey.

qualityprogress.com ❘ February 2018

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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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