DESCRIBING DATA

Frequency Tables, Frequency

Distributions, and Graphic

Presentation

A raw data is the data obtained before

it is being processed or arranged.

Raw Data

2

A raw score is the score obtained by a

particular student in a particular test

before it is being processed or

arranged.

Example: Raw Score

3

78, 74, 65, 74, 74, 67, 63,

67, 80, 58 74, 50, 65, 74,

86, 78, 63, 65, 80, 89

The raw scores for 20 students

in a test

The Raw Score is a Variable

4

Data in raw form are usually not easy to use for decision making Some type of organization is needed

Table

Graph

Techniques reviewed here: Ordered Array

Stem-and-Leaf Display

Frequency Distributions and Histograms

Bar charts and pie charts

Contingency tables

Organizing and Presenting Data Graphically

5

Interval Data

Ordered Array

Stem-and-Leaf

Display Histogram Polygon Ogive

Frequency Distributions

and

Cumulative Distributions

Tables and Charts for Interval Data

6

A sorted list of data:

Shows range (minimum to maximum)

Provides some signals about variability

within the range

May help identify outliers (unusual observations)

If the data set is large, the ordered array is

less useful

The Ordered Array

7

Data in raw form (as collected):

24, 26, 24, 21, 27, 27, 30, 41, 32, 38

Data in ordered array from smallest to

largest:

21, 24, 24, 26, 27, 27, 30, 32, 38, 41

(continued)

The Ordered Array

8

A simple way to see distribution details in a

data set

METHOD: Separate the sorted data series

into leading digits (the stem) and

the trailing digits (the leaves)

Stem-and-Leaf Diagram

9

Stem-and-Leaf

The major advantage to organizing the data into stem-and-leaf display is that we get a quick visual picture of the shape of the distribution.

Stem-and-leaf display is a statistical technique to present a set of data. Each numerical value is divided into two parts. The leading digit(s) becomes the stem and the trailing digit the leaf. The stems are located along the vertical axis, and the leaf values are stacked against each other along the horizontal axis.

Advantage of the stem-and-leaf display over a frequency distribution - the identity of each observation is not lost.

10

Here, use the 10’s digit for the stem

unit:

Data in ordered array: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41

21 is shown as

38 is shown as

Stem Leaf

2 1

3 8

Example

11

Example

Suppose the seven observations in the 90 up to 100 class are: 96, 94, 93, 94, 95, 96, and 97.

The stem value is the leading digit or digits, in this case 9. The leaves are the trailing digits. The stem is placed to the left of a vertical line and the leaf values to the right. The values in the 90 up to 100 class would appear as

Then, we sort the values within each stem from smallest to largest. Thus, the second row of the stem-and-leaf display would appear as follows:

12

Completed stem-and-leaf

diagram: Stem Leaves

2 1 4 4 6 7 7

3 0 2 8

4 1

(continued)

Data in ordered array: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41

Example

13

Using the 100’s digit as the stem:

Round off the 10’s digit to form the

leaves

613 would become 6 1

776 would become 7 8

. . .

1224 becomes 12 2

Stem Leaf

Using other stem units

14

Using the 100’s digit as the

stem:

The completed stem-and-leaf

display:

Stem Leaves

(continued)

6 1 3 6

7 2 2 5 8

8 3 4 6 6 9 9

9 1 3 3 6 8

10 3 5 6

11 4 7

12 2

Data:

613, 632, 658, 717,

722, 750, 776, 827,

841, 859, 863, 891,

894, 906, 928, 933,

955, 982, 1034,

1047,1056, 1140,

1169, 1224

Using other stem units

15

Stem-and-leaf: Another Example

Listed in Table 4–1 is the number of 30-second radio advertising spots purchased by each of the 45 members of the Greater Buffalo Automobile Dealers Association last year. Organize the data into a stem-and-leaf display. Around what values do the number of advertising spots tend to cluster? What is the fewest number of spots purchased by a dealer? The largest number purchased?

16

Stem-and-leaf: Another Example

17

What is a Frequency Distribution?

A frequency distribution is a list or a table …

containing class groupings (categories or ranges within which the data fall) ...

and the corresponding frequencies with which data fall within each grouping or category

Tabulating Numerical Data: Frequency Distributions

18

A frequency distribution is a way to

summarize data

The distribution condenses the raw data

into a more useful form...

and allows for a quick visual interpretation

of the data

Why Use Frequency

Distributions?

19

Score

(X)

Frequency

(f)

50

58

63

65

67

74

78

80

86

89

1

1

2

3

2

5

2

2

1

1

Total

20

Frequency distribution table

for ungrouped data

Frequency Distribution (ungrouped

data)

20

A Frequency

Distribution is a

grouping of data into

mutually exclusive

categories showing

the number of

observations in

each class.

Frequency Distribution (grouped

data)

21

EXAMPLE – Creating a Frequency

Distribution Table

Ms. Kathryn Ball of AutoUSA

wants to develop tables, charts,

and graphs to show the typical

selling price on various dealer

lots. The table on the right

reports only the price of the 80

vehicles sold last month at

Whitner Autoplex.

22

Constructing a Frequency Table -

Example

Step 1: Decide on the number of classes.

A useful recipe to determine the number of classes (k) is the “2 to the k rule.” such that 2k > n.

There were 80 vehicles sold. So n = 80. If we try k = 6, which means we would use 6 classes, then 26 = 64, somewhat less than 80. Hence, 6 is not enough classes. If we let k = 7, then 27 128, which is greater than 80. So the recommended number of classes is 7.

Step 2: Determine the class interval or width.

The formula is: i (H-L)/k where i is the class interval, H is the highest observed value, L is the lowest observed value, and k is the number of classes.

($35,925 - $15,546)/7 = $2,911

Round up to some convenient number, such as a multiple of 10 or 100. Use a class width of $3,000

23

Largest

observation

Collect data

Bills

42.19

38.45

29.23

89.35

118.04

110.46

0.00

72.88

83.05

.

.

(There are 200 data points

Prepare a frequency distribution How many classes to use?

Number of observations Number of classes

Less then 50 5-7

50 - 200 7-9

200 - 500 9-10

500 - 1,000 10-11

1,000 – 5,000 11-13

5,000- 50,000 13-17

More than 50,000 17-20

Class width = [Range] / [# of classes]

[119.63 -0] / [8] = 14.95 15

Largest

observation Largest

observation

Smallest

observation Smallest

observation Smallest

observation

Smallest

observation Largest

observation

NO of Class= 1 +3.3 log (n)

n: No of data/observation

Guide

line

Or Use No. of Class = 1 + 3.3log(n)

Guide line

24

Step 3: Set the individual class limits

Constructing a Frequency Table -

Example

25

Step 4: Tally the

vehicle selling prices

into the classes.

Step 5: Count the

number of items in

each class.

Constructing a Frequency Table -

Example

26

Frequency Distribution –

Characteristics

Class midpoint: A point that divides a class into

two equal parts. This is the average of the

upper and lower class limits.

Class frequency: The number of observations

in each class.

Class interval: The class interval is obtained by

subtracting the lower limit of a class from the

lower limit of the next class.

27

Relative Frequencies

Class frequencies can be converted to relative class frequencies to show the fraction of the total number of observations in each class.

A relative frequency captures the relationship between a class total and the total number of observations.

28

28

Relative Frequency Distribution

To convert a frequency distribution to a relative frequency

distribution, each of the class frequencies is divided by the

total number of observations.

29

Score

(X)

Frequency

(f)

Relative

Frequency

Percentage

(%)

50

1

0.05

5

58

1

0.05

5

63

2

0.10

10

65

3

0.15

15

67

2

0.10

10

74

5

0.25

25

78

2

0.10

10

80

2

0.10

10

86

1

0.05

5

89

1

0.05

5

f = 20

Example

30

Relative frequency =

Example: At score 65

Relative frequency =

Percentage = Relative frequency x 100

Example: At score 65

Percentage = 0.15 x 100 = 15%

f

f

students ofnumber Total

score particular aat students ofNumber

15.020

3

Relative Frequencies – Definition

31

The three commonly used graphic forms are:

Histograms

Frequency polygons

Cumulative frequency distributions

Ogive

Graphic Presentation of a Frequency

Distribution

32

Histogram for a frequency distribution based on quantitative data

is very similar to the bar chart showing the distribution of

qualitative data. The classes are marked on the horizontal axis

and the class frequencies on the vertical axis. The class

frequencies are represented by the heights of the bars.

Histogram

33

Histogram Using Excel

34

Frequency Polygon

A frequency polygon also shows the shape of a distribution and is similar to a histogram.

It consists of line segments connecting the points formed by the intersections of the class midpoints and the class frequencies.

35

Frequency Polygon: Daily High Temperature

0

1

2

3

4

5

6

7

5 15 25 35 45 55 More

Fre

qu

en

cy

Class Midpoints

Class

10 but less than 20 15 3

20 but less than 30 25 6

30 but less than 40 35 5

40 but less than 50 45 4

50 but less than 60 55 2

Frequency Class

Midpoint

(In a percentage

polygon the vertical axis

would be defined to

show the percentage of

observations per class)

Example: Frequency Polygon

36

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

Skor

Kekera

pan

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

Score

Fre

qu

en

cy

50 60 70 80 90

Example: Frequency Polygon

37

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1 2 3 4 5 6 7 8 9 10

Score

Rela

tive F

req

uen

cy

50 60 70 80 90

Example: Relative Frequency

Polygon

38

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11

Score

Perc

en

t (%

)

50 60 70 80 90

Example: Percent Graph

39

Frequency Polygon and Histogram

40

Cumulative Frequency

Distribution

41

Cumulative Frequency

Distribution

42

Class

10 but less than 20 3 15 3 15

20 but less than 30 6 30 9 45

30 but less than 40 5 25 14 70

40 but less than 50 4 20 18 90

50 but less than 60 2 10 20 100

Total 20 100

Percentage Cumulative Percentage

Data in ordered array:

12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

Frequency Cumulative

Frequency

Cumulative Frequency Distribution

43

Ogive: Daily High Temperature

0

20

40

60

80

100

10 20 30 40 50 60

Cu

mu

lati

ve

Pe

rce

nta

ge

Class Boundaries (Not Midpoints)

Class

Less than 10 10 0

10 but less than 20 20 15

20 but less than 30 30 45

30 but less than 40 40 70

40 but less than 50 50 90

50 but less than 60 60 100

Cumulative

Percentage

Lower class

boundary

The Ogive (Cumulative % Polygon)

44

Score

(X)

Frequency

(f)

Cumulative

Frequency (cf)

Cumulative Relative

Frequency (crf)

Cumulative

Percent (cp)

50

1

1

0.05

5

58

1

2

0.10

10

63

2

4

0.20

20

65

3

7

0.35

35

67

2 9

0.45

45

74

5

14

0.70

70

78

2

16

0.80

80

80

2

18

0.90

90

86

1

19

0.95

95

89

1

20

1.00

100

f = 20

Cumulative Relative Frequency

Distribution

45

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10

Score

Cu

mu

lati

ve

Fre

qu

en

cy

18 students obtain

score 85 or less

50 60 70 80 90

Cumulative Frequency Curve

46

Grouped Data – Cumulative Frequency

Distribution and Cumulative Percent

Class

Interval

(CI)

(score X)

Class Limit

(CL)

(score X)

Class Mid

Point

(m)

Frequency

(less than

Upper

Class Limit

(UCL))

(f)

Relative

Frequency

(less than

UCL)

(cf)

Cumulative

Relative

Frequency

(less than

UCL)

(crf)

Cumulative

Percent (less

than UCL)

(cp)

50 – 54

55 – 59

60 – 64

65 – 69

70 – 74

75 – 79

80 – 84

85 – 89

49.5 – 54.5

54.5 – 59.5

59.5 – 64.5

64.5 – 69.5

69.5 – 74.5

74.5 – 79.5

79.5 – 84.5

84.5 – 89.5

52

57

62

67

72

77

82

87

1

1

2

5

5

2

2

2

1

2

4

9

14

16

18

20

0.05

0.10

0.20

0.45

0.70

0.80

0.90

1.00

5

10

20

45

70

80

90

100

47

0

5

10

15

20

1 2 3 4 5 6 7 8 9

Score

Cu

mu

lati

ve F

req

uen

cy

100%

75%

50%

25%

0

Cumulative

Percent

49.5 54.5 59.5 64.5 69.5 74.5 79.5 84.5 89.5

Cumulative Frequency Curve and

Cumulative Percent for Grouped Data

48

Ogive

49

Orgive is a smooth cumulative frequency curve.

The curve moves from the left and increases

smoothly to the right.

The smooth increase is called monotonic.

OGIVE

Ogive

50

Categorical

Data

Graphing Data

Pie

Charts

Pareto

Diagram

Bar

Charts

Tabulating Data

Summary Table

Tables and Charts for Categorical Data

51

Investment Amount Percentage Type (in thousands $) (%) Stocks 46.5 42.27 Bonds 32.0 29.09 CD 15.5 14.09 Savings 16.0 14.55 Total 110.0 100.0

(Variables are

Categorical)

Summarize data by category

The Summary Table

52

Bar charts and Pie charts are often

used for qualitative

(category/nominal) data

Height of bar or size of pie slice

shows the frequency or percentage

for each category

Bar and Pie Charts

53

Bar Charts

54

Bar Chart: Example

Investor's Portfolio

0 10 20 30 40 50

Stocks

Bonds

CD

Savings

Amount in $1000's

Investment Amount Percentage Type (in thousands $) (%) Stocks 46.5 42.27 Bonds 32.0 29.09 CD 15.5 14.09 Savings 16.0 14.55 Total 110.0 100.0

Current Investment Portfolio

55

Pie Charts

56

Percentages

are rounded to

the nearest

percent

Current Investment Portfolio

Savings

15%

CD

14%

Bonds

29%

Stocks

42%

Investment Amount Percentage Type (in thousands $) (%) Stocks 46.5 42.27 Bonds 32.0 29.09 CD 15.5 14.09 Savings 16.0 14.55 Total 110.0 100.0

Pie Charts: Example

57

PIE CHART USING EXCEL

58

Pareto Diagram

Used to portray categorical data

A bar chart, where categories are shown in

descending order of frequency

A cumulative polygon is often shown in the

same graph

Used to separate the “vital few” from the “trivial

many”

59

cu

mu

lativ

e %

investe

d

(line g

rap

h)

% i

nveste

d i

n e

ach

cate

go

ry

(bar

gra

ph

)

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

Stocks Bonds Savings CD

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Current Investment Portfolio

Pareto Diagram: Example

60

Contingency Tables

A scatter diagram requires that both of the

variables be at least interval scale.

What if we wish to study the relationship

between two variables when one or both are

nominal or ordinal scale? In this case we tally

the results in a contingency table.

61

Contingency Tables – Example

A manufacturer of preassembled windows produced 50 windows yesterday. This morning the quality assurance inspector reviewed each window for all quality aspects. Each was classified as acceptable or unacceptable and by the shift on which it was produced. Thus we reported two variables on a single item. The two variables are shift and quality. The results are reported in the following table.

62

Contingency Table for Investment Choices

($1000’s) Investment Investor A Investor B Investor C Total Category

Stocks 46.5 55 27.5 129 Bonds 32.0 44 19.0 95 CD 15.5 20 13.5 49 Savings 16.0 28 7.0 51 Total 110.0 147 67.0 324

(Individual values could also be expressed as percentages of the overall total,

percentages of the row totals, or percentages of the column totals)

Contingency Table

63

Example

To conduct an efficient advertisement

campaign the relationship between

occupation and newspapers readership is

studied. The following table was created

Blue Collar White collar Professional

G&M 27 29 33

Post 18 43 51

Star 38 15 24

Sun 37 21 18

Contingency Table: Example

64

Solution

If there is no relationship between

occupation and newspaper read, the bar

charts describing the frequency of

readership of newspapers should look

similar across occupations.

Contingency Table: Example

65

Blue

0

10

20

30

40

1 2 3 4

Blue-collar workers prefer

the “Star” and the “Sun”.

White-collar workers and

professionals mostly read

the“Post” and the “Globe

and Mail”

White

0

10

20

30

40

50

1 2 3 4

Prof

0

10

20

30

40

50

60

1 2 3 4

Contingency Table: Example

66

We create a contingency table.

This table lists the frequency for each

combination of values of the two

variables.

We can create a bar chart that

represent the frequency of occurrence

of each combination of values.

Graphing the Relationship Between

Two Nominal Variables

67

Data can be classified according to the time it is collected.

Cross-sectional data are all collected at the same time.

Time-series data are collected at successive points in time.

Time-series data is often depicted on a line chart (a plot of the variable over time).

Describing Time-Series Data

68

Example

The total amount of income tax paid by

individuals in 1987 through 1999 are listed

below.

Draw a graph of this data and describe the

information produced.

Line Chart

69

Line Chart

0

200,000

400,000600,000

800,000

1,000,000

1,200,000

87 88 89 90 91 92 93 94 95 96 97 98 99

For the first five years – total tax was relatively flat From 1993 there was a rapid increase in tax revenues.

Line charts can be used to describe nominal data time series.

Line Chart

70

Present data in a way that provides substance, statistics and design

Communicate complex ideas with clarity, precision and efficiency

Give the largest number of ideas in the most efficient manner

Excellence almost always involves several dimensions

Tell the truth about the data

Principles of Graphical Excellence

71

Providing information concerning the monthly bills of new subscribers in the first month after signing on with a telephone company. (Refer to file) Collect data

Prepare a frequency distribution

Draw a histogram

APPLICATION EXAMPLE

72

42.19 103.15 39.21 89.5 75.71 2.42 8.37 77.21 1.62 109.08 28.77 104.4 35.32 115.78 13.9 6.95

38.45 94.52 48.54 13.36 88.62 1.08 7.18 72.47 91.1 2.45 9.12 2.88 117.69 0.98 9.22 6.48

29.23 26.84 93.31 44.16 99.5 76.69 11.07 0 10.88 21.97 118.75 65.9 106.84 19.45 109.94 11.64

89.35 93.93 104.88 92.97 85 13.62 1.47 5.64 30.62 17.12 0 20.55 8.4 0 10.7 83.26

118.04 90.26 30.61 99.56 0 88.51 26.4 6.48 100.05 19.7 13.95 3.43 90.04 27.21 0 15.42

110.46 72.78 22.57 92.62 8.41 55.99 13.26 6.95 26.97 6.93 14.34 10.44 3.85 89.27 11.27 24.49

0 101.36 63.7 78.89 70.48 12.24 21.13 19.6 15.43 10.05 79.52 21.36 91.56 14.49 72.02 89.13

72.88 104.8 104.84 87.71 92.88 119.63 95.03 8.11 29.25 99.03 2.72 24.42 10.13 92.17 7.74 111.14

83.05 74.01 6.45 93.57 3.2 23.31 29.04 9.01 1.88 29.24 9.63 95.52 5.72 21 5.04 92.64

95.73 56.01 16.47 0 115.5 11.05 5.42 84.77 16.44 15.21 21.34 6.72 33.69 106.59 33.4 53.9

114.67 19.34 15.3 112.94

27.57 13.54 75.49 20.12

64.78 18.89 68.69 53.21

45.81 1.57 35 15.3

56.04 0 9.12 49.24

20.39 5.2 18.49 9.44

31.77 2.8 84.12 2.67

94.67 5.1 13.68 4.69

44.32 3.03 20.84 41.38

3.69 9.16 100.04 45.77

Data of Bill

73

Largest

observation

Collect data

Bills

42.19

38.45

29.23

89.35

118.04

110.46

0.00

72.88

83.05

.

.

(There are 200 data points

Prepare a frequency distribution How many classes to use?

Number of observations Number of classes

Less then 50 5-7

50 - 200 7-9

200 - 500 9-10

500 - 1,000 10-11

1,000 – 5,000 11-13

5,000- 50,000 13-17

More than 50,000 17-20

Class width = [Range] / [# of classes]

[119.63 -0] / [8] = 14.95 15

Largest

observation Largest

observation

Smallest

observation Smallest

observation Smallest

observation

Smallest

observation Largest

observation

NO of Class= 1 +3.3 log (n)

n: No of data/observation

Guide

line

Preparing Frequency Distribution

74

Draw a Histogram Bill Frequency

15 71

30 37

45 13

60 9

75 10

90 18

105 28

120 14

0

20

40

60

80

15 30 45 60 75 90 105 120

Bills

Fre

qu

en

cy

Draw Histogram

75

0

20

40

60

80 1

5

30

45

60

75

90

10

5

12

0

Bills

Fre

qu

en

cy

What information can we extract from this histogram

About half of all

the bills are small

71+37=108 13+9+10=32

A few bills are in

the middle range

Relatively,

large number

of large bills

18+28+14=60

Extracting Information

76

There are four typical shape characteristics

Shapes of Histograms

77

Positively skewed Negatively skewed

•One with the long tail

extending to either right or

left side

Shapes of Histograms

78

A modal class is the one with the largest number of observations.

A unimodal histogram

The modal class

Modal classes

79

A bimodal histogram

A modal class A modal class

Modal classes

80

• A special type of symmetric unimodal histogram is bell shaped

• Many statistical techniques require that the population be bell

shaped.

• Drawing the histogram helps verify the shape of the population in

question

Bell shaped histograms

81

Example 2: Comparing students’ performance

Students’ performance in two statistics classes were compared.

The two classes differed in their teaching emphasis

Class A – mathematical analysis and development of theory.

Class B – applications and computer based analysis.

The final mark for each student in each course was recorded.

Draw histograms and interpret the results.

Interpreting Histograms

82

Marks (Manual) Marks (Computer)

77 59 75 60 65 81 72 59

74 83 71 50 71 53 85 66

75 77 75 52 66 70 72 71

75 74 74 47 79 76 77 68

67 78 53 46 65 73 64 72

72 67 49 50 82 73 77 75

81 82 56 51 80 85 89 74

76 55 61 44 86 83 87 77

79 73 61 52 67 80 78 69

73 92 54 53 64 67 79 60

75 71 44 56 62 78 59 92

52 53 54 53 74 68 63 69

72 75 78 76 67 67 84 69

72 70 73 82 72 62 74 73

83 59 81 82 68 83 74 65

Data

83

Histogram

02040

50 60 70 80 90 100

Marks(Manual)

Fre

qu

en

cy

Histogram

02040

50 60 70 80 90 100

Marks(Computer)

Fre

qu

en

cy

The mathematical emphasis

creates two groups, and a

larger spread.

Interpreting Histograms

84