Sample Pie Charts

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```					      STATISTICS for the Utterly
1-1
Confused, 2nd ed.

SLIDES PREPARED
By
Lloyd R. Jaisingh Ph.D.
1-2   Part 1 DESCRIPTIVE STATISTICS

Chapter 1
Graphical Displays of Univariate
Data
1-3   Outline
   Do I Need to Read This Chapter?
   1-1 Introduction
   1-2 Frequency Distributions
   1-3 Dot Plots
   1-4 Bar Charts or Bar Graphs
   1-5 Histograms
1-4   Outline
   1-6 Frequency Polygons
   1-7 Stem-and-Leaf Displays or Plots
   1-8 Time Series Graphs
   1-9 Pie Graphs or Pie Charts
   1-10 Pareto Charts
   It’s a Wrap
1-5   Objectives
   Introduction of some basic statistical
terms.
   Introduction of some graphical displays.
1-6   Introduction
   What is statistics? Statistics is the science of
collecting, organizing, summarizing,
analyzing, and making inferences from data.
   The subject of statistics is divided into two
inferential statistics.
Breakdown of the subject of statistics
1-7
Statistics

Descriptive                      Inferential
Statistics                        Statistics

Includes                        Includes
 Collecting                     Making inferences
 Organizing                     Hypothesis testing
 Summarizing                    Determining
 Presenting                      relationships
data                          Making predictions
1-8   Introduction (contd.)
   Explanation of the term data: Data are the
values or measurements that variables
describing an event can assume.
   Variables whose values are determined by
chance are called random variables.
   Types of variables – there are two types:
qualitative and quantitative.
1-9   Introduction (contd.)
   What are qualitative variables: These are
variables that are nonnumeric in nature.
   What are quantitative variables: These are
variables that can assume numeric values.
   Quantitative variables can be classified into
two groups – discrete variables and
continuous variables.
Breakdown of the types of variables
1-10
Variables

Quantitative                     Qualitative

Includes
 Discrete
 Continuous
variables
1-11     Introduction (contd.)
• What are quantitative data: These are data
values that are numeric.
• Example: the heights of female basketball
players.
• What are qualitative data: These are data
values that can be placed into distinct
categories according to some characteristic or
attribute.
• Example: the eye color of female basketball
players.
1-12   Introduction (contd.)
• What are discrete variables: These are
variables that can assume values that can be
counted.
• Example: the number of days it rained in your
neighborhood for the month of March.
• What are continuous variables: These are
variables that can assume all values between
any two values.
• Example: the time it takes to complete a quiz.
1-13    Introduction (contd.)
• In order for statisticians to do any analysis,
data must be collected or sampled.
• We can sample the entire population or just a
portion of the population.
• What is a population? A population consists of
all elements that are being studied.
• What is a sample?: A sample is a subset of the
population.
1-14     Introduction (contd.)
• Example: If we are interested in studying the
distribution of ACT math scores of freshmen at
a college, then the population of ACT math
scores will be the ACT math scores of all
freshmen at that particular college.
• Example: If we selected every tenth ACT math
scores of freshmen at a college, then this
selected set will represent a sample of ACT
math scores for the freshmen at that particular
college.
1-15   Introduction (contd.)

Population – all freshmen ACT math scores

Sample – every
10th ACT
math score
1-16   Introduction (contd.)
• What is a census? A census is a sample of
the entire population.
• Example: Every 10 years the U.S.
government gathers information from the
entire population. Since the entire
population is sampled, this is referred to
as a census.
1-17    Introduction (contd.)
• Both populations and samples have
characteristics that are associated with them.
• These are called parameters and statistics
respectively.
• A parameter is a characteristic of or a fact
• Example: The average age for the entire
student population on a campus is an example
of a parameter.
1-18    Introduction (contd.)
• A statistic is a characteristic of or a fact about
a sample.
• Example: The average ACT math score for a
sample of students on a campus is an example
of a statistic.
1-19   Introduction (contd.)

Population – Described by Parameters

Sample – Described
by Statistics
1-20   1-2 Frequency Distributions
• What is a frequency distribution? A
frequency distribution is an organization
of raw data in tabular form, using classes
(or intervals) and frequencies.
• What is a frequency count? The
frequency or the frequency count for a
data value is the number of times the
value occurs in the data set.
1-21
Categorical or Qualitative Frequency
Distributions
• NOTE: We will consider categorical,
ungrouped, and grouped frequency
distributions.
 What is a categorical frequency distribution? A

categorical frequency distribution represents
data that can be placed in specific categories,
such as gender, hair color, political affiliation
etc.
1-22
Categorical or Qualitative Frequency
Distributions -- Example
• Example: The blood types of 25 blood donors
are given below. Summarize the data using a
frequency distribution.

AB    B     A  O       B
O     B     O  A       O
B     O     B  B       B
A     O    AB AB       O
A     B    AB O        A
1-23
Categorical Frequency Distribution
for the Blood Types -- Example Continued

Note: The classes for the distribution are the blood types.
1-24
Quantitative Frequency Distributions
-- Ungrouped
• What is an ungrouped frequency
distribution? An ungrouped frequency
distribution simply lists the data values
with the corresponding frequency counts
with which each value occurs.
1-25
Quantitative Frequency Distributions
– Ungrouped -- Example
• Example: The at-rest pulse rate for 16
athletes at a meet were 57, 57, 56, 57,
58, 56, 54, 64, 53, 54, 54, 55, 57, 55,
60, and 58. Summarize the information
with an ungrouped frequency
distribution.
1-26
Quantitative Frequency Distributions
– Ungrouped -- Example Continued

Note: The (ungrouped)
classes are the
observed values
themselves.
1-27              Relative Frequency

• NOTE: Sometimes frequency distributions are
displayed with relative frequencies as well.

• What is a relative frequency for a class? The
relative frequency of any class is obtained
dividing the frequency (f) for the class by the
total number of observations (n).
1-28              Relative Frequency

Relative      for     
Frequency aclass

for
frequency theclass
number observatiointhedistributi
total    of        ns             on

Example: The relative frequency for the ungrouped
class of 57 will be 4/16 = 0.25.
1-29   Relative Frequency Distribution

Note: The relative
frequency for a
class is obtained
by computing f/n.
1-30       Cumulative Frequency and
Cumulative Relative Frequency
• NOTE: Sometimes frequency
distributions are displayed with
cumulative frequencies and cumulative
relative frequencies as well.
1-31       Cumulative Frequency and
Cumulative Relative Frequency
• What is a cumulative frequency for a
class? The cumulative frequency for a
specific class in a frequency table is the
sum of the frequencies for all values at or
below the given class.
1-32       Cumulative Frequency and
Cumulative Relative Frequency
• What is a cumulative relative frequency
for a class? The cumulative relative
frequency for a specific class in a
frequency table is the sum of the relative
frequencies for all values at or below the
given class.
1-33     Cumulative Frequency and
Cumulative Relative Frequency
Note: Table with
relative and
cumulative
relative
frequencies.
1-34
Quantitative Frequency Distributions
-- Grouped
• What is a grouped frequency distribution?
A grouped frequency distribution is
obtained by constructing classes (or
intervals) for the data, and then listing
the corresponding number of values
(frequency counts) in each interval.
1-35
Quantitative Frequency Distributions
-- Grouped
• There are several procedures that one can use
to construct a grouped frequency distribution.
• However, because of the many statistical
software packages (MINITAB, SPSS etc.) and
graphing calculators (TI-83 etc.) available
today, it is not necessary to try to construct
such distributions using pencil and paper.
1-36
Quantitative Frequency Distributions
-- Grouped
• Later, we will encounter a graphical
display called the histogram. We will
see that one can directly construct
grouped frequency distributions from
these displays.
1-37
Quantitative Frequency Distributions
– Grouped -- Quick Tip
• A frequency distribution should have a
minimum of 5 classes and a maximum of 20
classes.
• For small data sets, one can use between 5
and 10 classes.
• For large data sets, one can use up to 20
classes.
1-38
Quantitative Frequency Distributions
– Grouped -- Example
• Example: The weights of 30 female students
majoring in Physical Education on a college
campus are as follows: 143, 113, 107, 151, 90,
139, 136, 126, 122, 127, 123, 137, 132, 121,
112, 132, 133, 121, 126, 104, 140, 138, 99,
134, 119, 112, 133, 104, 129, and 123.
Summarize the data with a frequency distribution
using seven classes.
1-39
Quantitative Frequency Distributions
– Grouped -- Example Continued
• NOTE: We will introduce the histogram
here to help us construct a grouped
frequency distribution.
1-40
Quantitative Frequency Distributions
– Grouped -- Example Continued
• What is a histogram? A histogram is a
graphical display of a frequency or a relative
frequency distribution that uses classes and
vertical (horizontal) bars (rectangles) of
various heights to represent the frequencies.
1-41
Quantitative Frequency Distributions
– Grouped -- Example Continued
• The MINITAB statistical software was used to
generate the histogram.
• The histogram has seven classes.
• Classes for the weights are along the x-axis and
frequencies are along the y-axis.
• The number at the top of each rectangular box,
represents the frequency for the class.
1-42
Quantitative Frequency Distributions
– Grouped -- Example Continued

Histogram
with 7
classes
for the
weights.
1-43
Quantitative Frequency Distributions
– Grouped -- Example Continued
• Observations
• From the histogram, the classes
(intervals) are 85 – 95, 95 – 105,
105 – 115 etc. with corresponding
frequencies of 1, 3, 4, etc.
• We will use this information to construct
the group frequency distribution.
1-44
Quantitative Frequency Distributions
– Grouped -- Example Continued
• Observations (continued)
• Observe that the upper class limit of 95 for the
class 85 – 95 is listed as the lower class limit
for the class 95 – 105.
• Since the value of 95 cannot be included in
both classes, we will use the convention that
the upper class limit is not included in the
class.
1-45
Quantitative Frequency Distributions
– Grouped -- Example Continued
• Observations (continued)
• That is, the class 85 – 95 should be
interpreted as having the values 85 and up
to 95 but not including the value of 95.
• Using these observations, the grouped
frequency distribution is constructed from
the histogram and is given on the next slide.
1-46
Quantitative Frequency Distributions
– Grouped -- Example Continued
1-47
Quantitative Frequency Distributions
– Grouped -- Example Continued
• Observations (continued)
• In the previous slide with the grouped
frequency distribution, the sum of the
relative frequencies did not add up to 1.
This is due to rounding to four decimal
places.
• The same observation should be noted for
the cumulative relative frequency column.
1-48   Dot Plots
• What is a dot plot? A dot plot is a plot that
displays a dot for each value in a data set along
a number line. If there are multiple
occurrences of a specific value, then the dots
will be stacked vertically.
• Example: The following frequency distribution
shows the number of defectives observed by a
quality control officer over a 30 day period.
Construct a dot plot for the data.
1-49   Dot Plots – Example Continued

The next slide
shows the dot
plot for the
number of
defectives.
1-50
Dot Plots – Example Continued
1-51   Bar Charts or Bar Graphs
• What is a bar chart (graph)? A bar chart or a
bar graph is a graph that uses vertical or
horizontal bars to represent the frequencies of
the categories in a data set.
• Example: A sample of 300 college students was
asked to indicate their favorite soft drink. The
results of the survey are shown on the next
slide. Display the information using a bar
chart.
1-52   Bar Charts – Example Continued

The next slide
shows the bar
chart for the
soft drink
preferences of
the students.
1-53   Bar Chart – Example Continued
1-54   Bar Charts -- Quick Tip

• Bar charts are effective at reinforcing
differences in magnitude.
• Bar charts are useful when the data set has
categories (for example, hair color, gender,
etc.).
• Bar charts are useful when the data are
qualitative in nature.
• Note: The bars are equally separated.
1-55   Histograms Revisited

Histogram
with 7
classes
for the
weights.
1-56    Histograms -- Quick Tip
• Histograms are useful when the data values are
quantitative.
• A histogram gives an estimate of the shape of
the distribution of the population from which
the sample was taken.
• If the relative frequencies were plotted along
the vertical axis to produce the histogram, the
shape will be the same as when the frequencies
are used.
1-57   Frequency Polygons
• What is a frequency polygon? A frequency
polygon is a graph that displays the data using
lines to connect points plotted for the frequencies.
• Note: The frequencies represent the heights of the
vertical bars in the histogram.
• Example: Display a frequency polygon for the
weights of the 30 female students (presented
previously).
1-58   Frequency Polygons -- Example Continued

Frequency
Polygon
1-59   Frequency Polygons – Observations
• The frequency polygon is superimposed on
the histogram.
• The polygon is mound-shaped.
• This indicates that the shape of the
population from which the sample was taken
is mound shaped.
• The line segments pass through the mid
points at the top of the rectangles.
• The polygon is tied down at both ends.
1-60   Stem-and-Leaf Displays or Plots
• What is a stem-and-leaf plot? A stem-and-
leaf plot is a data plot that uses part of a
data value as the stem to form groups or
classes and part of the data value as the leaf.
• Note: A stem-and-leaf plot has an advantage
over a grouped frequency distribution, since
a stem-and-leaf plot retains the actual data
by showing them in graphic form.
1-61   Stem-and-Leaf Displays or Plots --
Example

• Example: Consider the following values
– 96, 98, 107, 110, and 112. Construct
a stem-and-leaf plot by using the units
digits as the leaves.
1-62    Stem-and-Leaf Plot – Example Continued

Stems and leaves for the   Stem-and-leaf plot for the
data values.               data values.

Stem          Leaf

09           6 8
10           7
11           0 2
1-63   Stem-and-Leaf Displays or Plots -- Example

• Example: A sample of the number of
admissions to a psychiatric ward at a local
hospital during the full phases of the moon is
as follows: 22, 30, 21, 27, 31, 36, 20, 28, 25,
33, 21, 38, 32, 35, 26, 19, 43, 30, 30, 34,
27, and 41.
• Display the data in a stem-and-leaf plot with
the leaves represented by the unit digits.
1-64   Stem-and-Leaf Plot – Example Continued

Stem        Leaf

1     9
2     0 1 1 2 5 6 7 7 8
3     0 0 0 1 2 3 4 5 6 8
4     1 3
1-65   Time Series Graphs
• What is a time series graph? A time series
graph is a plot which displays data that are
observed over a given period of time.
• Note: From a time series graph, one can
observe and analyze the behavior of the data
over time.
1-66   Time Series Graphs -- Example
• Example: The following table gives the
number of hurricanes for the years 1981 to
1990.

• Display the data with a time series graph.
1-67   Time Series Graphs – Example Continued
Graph
seem to
display an
upward
trend
over the
years.
 Highest
number
was in
1990.
Time Series Graph for the Dow Jones Industrial Average
1-68   from October 1999 to October 2000 – Example
1-69   Pie Graphs or Pie Charts
• What is a pie graph (chart)? A pie graph is a
circular display that is divided into sectors
(classes) according to the percentage of data
values in each class.
• Note: A pie chart allows us to observe the
proportions of the classes relative to the entire
data set.
• Pie charts are readily used to display qualitative
data.
1-70   Pie Graphs or Pie Charts -- Example
• Example: present a pie chart for the blood
type data given earlier.
• The pie chart is presented on the next slide.
• Note: Each sector (slice) is proportional to
the frequency count or percentage relative
to the total sample size.
1-71   Pie Graphs or Pie Charts – Example Continued
1-72   Pareto Charts
• What is a Pareto chart? A Pareto chart is a type
of bar chart in which the horizontal axis
represents the categories of interest. The bars
are ordered from largest to smallest in terms of
the frequency counts for the categories.
which of the categories make up the critical
few and which are the insignificant many.
1-73   Pareto Charts -- Example
• A cumulative percentage line helps you judge
the added contribution of each category.
• Example: The following information over a six
month period, relating to the number of defects
in a manufacturing process in a company, was
obtained by the quality control team.
• The data is given on the next slide.
• Present a Pareto chart for the data.
1-74   Pareto Charts – Example Continued

Note: The Pareto chart,
on the next slide, combined
the information for the scrap,
unconnected wire, and missing
studs as an Others category.
1-75   Pareto Charts – Example Continued
1-76                   It’s a Wrap
• All of the graphs that were presented in this
chapter can be generated or constructed with
many of the statistical software packages which
are on the market today.
• All of the graphs presented were constructed
using the MINITAB for Windows software.
• Other packages which can be used are SPSS,
SAS, and EXCEL.

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