Transcript Document

7/18/2015
Understanding Variability
Instructor: Ron S. Kenett
Email: [email protected]
Course Website: www.kpa.co.il/biostat
Course textbook: MODERN INDUSTRIAL STATISTICS,
Kenett and Zacks, Duxbury Press, 1998
(c) 2000, Ron S. Kenett, Ph.D.
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Course Syllabus
•Understanding Variability
•Variability in Several Dimensions
•Basic Models of Probability
•Sampling for Estimation of Population Quantities
•Parametric Statistical Inference
•Computer Intensive Techniques
•Multiple Linear Regression
•Statistical Process Control
•Design of Experiments
(c) 2000, Ron S. Kenett, Ph.D.
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Discrete Data
A set of data is said to be discrete if the values / observations
belonging to it are distinct and separate. That is, they can be counted
(1,2,3,.......). For example, the number of kittens in a litter; the
number of patients in a doctors surgery; the number of flaws in one
metre of cloth; gender (male, female); blood group (O, A, B, AB).
(c) 2000, Ron S. Kenett, Ph.D.
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Continuous Data
A set of data is said to be continuous if the values / observations
belonging to it may take on any value within a finite or infinite interval.
You can count, order and measure continuous data. For example,
height; weight; temperature; the amount of sugar in an orange; the
time required to run a mile.
(c) 2000, Ron S. Kenett, Ph.D.
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Types of Variables

Qualitative Variables

Attributes, categories

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Examples: male/female, registered to vote/not,
ethnicity, eye color....
Quantitative Variables

Discrete - usually take on integer values but can
take on fractions when variable allows - counts, how
many

Continuous - can take on any value at any point
along an interval - measurements, how much
(c) 2000, Ron S. Kenett, Ph.D.
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Self Assessment Test
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For each of the following,
indicate whether the appropriate variable
would be qualitative or quantitative.
If the variable is quantitative, indicate
whether it would be discrete or continuous.
(c) 2000, Ron S. Kenett, Ph.D.
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Self Assessment Test



a) Whether you own
an RCA Colortrak
television set
b) Your status as a
full-time or a parttime student
c) Number of people
who attended your
school’s graduation
last year
(c) 2000, Ron S. Kenett, Ph.D.
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Qualitative Variable



two levels: yes/no
no measurement
Qualitative Variable

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two levels: full/part
no measurement
Quantitative, Discrete
Variable


a countable number
only whole numbers
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Self Assessment Test

d) The price of
your most recent
haircut

Quantitative, Discrete
Variable



e) Sam’s travel
time from his
dorm to the
Student Union

a countable number
only whole numbers
Quantitative, Continuous
Variable



(c) 2000, Ron S. Kenett, Ph.D.
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any number
time is measured
can take on any value greater
than zero
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Self Assessment Test

f) The number of
students on
campus who
belong to a social
fraternity or
sorority
(c) 2000, Ron S. Kenett, Ph.D.
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Quantitative, Discrete
Variable


a countable number
only whole numbers
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Scales of Measurement

Nominal Scale -
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Labels represent various levels of a
categorical variable.

Ordinal Scale -

Interval Scale -

Ratio Scale -
Labels represent an order that
indicates either preference or ranking.
Numerical labels indicate order and
distance between elements. There is no absolute zero
and multiples of measures are not meaningful.
Numerical labels indicate order and
distance between elements. There is an absolute zero
and multiples of measures are meaningful.
(c) 2000, Ron S. Kenett, Ph.D.
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Self Assessment Test
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Bill scored 1200 on the Scholastic Aptitude
Test and entered college as a physics
major. As a freshman, he changed to
business because he thought it was more
interesting. Because he made the dean’s
list last semester, his parents gave him $30
to buy a new Casio calculator. Identify at
least one piece of information in the:
(c) 2000, Ron S. Kenett, Ph.D.
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Self Assessment Test

a) nominal
scale of
measurement.
(c) 2000, Ron S. Kenett, Ph.D.
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1. Bill is going to college.
2. Bill will buy a Casio
calculator.
3. Bill was a physics major.
4. Bill is a business major.
5. Bill was on the dean’s list.
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Self Assessment Test



b) ordinal scale of
measurement
c) interval scale of
measurement
d) ratio scale of
measurement
(c) 2000, Ron S. Kenett, Ph.D.
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
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Bill is a freshman.
Bill earned a 1200 on the
SAT.
Bill’s parents gave him $30.
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Self Assessment Test



b) ordinal scale of
measurement
c) interval scale of
measurement
d) ratio scale of
measurement
(c) 2000, Ron S. Kenett, Ph.D.


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Bill is a freshman.
Bill earned a 1200 on the
SAT.
Bill’s parents gave him $30.
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Histogram
A histogram is a way of summarising data that are measured on
an interval scale (either discrete or continuous). It is often used in
exploratory data analysis to illustrate the major features of the
distribution of the data in a convenient form. It divides up the range
of possible values in a data set into classes or groups. For each
group, a rectangle is constructed with a base length equal to the
range of values in that specific group, and an area proportional to
the number of observations falling into that group. This means that
the rectangles might be drawn of non-uniform height.
(c) 2000, Ron S. Kenett, Ph.D.
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Key Terms

Data array
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An orderly presentation of data in either
ascending or descending numerical order.
Frequency Distribution

A table that represents the data in classes
and that shows the number of observations
in each class.
(c) 2000, Ron S. Kenett, Ph.D.
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Key Terms

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Frequency Distribution





Class - The category
Frequency - Number in each class
Class limits - Boundaries for each class
Class interval - Width of each class
Class mark - Midpoint of each class
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Sturge’s Rule

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How to set the approximate number of
classes to begin constructing a
frequency distribution.
where k = approximate number of classes to use and
n = the number of observations in the data set .
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Frequency Distributions
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1. Number of classes
Choose an approximate number of classes for your data.
Sturges’ rule can help.
2. Estimate the class interval
Divide the approximate number of classes (from Step 1)
into the range of your data to find the approximate class
interval, where the range is defined as the largest data value
minus the smallest data value.
3. Determine the class interval
Round the estimate (from Step 2) to a convenient value.
(c) 2000, Ron S. Kenett, Ph.D.
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Frequency Distributions
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4. Lower Class Limit
Determine the lower class limit for the first class by
selecting a convenient number that is smaller than the
lowest data value.
5. Class Limits
Determine the other class limits by repeatedly adding the
class width (from Step 2) to the prior class limit, starting
with the lower class limit (from Step 3).
6. Define the classes
Use the sequence of class limits to define the classes.
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Relative Frequency Distributions
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1. Retain the same classes defined in the
frequency distribution.
2. Sum the total number of observations
across all classes of the frequency
distribution.
3. Divide the frequency for each class by the
total number of observations, forming the
percentage of data values in each class.
(c) 2000, Ron S. Kenett, Ph.D.
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Cumulative Relative Frequency
Distributions
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1. List the number of observations in the lowest
class.
2. Add the frequency of the lowest class to the
frequency of the second class. Record that
cumulative sum for the second class.
3. Continue to add the prior cumulative sum to the
frequency for that class, so that the cumulative
sum for the final class is the total number of
observations in the data set.
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Cumulative Relative Frequency
Distributions
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4. Divide the accumulated frequencies for each
class by the total number of observations -giving you the percent of all observations that
occurred up to an including that class.

An Alternative: Accrue the relative frequencies
for each class instead of the raw frequencies.
Then you don’t have to divide by the total to get
percentages.
(c) 2000, Ron S. Kenett, Ph.D.
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Example

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The average daily cost to community hospitals for
patient stays during 1993 for each of the 50 U.S.
states was given in the next table.





a) Arrange these into a data array.
b) Construct a stem-and-leaf display.
*) Approximately how many classes would be appropriate
for these data?
c & d) Construct a frequency distribution. State interval
width and class mark.
e) Construct a histogram, a relative frequency distribution,
and a cumulative relative frequency distribution.
(c) 2000, Ron S. Kenett, Ph.D.
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Example –Data List
AL
AK
AZ
AR
CA
CO
CT
DE
FL
GA
$775
1,136
1,091
678
1,221
961
1,058
1,024
960
775
HI
ID
IL
IN
IA
KS
KY
LA
ME
MD
(c) 2000, Ron S. Kenett, Ph.D.
823
659
917
898
612
666
703
875
738
889
MA 1,036
MI
902
MN 652
MS
555
MO 863
MT
482
NE
626
NV 900
NH 976
NJ
829
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NM 1,046
NY
784
NC
763
ND
507
OH
940
OK
797
OR 1,052
PA
861
RI
885
SC
838
SD
506
TN
859
TX 1,010
UT 1,081
VT
676
VA
830
WA 1,143
WV
701
WI
744
WY 537
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Example – Data Array
CA 1,221
WA 1,143
AK 1,136
AZ 1,091
UT 1,081
CT 1,058
OR 1,052
NM 1,046
MA 1,036
DE 1,024
TX 1,010
NH 976
CO
961
FL
960
CH 940
IL
917
MI
902
NV 900
IN
898
MD 889
(c) 2000, Ron S. Kenett, Ph.D.
RI
LA
MO
PA
TN
SC
VA
NJ
HI
OK
885
875
863
861
859
838
830
829
823
797
NY
AL
GA
NC
WI
ME
KY
WV
AR
VT
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784
775
775
763
744
738
703
701
678
676
KS
666
ID
659
MN 652
NE 626
IA
612
MS 555
WY 537
ND 507
SD
506
MT 482
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Example – Stem and Leaf Display
Stem-and-Leaf Display
Leaf Unit: 100
1 12
2 11
8 10
7
9
(11) 8
9
7
7
6
4
5
1
4
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N = 50
21
43, 36
91, 81, 58, 52, 46, 36, 24, 10
76, 61, 60, 40, 17, 02, 00
98, 89, 85, 75, 63, 61, 59, 38, 30, 29, 23
97, 84, 75, 75, 63, 44, 38, 03, 01
78, 76, 66, 59, 52, 26, 12
55, 37, 07, 06
82
Range: $482 - $1,221
(c) 2000, Ron S. Kenett, Ph.D.
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Example – Frequency Distribution

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To approximate the number of classes
we should use in creating the frequency
distribution, use Sturges’ Rule, n = 50:
k 13.322(log n)13.322(log 50)
10
10
13.322(1.69897)15.6446.6447
Sturges’ rule suggests we use
approximately 7 classes.
(c) 2000, Ron S. Kenett, Ph.D.
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Example – Frequency Distribution

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Step 1. Number of classes

Sturges’ Rule: approximately 7 classes.
The range is: $1,221 – $482 = $739
$739/7 $106 and $739/8 $92

Steps 2 & 3. The Class Interval

So, if we use 8 classes, we can make each
class $100 wide.
(c) 2000, Ron S. Kenett, Ph.D.
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Example – Frequency Distribution

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Step 1. Number of classes

Sturges’ Rule: approximately 7 classes.
The range is: $1,221 – $482 = $739
$739/7 $106 and $739/8 $92

Steps 2 & 3. The Class Interval

So, if we use 8 classes, we can make each
class $100 wide.
(c) 2000, Ron S. Kenett, Ph.D.
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Example – Frequency Distribution

Step 4. The Lower Class Limit


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If we start at $450, we can cover the range in 8 classes,
each class $100 in width.
The first class : $450 up to $550
Steps 5 & 6. Setting Class Limits
$450
$550
$650
$750
up
up
up
up
(c) 2000, Ron S. Kenett, Ph.D.
to
to
to
to
$550
$650
$750
$850
$850 up to $950
$950 up to $1,050
$1,050 up to $1,150
$1,150 up to $1,250
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Example – Frequency Distribution
Average daily cost
$450 – under $550
$550 – under $650
$650 – under $750
$750 – under $850
$850 – under $950
$950 – under $1,050
$1,050 – under $1,150
$1,150 – under $1,250
Number
4
3
9
9
11
7
6
1
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Mark
$500
$600
$700
$800
$900
$1,000
$1,100
$1,200
Interval width: $100
(c) 2000, Ron S. Kenett, Ph.D.
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Example – Histogram
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12
10
8
6
4
2
0
500
(c) 2000, Ron S. Kenett, Ph.D.
600
700
800
900
1000 1100 1200
33
Example – Relative Frequency
Distribution
Average daily cost
$450 – under $550
$550 – under $650
$650 – under $750
$750 – under $850
$850 – under $950
$950 – under $1,050
$1,050 – under $1,150
$1,150 – under $1,250
(c) 2000, Ron S. Kenett, Ph.D.
Number
4
3
9
9
11
7
6
1
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Rel. Freq.
4/50 = .08
3/50 = .06
9/50 = .18
9/50 = .18
11/50 = .22
7/50 = .14
6/50 = .12
1/50 = .02
34
Example – Polygon
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0.25
0.2
0.15
0.1
0.05
0
0
200
(c) 2000, Ron S. Kenett, Ph.D.
400
600
800
1000
1200
1400
35
Example – Cumulative Frequency
Distribution
Average daily cost
Number
$450 – under $550
4
$550 – under $650
3
$650 – under $750
9
$750 – under $850
9
$850 – under $9
11
$950 – under $1,050
7
$1,050 – under $1,150
6
$1,150 – under $1,250
1
(c) 2000, Ron S. Kenett, Ph.D.
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Cum. Freq.
4
7
16
25
36
43
49
50
36
Example – Cumulative Relative
Frequency Distribution
Average daily cost
Cum.Freq.
$450 – under $550
4
$550 – under $650
7
$650 – under $750
16
$750 – under $850
25
$850 – under $950
36
$950 – under $1,050
43
$1,050 – under $1,150
49
$1,150 – under $1,250
50
(c) 2000, Ron S. Kenett, Ph.D.
7/18/2015
Cum.Rel.Freq.
4/50 = .02
7/50 = .14
16/50 = .32
25/50 = .50
36/50 = .72
43/50 = .86
49/50 = .98
50/50 = 1.00
37
Example – Percentage Ogive
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50
45
40
35
30
25
20
15
10
5
0
0
200
(c) 2000, Ron S. Kenett, Ph.D.
400
600
800
1000
1200
38
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(c) 2000, Ron S. Kenett, Ph.D.
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Key Terms

Measures of
Central
Tendency,
The Center
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




(c) 2000, Ron S. Kenett, Ph.D.
Mean
µ, population; x , sample
Weighted Mean
Median
Mode
40
Key Terms

Measures of
Dispersion,
The
Spread
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






(c) 2000, Ron S. Kenett, Ph.D.
Range
Mean absolute deviation
Variance
Standard deviation
Interquartile range
Interquartile deviation
Coefficient of variation
41
Key Terms

Measures of
Relative
Position
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
Quantiles





(c) 2000, Ron S. Kenett, Ph.D.
Quartiles
Deciles
Percentiles
Residuals
Standardized values
42
The Mean

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Mean

Arithmetic average = (sum all values)/# of values


Population: µ = (Sxi)/N
Sample: x = (Sxi)/n
Problem: Calculate the average number of truck shipments
from the United States to five Canadian cities for the
following data given in thousands of bags:
Montreal, 64.0; Ottawa, 15.0; Toronto, 285.0;
Vancouver, 228.0; Winnipeg, 45.0
(Ans: 127.4)
(c) 2000, Ron S. Kenett, Ph.D.
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The Weighted Mean

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When what you have is grouped data,
compute the mean using µ = (Swixi)/Swi
Problem: Calculate the average profit from truck shipments,
United States to Canada, for the following data given in
thousands of bags and profits per thousand bags:
Montreal 64.0
Ottawa 15.0
Toronto 285.0
$15.00
$13.50
$15.50
Vancouver 228.0
Winnipeg 45.0
$12.00
$14.00
(Ans: $14.04 per thous. bags)
(c) 2000, Ron S. Kenett, Ph.D.
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The Median

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To find the median:
1. Put the data in an array.
2A. If the data set has an ODD number of numbers, the
median is the middle value.
2B. If the data set has an EVEN number of numbers, the
median is the AVERAGE of the middle two values.
(Note that the median of an even set of data values is not
necessarily a member of the set of values.)

The median is particularly useful if there are outliers
in the data set, which otherwise tend to sway the
value of an arithmetic mean.
(c) 2000, Ron S. Kenett, Ph.D.
45
The Mode


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The mode is the most frequent value.
While there is just one value for the mean
and one value for the median, there may be
more than one value for the mode of a data
set.
The mode tends to be less frequently used
than the mean or the median.
12
10
8
6
4
2
0
500
(c) 2000, Ron S. Kenett, Ph.D.
600
700
800
900
1000
1100
1200
46
Comparing Measures of Central
Tendency


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If mean = median = mode, the shape of the distribution is
symmetric.
If mode < median < mean or if mean > median > mode,
the shape of the distribution trails to the right,
is positively skewed.
If mean < median < mode or if mode > median > mean,
the shape of the distribution trails to the left,
is negatively skewed.
(c) 2000, Ron S. Kenett, Ph.D.
47
The Range



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The range is the distance between the
smallest and the largest data value in the
set.
Range = largest value – smallest value
Sometimes range is reported as an
interval, anchored between the smallest
and largest data value, rather than the
actual width of that interval.
(c) 2000, Ron S. Kenett, Ph.D.
48
Residuals
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Residuals are the differences between
each data value in the set and the
group mean:


for a population, xi – µ
for a sample, xi – x
(c) 2000, Ron S. Kenett, Ph.D.
49
The MAD

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The mean absolute deviation is found
by summing the absolute values of all
residuals and dividing by the number of
values in the set:
for a population, MAD = (S|xi – µ|)/N
for a sample, MAD = (S|xi – x |)/n
(c) 2000, Ron S. Kenett, Ph.D.
50
The Variance

Variance is one of the most frequently used
measures of spread,



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S(x –)2 S(x )2 – N2
 i
for population,  2  i
N
N
S(x – x)2 S(x )2 – nx 2
i
i
for sample, s2 

n –1
n–1
The right side of each equation is often used
as a computational shortcut.
(c) 2000, Ron S. Kenett, Ph.D.
51
The Standard Deviation

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Since variance is given in squared units,
we often find uses for the standard
deviation, which is the square root of
variance:
2
 for a population,   

for a sample, s s2
(c) 2000, Ron S. Kenett, Ph.D.
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Quartiles



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One of the most frequently used quantiles is the
quartile.
Quartiles divide the values of a data set into four
subsets of equal size, each comprising 25% of the
observations.
To find the first, second, and third quartiles:




1.
2.
3.
4.
Arrange the N data values into an array.
First quartile, Q1 = data value at position (N + 1)/4
Second quartile, Q2 = data value at position 2(N + 1)/4
Third quartile, Q3 = data value at position 3(N + 1)/4
(c) 2000, Ron S. Kenett, Ph.D.
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Quartiles
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Cumulative Frequency
100
75
50
25
0
0.0
Q1
1.5
Q2
Q3
3.0
4.5
6.0
Ln_YarnS
(c) 2000, Ron S. Kenett, Ph.D.
54
Standardized Values

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How far above or below the individual value is
compared to the population mean in units of
standard deviation


“How far above or below” (data value – mean)
which is the residual...
“In units of standard deviation” divided by 
Standardized individual value:
x –  z

A negative z means the data value falls below the mean.
(c) 2000, Ron S. Kenett, Ph.D.
55