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Introduction to
Basic Statistical
Methodology
CHAPTER 1
~ Introduction ~
What is “random variation” in the distribution of a population?
Examples: Toasting time, Temperature settings, etc.…
POPULATION 1: Little to no variation (e.g., product manufacturing)
In engineering situations such as this, we
try to maintain “quality control”… i.e.,
“tight tolerance levels,” high precision,
low variability.
But what about a population of, say, people?
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What is “random variation” in the distribution of a population?
Examples: Body Temperature (F)
POPULATION 1: Little to no variation (e.g., clones)
Most individual values ≈ population mean value
Density
Very little variation
about the mean!
98.6 F
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What is “random variation” in the distribution of a population?
Examples: Gender,
Race, Age,(Height,
Drug Response (e.g., cholesterol level),…
Body Temperature
F)
POPULATION 2: Much variation (more realistic)
Density
Much more
variation about
the mean!
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What are “statistics,” and how can they be applied to real issues?
•
Example: Suppose a certain company insists that it complies with “gender equality”
regulations among its employee population, i.e., approx. 50% male and 50% female.
To test this claim, let us select a random sample of
n = 100 employees, and count X = the number of
males. (If the claim is true, then we expect X  50.)


etc.
 


X = 64 males
(+ 36 females)
Questions:
If the claim is true, how likely is this
experimental result? (“p-value”)
Could the difference (14 males) be
due to random chance variation, or
is it statistically significant?
GLOBAL OPERATION
DYNAMICS, INC.
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The experiment in this problem can be modeled by a random sequence of n = 100
......
independent coin tosses (Heads = Male, Tails = Female).
It can be mathematically proved that, if the coin is “fair” (“unbiased”), then in 100 tosses:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
probability of obtaining at…..from
least 0 Heads
away
from 50 is = 1.0000 “certainty”
0 to 100
Heads…..
probability of obtaining at least 1 Head away from 50 is = 0.9204
probability of obtaining at least 2 Heads away from 50 is = 0.7644
probability of obtaining at least 3 Heads away from 50 is = 0.6173
probability of obtaining at least 4 Heads away from 50 is = 0.4841
The  = .05
probability of obtaining at least 5 Heads away from 50 is = 0.3682
cutoff is
probability of obtaining at least 6 Heads away from 50 is = 0.2713
called the
probability of obtaining at least 7 Heads away from 50 is = 0.1933
significance
probability of obtaining at least 8 Heads away from 50 is = 0.1332
level.
probability of obtaining at least 9 Heads away from 50 is = 0.0886
probability of obtaining at least 10 Heads away from 50 is = 0.0569
probability of obtaining at least 11 Heads away from 50 is = 0.0352
probability of obtaining at least 12 Heads away from 50 is = 0.0210
probability of obtaining at least 13 Heads away from 50 is = 0.0120
0.0066 is called
probability of obtaining at least 14 Heads away from 50 is = 0.0066
the p-value of
etc.  0
the sample.
Because our p-value (.0066) is less than the significance level (.05),
our data suggest that the coin is indeed biased, in favor of Heads.
Likewise, our evidence suggests that employee gender in this
company is biased, in favor of Males.
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What are “statistics,” and how can they be applied to real issues?
•
Example: Suppose a certain company insists that it complies with “gender equality”
regulations among its employee population, i.e., approx. 50% male and 50% female.
HYPOTHESIS
EXPERIMENT
To test this claim, let us select a random sample of
n = 100 employees, and count X = the number of
males. (If the claim is true, then we expect X  50.)


etc. 





OBSERVATIONS
X = 64 males
(+ 36 females)
Questions:
If the claim is true, how likely is this
experimental result? (“p-value”)
Could the difference (14 males) be
due to random chance variation, or
is it statistically significant?
GLOBAL OPERATION
DYNAMICS, INC.
8
The experiment in this problem can be modeled by a random sequence of n = 100
......
independent coin tosses (Heads = Male, Tails = Female).
It can be mathematically proved that, if the coin is “fair” (“unbiased”), then in 100 tosses:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
probability of obtaining at least 0 Heads away from 50 is = 1.0000 “certainty”
probability of obtaining at least 1 Head away from 50 is = 0.9204
probability of obtaining at least 2 Heads away from 50 is = 0.7644
probability of obtaining at least 3 Heads away from 50 is = 0.6173
probability of obtaining at least 4 Heads away from 50 is = 0.4841
The  = .05
probability of obtaining at least 5 Heads away from 50 is = 0.3682
cutoff is
probability of obtaining at least 6 Heads away from 50 is = 0.2713
called the
probability of obtaining at least 7 Heads away from 50 is = 0.1933
significance
probability of obtaining at least 8 Heads away from 50 is = 0.1332
level.
probability of obtaining at least 9 Heads away from 50 is = 0.0886
probability of obtaining at least 10 Heads away from 50 is = 0.0569
probability of obtaining at least 11 Heads away from 50 is = 0.0352
ANALYSIS
probability of obtaining at least 12 Heads away from 50 is = 0.0210
probability of obtaining at least 13 Heads away from 50 is = 0.0120
0.0066 is called
probability of obtaining at least 14 Heads away from 50 is = 0.0066
the p-value of
etc.  0
the sample.
Because our p-value (.0066) is less than the significance level (.05),
our data suggest that the coin is indeed biased, in favor of Heads.
Likewise, our evidence suggests that employee gender in this
company is biased, in favor of Males.
CONCLUSION
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“Classical Scientific Method”

Hypothesis – Define the study population...
What’s the question?

Experiment – Designed to test hypothesis

Observations – Collect sample measurements

Analysis – Do the data formally tend to
support or refute the hypothesis, and with
what strength? (Lots of juicy formulas...)

Conclusion – Reject or retain hypothesis; is
the result statistically significant?

Interpretation – Translate findings in context!
Statistics is implemented in each step of the
classical scientific method!
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