1. Understanding Variability
3/25/2017
Understanding Variability
Instructor: Ron S. Kenett
Course Website:
Course textbook: MODERN INDUSTRIAL STATISTICS,
Kenett and Zacks, Duxbury Press, 1998
(c) 2000, Ron S. Kenett, Ph.D.

2. Understanding Variability Variability in Several Dimensions
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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.

3. A set of data is said to be discrete if the values / observations
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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.

4. A set of data is said to be continuous if the values / observations
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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.

5. Types of Variables Qualitative Variables Quantitative Variables
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Types of Variables
Qualitative Variables
Attributes, categories
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.

6. Self Assessment Test For each of the following,
3/25/2017
Self Assessment Test
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.

7. 3/25/2017
Self Assessment Test
a) Whether you own an RCA Colortrak television set
b) Your status as a full-time or a part-time student
c) Number of people who attended your school’s graduation last year
Qualitative Variable
two levels: yes/no
no measurement
two levels: full/part
Quantitative, Discrete Variable
a countable number
only whole numbers
(c) 2000, Ron S. Kenett, Ph.D.

8. Self Assessment Test d) The price of your most recent haircut
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Self Assessment Test
d) The price of your most recent haircut
e) Sam’s travel time from his dorm to the Student Union
Quantitative, Discrete Variable
a countable number
only whole numbers
Quantitative, Continuous Variable
any number
time is measured
can take on any value greater than zero
(c) 2000, Ron S. Kenett, Ph.D.

9. 3/25/2017
Self Assessment Test
f) The number of students on campus who belong to a social fraternity or sorority
Quantitative, Discrete Variable
a countable number
only whole numbers
(c) 2000, Ron S. Kenett, Ph.D.

10. 3/25/2017
Scales of Measurement
Nominal Scale - Labels represent various levels of a categorical variable.
Ordinal Scale - Labels represent an order that indicates either preference or ranking.
Interval Scale - Numerical labels indicate order and distance between elements. There is no absolute zero and multiples of measures are not meaningful.
Ratio Scale - 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.

11. 3/25/2017
Self Assessment Test
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.

12. Self Assessment Test a) nominal scale of measurement.
3/25/2017
Self Assessment Test
a) nominal scale of measurement.
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.
(c) 2000, Ron S. Kenett, Ph.D.

13. Self Assessment Test b) ordinal scale of measurement
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Self Assessment Test
b) ordinal scale of measurement
c) interval scale of measurement
d) ratio scale of measurement
Bill is a freshman.
Bill earned a 1200 on the SAT.
Bill’s parents gave him $30.
(c) 2000, Ron S. Kenett, Ph.D.

14. Self Assessment Test b) ordinal scale of measurement
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Self Assessment Test
b) ordinal scale of measurement
c) interval scale of measurement
d) ratio scale of measurement
Bill is a freshman.
Bill earned a 1200 on the SAT.
Bill’s parents gave him $30.
(c) 2000, Ron S. Kenett, Ph.D.

15. A histogram is a way of summarising data that are measured on
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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.

16. Key Terms Data array Frequency Distribution
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Key Terms
Data array
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.

17. Key Terms Frequency Distribution Class - The category
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Key Terms
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
(c) 2000, Ron S. Kenett, Ph.D.

18. 3/25/2017
Sturge’s Rule
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 .
(c) 2000, Ron S. Kenett, Ph.D.

19. Frequency Distributions
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Frequency Distributions
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.

20. Frequency Distributions
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Frequency Distributions
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.
(c) 2000, Ron S. Kenett, Ph.D.

21. Relative Frequency Distributions
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Relative Frequency Distributions
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.

22. Cumulative Relative Frequency Distributions
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Cumulative Relative Frequency Distributions
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.
(c) 2000, Ron S. Kenett, Ph.D.

23. Cumulative Relative Frequency Distributions
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Cumulative Relative Frequency Distributions
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.

24. 3/25/2017
Example
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.

25. Example –Data List AL $775 HI 823 MA 1,036 NM 1,046 SD 506
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Example –Data List
AL $775 HI MA 1,036 NM 1,046 SD
AK 1,136 ID MI NY TN
AZ 1,091 IL MN NC TX 1,010
AR IN MS ND UT 1,081
CA 1,221 IA MO OH VT
CO KS MT OK VA
CT 1,058 KY NE OR 1,052 WA 1,143
DE 1,024 LA NV PA WV
FL ME NH RI WI
GA MD NJ SC WY
(c) 2000, Ron S. Kenett, Ph.D.

26. Example – Data Array CA 1,221 TX 1,010 RI 885 NY 784 KS 666
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Example – Data Array
CA 1,221 TX 1,010 RI NY KS
WA 1,143 NH LA AL ID
AK 1,136 CO MO GA MN 652
AZ 1,091 FL PA NC NE
UT 1,081 CH TN WI IA
CT 1,058 IL SC ME MS
OR 1,052 MI VA KY WY 537
NM 1,046 NV NJ WV ND
MA 1,036 IN HI AR SD
DE 1,024 MD OK VT MT
(c) 2000, Ron S. Kenett, Ph.D.

27. Example – Stem and Leaf Display
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Example – Stem and Leaf Display
Stem-and-Leaf Display N = 50
Leaf Unit: 100
, 36
, 81, 58, 52, 46, 36, 24, 10
, 61, 60, 40, 17, 02, 00
(11) , 89, 85, 75, 63, 61, 59, 38, 30, 29, 23
, 84, 75, 75, 63, 44, 38, 03, 01
, 76, 66, 59, 52, 26, 12
, 37, 07, 06
Range: $482 - $1,221
(c) 2000, Ron S. Kenett, Ph.D.

28. Example – Frequency Distribution
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Example – Frequency Distribution
To approximate the number of classes we should use in creating the frequency distribution, use Sturges’ Rule, n = 50:
Sturges’ rule suggests we use approximately 7 classes.
(c) 2000, Ron S. Kenett, Ph.D.

29. Example – Frequency Distribution
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Example – Frequency Distribution
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.

30. Example – Frequency Distribution
3/25/2017
Example – Frequency Distribution
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.

31. Example – Frequency Distribution
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Example – Frequency Distribution
Step 4. The Lower Class Limit
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 up to $550 $850 up to $950
$550 up to $650 $950 up to $1,050
$650 up to $ $1,050 up to $1,150
$750 up to $ $1,150 up to $1,250
(c) 2000, Ron S. Kenett, Ph.D.

32. Example – Frequency Distribution
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Example – Frequency Distribution
Average daily cost Number Mark
$450 – under $ $500
$550 – under $ $600
$650 – under $ $700
$750 – under $ $800
$850 – under $ $900
$950 – under $1, $1,000
$1,050 – under $1, $1,100
$1,150 – under $1, $1,200
Interval width: $100
(c) 2000, Ron S. Kenett, Ph.D.

33. 3/25/2017
Example – Histogram
(c) 2000, Ron S. Kenett, Ph.D.

34. Example – Relative Frequency Distribution
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Example – Relative Frequency Distribution
Average daily cost Number Rel. Freq.
$450 – under $ /50 = .08
$550 – under $ /50 = .06
$650 – under $ /50 = .18
$750 – under $ /50 = .18
$850 – under $ /50 = .22
$950 – under $1, /50 = .14
$1,050 – under $1, /50 = .12
$1,150 – under $1, /50 = .02
(c) 2000, Ron S. Kenett, Ph.D.

35. 3/25/2017
Example – Polygon
(c) 2000, Ron S. Kenett, Ph.D.

36. Example – Cumulative Frequency Distribution
3/25/2017
Example – Cumulative Frequency Distribution
Average daily cost Number Cum. Freq.
$450 – under $
$550 – under $
$650 – under $
$750 – under $
$850 – under $
$950 – under $1,
$1,050 – under $1,
$1,150 – under $1,
(c) 2000, Ron S. Kenett, Ph.D.

37. Example – Cumulative Relative Frequency Distribution
3/25/2017
Example – Cumulative Relative Frequency Distribution
Average daily cost Cum.Freq Cum.Rel.Freq.
$450 – under $ /50 = .02
$550 – under $ /50 = .14
$650 – under $ /50 = .32
$750 – under $ /50 = .50
$850 – under $ /50 = .72
$950 – under $1, /50 = .86
$1,050 – under $1, /50 = .98
$1,150 – under $1, /50 = 1.00
(c) 2000, Ron S. Kenett, Ph.D.

38. Example – Percentage Ogive
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Example – Percentage Ogive
(c) 2000, Ron S. Kenett, Ph.D.

39. Statistical Description of Data
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Statistical Description of Data
(c) 2000, Ron S. Kenett, Ph.D.

40. Key Terms Measures of Central Tendency, The Center Mean Weighted Mean
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Key Terms
Measures of Central Tendency,
The Center
Mean
µ, population; , sample
Weighted Mean
Median
Mode
(c) 2000, Ron S. Kenett, Ph.D.

41. Key Terms The Spread Measures of Dispersion, Range
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Key Terms
Measures of Dispersion,
The Spread
Range
Mean absolute deviation
Variance
Standard deviation
Interquartile range
Interquartile deviation
Coefficient of variation
(c) 2000, Ron S. Kenett, Ph.D.

42. Key Terms Measures of Relative Position Quantiles Residuals
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Key Terms
Measures of Relative Position
Quantiles
Quartiles
Deciles
Percentiles
Residuals
Standardized values
(c) 2000, Ron S. Kenett, Ph.D.

43. The Mean Mean Arithmetic average = (sum all values)/# of values
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The Mean
Mean
Arithmetic average = (sum all values)/# of values
Population: µ = (Sxi)/N
Sample: = (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.

44. 3/25/2017
The Weighted Mean
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 Ottawa Toronto
$ $ $15.50
Vancouver Winnipeg 45.0
$ $14.00
(Ans: $14.04 per thous. bags)
(c) 2000, Ron S. Kenett, Ph.D.

45. The Median To find the median:
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The Median
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.

46. The Mode The mode is the most frequent value.
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The Mode
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.
(c) 2000, Ron S. Kenett, Ph.D.

47. Comparing Measures of Central Tendency
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Comparing Measures of Central Tendency
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.

48. 3/25/2017
The Range
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.

49. 3/25/2017
Residuals
Residuals are the differences between each data value in the set and the group mean:
for a population, xi – µ
for a sample, xi –
(c) 2000, Ron S. Kenett, Ph.D.

50. 3/25/2017
The MAD
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 – |)/n
(c) 2000, Ron S. Kenett, Ph.D.

51. 3/25/2017
The Variance
Variance is one of the most frequently used measures of spread,
for population,
for sample,
The right side of each equation is often used as a computational shortcut.
(c) 2000, Ron S. Kenett, Ph.D.

52. The Standard Deviation
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The Standard Deviation
Since variance is given in squared units, we often find uses for the standard deviation, which is the square root of variance:
for a population,
for a sample,
(c) 2000, Ron S. Kenett, Ph.D.

53. Quartiles One of the most frequently used quantiles is the quartile.
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Quartiles
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. Arrange the N data values into an array.
2. First quartile, Q1 = data value at position (N + 1)/4
3. Second quartile, Q2 = data value at position 2(N + 1)/4
4. Third quartile, Q3 = data value at position 3(N + 1)/4
(c) 2000, Ron S. Kenett, Ph.D.

54. 3/25/2017
Quartiles
(c) 2000, Ron S. Kenett, Ph.D.

55. 3/25/2017
Standardized Values
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 s
Standardized individual value:
A negative z means the data value falls below the mean.
(c) 2000, Ron S. Kenett, Ph.D.