Download The Empirical Guidelines.

The Normal Distribution
I know of scarcely anything so apt to impress the
imagination as the wonderful form of cosmic order
expressed by the “Law of Frequency of Error.”
The law would have been personified by the
Greeks and deified, if they had known it. It reigns
with serenity and in complete self-effacement,
amidst the wildest confusion. The huger the mob,
and the greater the apparent anarchy, the more
perfect is its sway. It is the supreme law of
Unreason. Whenever a large sample of chaotic
elements are taken in hand and marshaled in the
order of their magnitude, an unsuspected and
most beautiful form of regularity proves to have
been latent all along.

Sir Francis Galton (1822–1911)
The Normal Distribution
THE
NORMAL
LAW OF ERROR
STANDS OUT IN THE
EXPERIENCE OF MANKIND
AS ONE OF THE BROADEST
GENERALIZATIONS OF NATURAL
PHILOSOPHY IT SERVES AS THE
GUIDING INSTRUMENT IN RESEARCHES
IN THE PHYSICAL AND SOCIAL SCIENCES AND
IN MEDICINE AGRICULTURE AND ENGINEERING
IT IS AN INDISPENSABLE TOOL FOR THE ANALYSIS AND THE
INTERPRETATION OF THE BASIC DATA OBTAINED BY OBSERVATION AND EXPERIMENT
W. J. Youden
Standard Deviation

Standard Deviation


A measure of variability
Populations below have mean=50 and
standard deviations 10 and 15 respectively.
The Empirical Guidelines
Empirical Guidelines for Bell-Shaped Curves
Chebyshev's Theorem
Example

In Brazil the average height of adult males
is 63.5 inches with a standard deviation of
just under 1.8 inches.

What percentage of adult males in Brazil
are between 5’ and 5’7”?

Someone who is 5’ 7” tall is taller than
___% of adult males in Brazil?

In Brazil the average height of adult males is
63.5 inches with a standard deviation of just
under 1.8 inches.

What percentage of adult males in Brazil are between 5’
and 5’ 7”?

This is very close to the interval x-2s to x+2s

According to Chebyshev’s at least 75%.

If the distribution of heights is bell-shaped (which is likely)
then the Emperical Guidelines suggest 95%

Someone who is 5’ 7” tall is taller than ___% of adult males
in Brazil?
Using Chebyshev’s at least 75% and likely and additional
(25/2) = 87%.
Using a normal curve only about two percent of the males
are taller so about 98%.


Open Worksheet in Excel, then enter
mean and standard deviation
mean
100
70
standard deviation
80
90
10
100
110
120
130
Adolphe Quetelet (1833)
Quetelet was one of the first to observe
the bell-shaped curve was “natural”.
 One of the data sets he worked with was
the data on 5732 Scottish Militia. A
portion of that data appears in the
activity.
 We will examine how accurately the data
can be approximated with the Empirical
Guidelines.

Adolphe Quetelet’s data of 5732 Scottish Militia
Height and Chest Diameter
72-73
70-71
68-69
66-67
64-65
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47