Unit 2.

2:
Graphs for
Qualitative and
Quantitative Data
Objectives:
• At the end of the lesson, you will be able to:
 Construct a frequency distribution from a set of data.
 Construct different types of quantitative data graphs,
including histograms, frequency polygons, ogives, dot
plots, and stem-and-leaf plots, in order to interpret the
data being graphed.
 Construct different types of qualitative data graphs,
including pie charts, bar graphs, and Pareto charts, in
order to interpret the data being graphed.
Lessons:
• Frequency Distributions
• Quantitative Data Graphs
• Qualitative Data Graphs
Okay. Let’s begin.
Lesson 3.2.1:
Frequency Distributions
Terminologies:
• Raw Data
- Data that have not been summarized in any
way, are sometimes referred to as
ungrouped data.
Terminologies:
• Range
- The difference between the largest and the
lowest numbers.
Frequency Distribution
• A summary of data presented in the form of class
intervals and frequencies.
• A tool to easily understand your data presentation.
Try to figure out this table…
Age of the top 50 wealthiest people in the world
Try to figure out this table…
Age of the top 50 wealthiest people in the world
How to construct frequency distribution
from a raw data?
How to construct frequency distribution
from a raw data?
How to construct frequency distribution
from a raw data?
How to construct frequency distribution
from a raw data?
How to construct frequency distribution
from a raw data?
How to construct frequency distribution
from a raw data?
How to construct frequency distribution
from a raw data?
Example
(With 6 number of classes)
Class Midpoint
• The midpoint of each class interval is called the
class midpoint and is sometimes referred to as the
class mark. It is the value halfway across the
class interval and can be calculated as the
average of the two class endpoints.
• Theclass midpoint is important, because it
becomes the representative value for each class in
most group statistics calculations.
Example:
• The table below are the scores of each student in a 50- item quiz:

Interval Frequency (f) Class Midpoint

1-10 3 5.5
11-20 15 15.5
21-30 22 25.5
31-40 13 35.5
41-50 7 45.5
Relative Frequency
• theproportion of the total frequency that is in any given
class interval in a frequency distribution.
• Relativefrequency is the individual class frequency
divided by the total frequency.
 Example:
• The table below are the scores of each student in a 50- item quiz:

Interval Frequency (f) Class Midpoint Relative Frequency

1-10 3 5.5 0.05
11-20 15 15.5 0.25
21-30 22 25.5 0.37
31-40 13 35.5 0.22
41-50 7 45.5 0.11
Total 60 100.00
Cumulative Frequency
•a running total of frequencies through the classes of a
frequency distribution.
• Theconcept of cumulative frequency is used in many
areas, including sales cumulated over a fiscal year, sports
scores during a contest (cumulated points), years of
service, points earned in a course, and costs of doing
business over a period of time.
 Example:
• The table below are the scores of each student in a 50- item quiz:

Interval f Class Midpoint Relative Frequency Cumulative f

1-10 3 5.5 0.05 3
11-20 15 15.5 0.25 18
21-30 22 25.5 0.37 40
31-40 13 35.5 0.22 53
41-50 7 45.5 0.11 60
Total 60 100.00
Lesson 2.2:
Quantitative Data Graphs
Quantitative Data Graphs
• One of the most important uses of graphical depiction in
statistics is to help the researcher determine the shape of
a distribution.
• Important
mechanism for presenting data in a form
meaningful to decision makers is graphical depiction.
• Through graphs and charts, the decision maker can often
get an overall picture of the data and reach some useful
conclusions merely by studying the chart or graph.
Types of Quantitative Data Graphs
• Histogram

• Frequency polygon
• Ogive

• Stem-and-leaf plot
Histogram
Histograms
• Oneof the more widely used types of graphs for
quantitative data.
• A histogram is a series of contiguous bars or rectangles
that represent the frequency of data in given class
intervals.
To construct histogram…
• Draw and label the x and y axes. The x axis is always the
horizontal axis, and the y axis is always the vertical axis.
• Representthe frequency on the y axis and the class
boundaries on the x axis.
• Using the frequencies as the heights, draw vertical bars
for each class.
Example of Histogram
Example of Histogram
Continue…
• A histogram is a useful tool for differentiating the
frequencies of class intervals. A quick glance at a
histogram reveals which class intervals produce the
highest frequency totals.
Frequency Polygon
Frequency Polygons
• A frequency polygon, like the histogram, is a graphical
display of class frequencies.
• Instead of using bars or rectangles like a histogram, in a
frequency polygon each class frequency is plotted as a
dot at the class midpoint, and the dots are connected by a
series of line segments.
Frequency Polygons
• Find the midpoints of each class. Recall that midpoints
are found by adding the upper and lower boundaries and
dividing by 2
• Draw the x and y axes. Label the x axis with the
midpoint of each class, and then use a suitable scale on
the y axis for the frequencies
Frequency Polygons
• Using the midpoints for the x values and the frequencies
as the y values, plot the points.
• Connect adjacent points with line segments. Draw a line
back to the x axis at the beginning and end of the graph,
at the same distance that the previous and next midpoints
would be located.
Example of Frequency Polygon
Ogives (o-jive)
• An ogive is a cumulative frequency polygon.
• Steepslopes in an ogive can be used to identify sharp
increases in frequencies
Ogive
Ogives (o-jives)
• Steps in creating Ogive
1. Find the cumulative frequency for each class.
2. Draw the x and y axes. Label the x axis with the class
boundaries. Use an appropriate scale for the y axis to
represent the cumulative frequencies. (Depending on the
numbers in the cumulative frequency columns, scales
such as 0, 1, 2, 3, . . . , or 5, 10, 15, 20, . . . , or 1000,
2000, 3000, . . . can be used. Do not label the y axis with
the numbers in the cumulative frequency column.) In
this example, a scale of 0, 5, 10, 15, . . . will be used.
Ogives (o-jives)
• Steps in creating Ogive
3. Plot the cumulative frequency at each upper class
boundary. Upper boundaries are used since the cumulative
frequencies represent the number of data values
accumulated up to the upper boundary of each class.
4. Starting with the first upper class boundary, 104.5,
connect adjacent points with line segments. Then extend
the graph to the first lower class boundary, 99.5, on the x
axis
Example of Ogive
Example of Ogive
Example of Ogive
Assignment
Stem-and-leaf Plot
• Thistechnique is simple and provides a unique view of
the data.
• A stem-and-leaf plot is constructed by separating the
digits for each number of the data into two groups, a
stem and a leaf.
• Theleftmost digits are the stem and consist of the higher
valued digits. The rightmost digits are the leaves and
contain the lower values.
Example of stem-and-leaf plot
Example of stem-and-leaf plot
Exercise
Lesson 2.3:
Qualitative Data Graphs
Qualitative Data Graphs
• Qualitative graphs are plotted using non-numerical
categories.
• Contrast to Quantitative data graphs.
Types of Qualitative Data Graphs
• Pie Charts
• Bar Charts
• Pareto Charts
Pie Charts
• A piechart is a circular depiction of data where the area
of the whole pie represents 100% of the data and slices
of the pie represent a percentage breakdown of the
sublevels.
• They are widely used in business, particularly to depict
such things as budget categories, market share, and
time/resource allocations.
• Constructionof the pie chart begins by determining the
proportion of the subunit to the whole.
Leading Petroleum Refining Companies
Leading Petroleum Refining Companies
Leading Petroleum Refining Companies
Leading Petroleum Refining Companies
Leading Petroleum Refining Companies
Continue…
• Toconstruct a pie chart from these data, first convert the
raw sales figures to proportions by dividing each sales
figure by the total sales figure. This proportion is
analogous to relative frequency computed for frequency
distributions. Because a circle contains 360°, each
proportion is then multiplied by 360 to obtain the correct
number of degrees to represent each item.
Bar Charts
• Another widely used qualitative data graphing technique
is the bar graph or bar chart.
• A bargraph or chart contains two or more categories
along one axis and a series of bars, one for each
category, along the other axis.
• Typically, the length of the bar represents the magnitude
of the measure (amount, frequency, money, percentage,
etc.) for each category. The bar graph is qualitative
because the categories are non-numerical, and it may be
either horizontal or vertical.
How Much is Spent on Back-to-College
Shopping by the Average Student
Using Bar Chart,
Demonstration Problem
Solution for Pie Chart
For Bar Graph,
Pareto Chart
• A thirdtype of qualitative data graph is a Pareto chart,
which could be viewed as a particular application of the
bar graph.
• Paretoanalysis is a quantitative tallying of the number
and types of defects that occur with a product or service.
• Analysts use this tally to produce a vertical bar chart that
displays the most common types of defects, ranked in
order of occurrence from left to right.
• Suppose the number of electric motors being rejected by
inspectors for a company has been increasing. Company
officials examine the records of several hundred of the
motors in which at least one defect was found to
determine which defects occurred more frequently. They
find that 40% of the defects involved poor wiring, 30%
involved a short in the coil, 25% involved a defective plug,
and 5% involved cessation of bearings.
• Company officials and workers would probably begin to
improve quality by examining the segments of the
production process that involve the wiring. Next, they
would study the construction of the coil, then examine
the plugs used and the plug-supplier process.
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THANK YOU!