1

GRAPHICAL DESCRIPTIVE TECHNIQUES

Content• Variable and data• Types of data

– Quantitative, qualitative, and ranked data– Cross sectional and time series data

• Graphical techniques for quantitative data– Histogram: frequency distribution, relative frequency

distribution, shapes of histograms– Ogives, stem and leaf displays and dot plots

2

VARIABLE AND DATA

• Variable– A characteristic of a population or sample that is of

interest to us.• Example

– In case of Hudson Auto a variable is the average cost of parts used in engine tune-ups.

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VARIABLE AND DATA

• Data – The actual values of the variables– Data may be quantitative, qualitative or ranked

• Example– In case of Hudson Auto data are the actual costs

of parts used in 50 engine tune-ups observed:

91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76

104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73

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Data

Quantitative Qualitative Ranked

Discrete Continuous

TYPES OF DATA

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QUANTITATIVE DATA

• Quantitative data may have a ratio scale– Data possessing a natural zero point and

organized into measures for which differences are meaningful.

– Examples: Money, income, sales, profits, losses, heights of NBA players

• Quantitative data may also have an interval scale– The distance between numbers is a known,

constant size, but the zero value is arbitrary.– Examples: Temperature on the Fahrenheit scale.

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QUALITATIVE DATA

• Qualitative data has a nominal scale– Data that can only be classified into categories

and cannot be arranged in an ordering scheme.– Examples: eye color, gender, marital status,

religious affiliation, etc. – Examples: Suppose that the responses to the

marital-status question is recorded as follows:

Single 1 Divorced3

Married 2 Widowed 4

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RANKED DATA

• Ranked data has an ordinal scale

– Data or categories that can be ranked; that is, one category is higher than another. However, numerical differences between data values cannot be determined.

– Examples: Maclean’s ranking (medical doctoral);

1. University of Toronto

2. University of British Columbia

3. Queen’s university– Examples: Opinion survey:

Excellent 4 Fair 2Good 3 Poor 1

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DISCRETE AND CONTINUOUS DATA

• Discrete – Assume values that can be counted.– Example: Number of defective items, number of

customers arrive in a bank• Continuous

– Can assume all values between any two specific values. They are obtained by measuring.

– Example: length, weight, volume

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CROSS-SECTIONAL AND TIME-SERIES DATA

• Cross-sectional data– Observations are measure at the same time– Examples: Marketing surveys and political opinion

polls• Time-series data

– Observations are measured at successive points in time

– Examples: Monthly sales data, daily temperature data

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HISTOGRAM

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 3975 56 71 51 99 68 95 8657 53 47 50 55 81 80 9851 31 63 66 85 79 83 70

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HISTOGRAM

• 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: See Table 2.3, p. 29Number of observation Number of classes

Less than 50 5-750-200 7-9200-500 9-10500-1,000 10-111,000-5,000 11-135,000-50,000 13-17More than 50,000 17-20

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HISTOGRAM

• Approximate class width is obtained as follows:

classes of Number

value Smallest-value Largest widthclass eApproximat

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HISTOGRAM

Classes and counts for the days-to-maturity data

Days toMaturity

TALLY Number ofInvestments

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HISTOGRAM

• Class relative frequency is obtained as follows:

nsobservatio of number Total

frequency Classfrequency relative Class

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HISTOGRAM

0

2

4

6

8

10

12

40 50 60 70 80 90 100

Number of Days to Maturity

Fre

qu

ency

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HISTOGRAMClasses: Categories for grouping data.Frequency: The number of observations that fall in a class.Frequency distribution: A listing of all classes along with theirfrequencies.Relative frequency: The ratio of the frequency of a class to the totalnumber of observations.Relative-frequency distribution: A listing of all classes along withtheir relative frequencies.Lower cutpoint: The smallest value that can go in a class.Upper cutpoint: The smallest value that can go in the next higherclass. 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 itslower and upper cutpoints.Width: The difference between the upper and lower cutpoints of aclass.

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HISTOGRAM

Frequency histogram: A graph that displays the classes onthe horizontal axis and the frequencies of the classes on thevertical axis. The frequency of each class is represented by avertical bar whose height is equal to the frequency of the class.

Relative-frequency histogram: A graph that displays theclasses on the horizontal axis and the relative frequencies ofthe classes on the vertical axis. The relative frequency of eachclass is represented by a vertical bar whose height is equal tothe relative frequency of the class.

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RELATIVE FREQUENCY HISTOGRAM

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

Days toMaturity

Relative Frequency

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RELATIVE FREQUENCY HISTOGRAM

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

40 50 60 70 80 90 100

Number of Days to Maturity

Rel

ativ

e F

req

uen

cy

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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.

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FREQUENCY POLYGON

Classes Mid-value Frequency

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FREQUENCY POLYGON

0

2

4

6

8

10

12

35 45 55 65 75 85 95

Number of Days to Maturity

Fre

qu

ency

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OGIVECUMULATIVE 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.

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OGIVECUMULATIVE RELATIVE FREQUENCY GRAPH

Class Frequency RelativeFrequency

CumulativeRelative

Frequency

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OGIVE

CUMULATIVE RELATIVE FREQUENCY GRAPH

0.075 0.100

0.300

0.550

0.725

0.9001.000

0.000

0.200

0.400

0.600

0.800

1.000

40 50 60 70 80 90 100

Number of Days to Maturity

Cu

mu

lati

ve F

req

uen

cy

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SYMMETRIC HISTOGRAM

0

2

4

6

8

10

12

14

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

Number of Units Sold

Fre

qu

ency

27

SYMMETRIC HISTOGRAM

0

2

4

6

8

10

12

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

Number of Units Sold

Fre

qu

ency

28

SYMMETRIC HISTOGRAM

0

1

2

3

4

5

6

7

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

Number of Units Sold

Fre

qu

ency

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POSITIVELY SKEWED HISTOGRAM

0

2

4

6

8

10

12

14

16

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

Number of Units Sold

Fre

qu

ency

30

NEGATIVELY SKEWED HISTOGRAM

0

2

4

6

8

10

12

14

16

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

Number of Units Sold

Fre

qu

ency

31

BIMODAL HISTOGRAM

0

1

2

3

4

5

6

7

8

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

Number of Units Sold

Fre

qu

ency

32

BELL-SHAPED HISTOGRAM

0

2

4

6

8

10

12

14

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

Number of Units Sold

Fre

qu

ency

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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.

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STEM-AND-LEAF DISPLAY

Diagrams for days-to-maturity data: (a) stem-and-leaf (b) ordered stem-and-leaf

Stem Leaves Stem Leaves 3 3 4 4 5 5 6 6 7 7 8 8 9 9 (a) (b)

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DOT PLOTS

Number of times oat yields was 60 bushels = 2 (count the dots)

Oat Yields

55 56 57 58 59 60 61 62 63 64 65 66 67

Yield (bushels)