ch02 extra1 frequency

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2

FREQUENCY DISTRIBUTION

Consider the following data that shows days to maturity for

40 short-term investments

70 64 99 55 64 89 87 6562 38 67 70 60 69 78 39

75 56 71 51 99 68 95 86

57 53 47 50 55 81 80 98

51 31 63 66 85 79 83 70



3

• 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 classesNumber of observation Number of classes

Less than 50 5-7

50-200 7-9

200-500 9-10500-1,000 10-11

1,000-5,000 11-13

5,000-50,000 13-17

More than 50,000 17-20




4

• Approximate class width is obtained as follows:

classesof Number

valueSmallest-valueLargest widthclasse Approximat




5

Classes and counts for the days-to-maturity data

Days toMaturity

TALLY Number ofInvestments




6

HISTOGRAM

0

2

4

6

8

10

12

40 50 60 70 80 90 100

Number of Days to Maturity

F r e q u e

n c y



7

HISTOGRAM

Classes: Categories for grouping data.Frequency: The number of observations that fall in a class.

Frequency distribution: A listing of all classes along with their

frequencies.

Relative frequency: The ratio of the frequency of a class to the total

number of observations. Relative-frequency distribution: A listing of all classes along with

their relative frequencies.

Lower cutpoint: The smallest value that can go in a class.

Upper cutpoint: The smallest value that can go in the next higher

class. The upper cutpoint of a class is the same as the lower cutpointof the next higher class.

Midpoint: The middle of a class, obtained by taking the average of its

lower and upper cutpoints.

Width: The difference between the upper and lower cutpoints of a

class.



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

Classes Mid-value Frequency



10

FREQUENCY POLYGON

0

2

4

6

8

10

12

35 45 55 65 75 85 95


F r e q u e n c y



11

Frequency histogram: A graph that displays the classes on

the horizontal axis and the frequencies of the classes on the

vertical axis. The frequency of each class is represented by a

vertical bar whose height is equal to the frequency of the class.

Relative-frequency histogram: A graph that displays the

classes on the horizontal axis and the relative frequencies of

the classes on the vertical axis. The relative frequency of eachclass is represented by a vertical bar whose height is equal to

the relative frequency of the class.

RELATIVE FREQUENCY HISTOGRAM



12

• Class relative frequency is obtained as follows:

nsobservatioof number Total

frequencyClassfrequencyrelativeClass




13


Relative-frequency distribution for the days-to-maturity data

Days toMaturity

Relative Frequency



14


0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

40 50 60 70 80 90 100


R e l a t i v e F r e q u e n c y



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


Class Frequency RelativeFrequency

CumulativeRelative

Frequency



17

OGIVE


0.075 0.100

0.300

0.550

0.725

0.900 1.000

0.000

0.200

0.400

0.600

0.800

1.000

40 50 60 70 80 90 100


C u

m u l a t i v e F r e q u e n c y



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

3 34 4

5 5

6 6

7 78 8

9 9

(a) (b)



20

SCATTTER DIAGRAM

• Often, we are interested in two variables. Forexample, we may want to know the relationship

between

– advertising and sales

– experience and time required to produce an unit ofa product



22

SCATTTER DIAGRAM

•In some cases, the value of one variable may dependon 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 thehorizontal axis ( x axis) and the dependent variable on

the vertical axis ( y axis)

• Sometimes, two variables show some relationships

– positive relationship: two variables move togetheri.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 ornon-linear. For example,

– the relationship between advertising and sales

may be linear.

–the relationship between number of unitsproduced and the production time/unit may be

nonlinear.



25

SCATTTER DIAGRAM (EXAMPLE)

Advertizing Sales

1,000 of dollars 1,000 of dollars

1 30

3 405 40

4 50

2 35

5 503 35

2 25



26

SCATTER DIAGRAM

20

40

60

0 2 4 6

Advertising

S a l e s



27

SCATTTER DIAGRAM (EXAMPLE)

Number of units Production time

produced hours/unit

10 9.22

25 4.8510 3.8

250 2.44

500 1.7

1000 1.035000 0.6

10000 0.5



28

SCATTER DIAGRAM

0

5

10

0 2000 4000 6000 8000 10000 12000

Number of units produced

P r o d u c t i o n t i m e

( h o u r s ) / u n i t



29

PIE CHART

•

A pie chart is the most popular graphical method forsummarizing 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

Code Area Number Proportion Angles on a

of Area of Graduates of Graduates Pie Chart

1 Accounting 73

2 Finance 523 General Mgmnt 36

4 Marketing 64

5 Other 28

PIE CHART (EXAMPLE)



31

PIE CHART

1

29%

2

21%3

14%

4

25%

5

11%

More

0%

1

23

4

5

More



32

BAR CHART

•

Bar charts graphically represent the frequency orrelative 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 particularorder such as years 1995, 1996, 1997, ...



33

Code Area Number

of Area of Graduates

1 Accounting 73

2 Finance 523 General Mgmnt 36

4 Marketing 64

5 Other 28

BAR CHART (EXAMPLE)



34

BAR CHART

0

10

20

30

40

50

60

7080

1 2 3 4 5

Area

C o u n t o

f A r e a



35

LINE CHART

•

Line charts are often used when the categories arepoints 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 pointabove and then points are joined by straight lines



36

LINE CHART (EXAMPLE)

Year Fatal Accidents Number of PassengersPer 1,000 Departures Millions

1983 0.10 21

1984 0.23 27

1985 0.24 281986 0.02 29

1987 0.31 32

1988 0.04 36

1989 0.12 38

1990 0.04 421991 0.24 42

1992 0.15 48

1993 0.10 53

1994 0.04 58



37

Line Chart

0.00

0.05

0.10

0.15

0.20

0.25

0.300.35

1 2 3 4 5 6 7 8 9 10 11 12

Year

F a t a l A c c i d e n t s

P e r 1 , 0 0 0 D

e p a r t u r e s



38

Line Chart

0

10

20

30

40

50

6070

1 2 3 4 5 6 7 8 9 10 11 12

Year

N

u m b e r o f P

a s s e n g e r s

( M i l l i o n s )



40

•

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 – Small/intermediate/large number of categories

– Can present categories in a particular order, if any

– Emphasizes trend, if any

CHOICE OF A CHART



41

SYMMETRIC HISTOGRAM

0

2

4

6

8

10

1214

14 15 16 17 18 19 20 21 22 23 24 25 26

Number of Units Sold

F r e q u

e n c y



42

SYMMETRIC HISTOGRAM

0

2

4

6

8

10

12

14 15 16 17 18 19 20 21 22 23 24 25 26


F r e q u

e n c y



43

POSITIVELY SKEWED HISTOGRAM

0

2

4

68

10

12

1416

14 15 16 17 18 19 20 21 22 23 24 25 26


F r e q u

e n c y



44

NEGATIVELY SKEWED HISTOGRAM

0

2

4

68

10

12

1416

14 15 16 17 18 19 20 21 22 23 24 25 26


F r e q u

e n c y



45

BIMODAL HISTOGRAM

0

1

2

34

5

6

78

14 15 16 17 18 19 20 21 22 23 24 25 26


F r e q u

e n c y



46

NORMAL DISTRIBUTION HISTOGRAM

0

2

4

6

8

10

1214

14 15 16 17 18 19 20 21 22 23 24 25 26


F r e q u e n c y



47

EXPONENTIAL DISTRIBUTION HISTOGRAM

0

5

10

15

20

25

30

35

40

45

14 15 16 17 18 19 20 21 22 23 24 25 26


F r e q u

e n c y



48

UNIFORM DISTRIBUTION HISTOGRAM

0

1

2

3

4

5

6

7

14 15 16 17 18 19 20 21 22 23 24 25 26


F r e q u

e n c y

ch02 extra1 frequency

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