1. LESSON 2: FREQUENCY DISTRIBUTION
Outline
Frequency distribution, histogram, frequency polygon
Relative frequency histogram
Cumulative relative frequency graph
Stem-and-leaf plots
Scatter diagram
Pie charts, bar chart, line chart
Some special frequency distribution forms

2. FREQUENCY DISTRIBUTION
Consider the following data that shows days to maturity for 40 short-term investments

3. FREQUENCY DISTRIBUTION
First, construct a frequency distribution
An arrangement or table that groups data into non-overlapping intervals called classes and records the number of observations in each class
Approximate number of classes
Number of observation Number of classes
Less than
500-1,
1,000-5,
5,000-50,
More than 50,

4. FREQUENCY DISTRIBUTION
Approximate class width is obtained as follows:

5. FREQUENCY DISTRIBUTION
Classes and counts for the days-to-maturity data

7. HISTOGRAM

8. FREQUENCY POLYGON
A frequency polygon is a graph that displays the data by using lines that connect points plotted for frequencies at the midpoint of classes. The frequencies represent the heights of the midpoints.

9. FREQUENCY POLYGON

11. RELATIVE FREQUENCY HISTOGRAM

12. RELATIVE FREQUENCY HISTOGRAM
Class relative frequency is obtained as follows:

13. RELATIVE FREQUENCY HISTOGRAM
Relative-frequency distribution for the days-to-maturity data

15. OGIVE CUMULATIVE RELATIVE FREQUENCY GRAPH
A cumulative relative frequency graph or ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution.

16. OGIVE CUMULATIVE RELATIVE FREQUENCY GRAPH

17. OGIVE
CUMULATIVE RELATIVE FREQUENCY GRAPH
1.000
1.000
0.900
0.800
0.725
0.600
Cumulative Frequency
0.550
0.400
0.300
0.200
0.100
0.075
0.000 40 50 60 70 80 90
100
Number of Days to Maturity

18. STEM-AND-LEAF DISPLAY
When summarizing the data by a group frequency distribution, some information is lost. The actual values in the classes are unknown. A stem-and-leaf display offsets this loss of information.
The stem is the leading digit.
The leaf is the trailing digit.

19. STEM-AND-LEAF DISPLAY
Diagrams for days-to-maturity data: (a) stem-and-leaf (b) ordered stem-and-leaf Stem Leaves Stem Leaves (a) (b)

20. SCATTTER DIAGRAM
Often, we are interested in two variables. For example, we may want to know the relationship between
advertising and sales
experience and time required to produce an unit of a product

21. SCATTTER DIAGRAM
Scatter diagrams show how two variables are related to one another
To draw a scatter diagram, we need a set of two variables
Label one variable x and the other y
Each pair of values of x and y constitute a point on the graph

22. SCATTTER DIAGRAM
In some cases, the value of one variable may depend on the value of the other variable. For example,
sales depend on advertising
time required to produce an item of a product depend on the number of units produced before
In such cases, the first variable is called dependent variable and the second variable is called independent variable. For example,
Independent variable Dependent variable
Advertising Sales
Number of units produced Production time/unit

23. SCATTTER DIAGRAM
Usually, independent variable is plotted on the horizontal axis (x axis) and the dependent variable on the vertical axis (y axis)
Sometimes, two variables show some relationships
positive relationship: two variables move together i.e., one variable increases (or decreases) whenever the other increases (or, decreases). Example: advertising and sales.
negative relationship: one variable increases (or, decreases) whenever the other decreases (increases). Example: number of units produced and production time/unit

24. SCATTTER DIAGRAM
Relationship between two variables may be linear or non-linear. For example,
the relationship between advertising and sales may be linear.
the relationship between number of units produced and the production time/unit may be nonlinear.

25. SCATTTER DIAGRAM (EXAMPLE)

27. SCATTTER DIAGRAM (EXAMPLE)

29. PIE CHART
A pie chart is the most popular graphical method for summarizing quantitative/nominal data
A pie chart is a circle is subdivided into a number of slices
Each slice represents a category
Angle allocated to a slice is proportional to the proportion of times the corresponding category is observed
Since the entire circle corresponds to 3600, every 1% of the observations corresponds to 0.01  3600 = 3.60

30. PIE CHART (EXAMPLE)

32. BAR CHART
Bar charts graphically represent the frequency or relative frequency of each category as a bar rising vertically
The height of each bar is proportional to the frequency or the relative frequency
All the bars must have the same width
A space may be left between bars
Bar charts may be used for qualitative data or categories that should be presented in a particular order such as years 1995, 1996, 1997, ...

33. BAR CHART (EXAMPLE)

35. LINE CHART
Line charts are often used when the categories are points in time. Such a chart is called a time-series chart. For example, consider a graph that shows monthly or weekly sales data.
Frequency of each category is represented by a point above and then points are joined by straight lines

36. LINE CHART (EXAMPLE)

39. CHOICE OF A CHART Pie chart Small / intermediate number of categories
Cannot show order of categories
Emphasizes relative values e.g., frequencies
Bar chart
Small / intermediate/large number of categories
Can present categories in a particular order, if any

40. CHOICE OF A CHART Bar chart
Small/intermediate/large number of categories
Can present categories in a particular order, if any
Emphasizes relative values e.g., frequencies
Line chart
Emphasizes trend, if any

49. READING AND EXERCISES Lesson 2 Reading: Section 2-1, pp. 22-33
2-1, 2-9, 2-13, 2-14