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Chapter 2 Descriptive Statistics: Tabular and Graphical Methods

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- Summarizing Qualitative Data
- Summarizing Quantitative Data
- Exploratory Data Analysis
- Crosstabulations
and Scatter Diagrams

- Frequency Distribution
- Relative Frequency
- Percent Frequency Distribution
- Bar Graph
- Pie Chart

- A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several nonoverlapping classes.
- The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data.

Guests staying at Marada Inn were asked to rate the

quality of their accommodations as being excellent,

above average, average, below average, or poor. The

ratings provided by a sample of 20 quests are shown

below.

Below Average Average Above Average

Above Average Above Average Above Average Above Average Below Average Below Average Average Poor Poor

Above Average ExcellentAbove Average

Average Above AverageAverage

Above Average Average

- Frequency Distribution
RatingFrequency

Poor 2

Below Average 3

Average 5

Above Average 9

Excellent 1

Total 20

- The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class.
- A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class.

- The percent frequency of a class is the relative frequency multiplied by 100.
- Apercent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class.

- Relative Frequency and Percent Frequency Distributions
RelativePercent

RatingFrequencyFrequency

Poor .1010

Below Average .1515

Average .2525

Above Average .4545

Excellent .05 5

Total 1.00 100

- A bar graph is a graphical device for depicting qualitative data.
- On the horizontal axis we specify the labels that are used for each of the classes.
- A frequency, relative frequency, or percent frequency scale can be used for the vertical axis.
- Using a bar of fixed width drawn above each class label, we extend the height appropriately.
- The bars are separated to emphasize the fact that each class is a separate category.

9

8

7

6

Frequency

5

4

3

2

1

Rating

Above

Average

Excellent

Poor

Below

Average

Average

- Bar Graph

- The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data.
- First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class.
- Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) =
90 degrees of the circle.

Exc.

5%

Poor

10%

Below

Average

15%

Above

Average

45%

Average

25%

Quality Ratings

- Pie Chart

- Insights Gained from the Preceding Pie Chart
- One-half of the customers surveyed gave Marada a quality rating of “above average” or “excellent” (looking at the left side of the pie). This might please the manager.
- For each customer who gave an “excellent” rating, there were two customers who gave a “poor” rating (looking at the top of the pie). This should displease the manager.

- Frequency Distribution
- Relative Frequency and Percent Frequency Distributions
- Dot Plot
- Histogram
- Cumulative Distributions
- Ogive

The manager of Hudson Auto would like to get a

better picture of the distribution of costs for engine

tune-up parts. A sample of 50 customer invoices has

been taken and the costs of parts, rounded to the

nearest dollar, are listed below.

- Guidelines for Selecting Number of Classes
- Use between 5 and 20 classes.
- Data sets with a larger number of elements usually require a larger number of classes.
- Smaller data sets usually require fewer classes.

- Guidelines for Selecting Width of Classes
- Use classes of equal width.
- Approximate Class Width =

- Frequency Distribution
If we choose six classes:

Approximate Class Width = (109 - 52)/6 = 9.5 10

Cost ($)Frequency

50-59 2

60-69 13

70-79 16

80-89 7

90-99 7

100-109 5

Total 50

- Relative Frequency and Percent Frequency Distributions
Relative Percent

Cost ($)FrequencyFrequency

50-59.04 4

60-69 .2626

70-79.3232

80-89 .1414

90-99.1414

100-109 .1010

Total 1.00 100

- Insights Gained from the Percent Frequency Distribution
- Only 4% of the parts costs are in the $50-59 class.
- 30% of the parts costs are under $70.
- The greatest percentage (32% or almost one-third) of the parts costs are in the $70-79 class.
- 10% of the parts costs are $100 or more.

- One of the simplest graphical summaries of data is a dot plot.
- A horizontal axis shows the range of data values.
- Then each data value is represented by a dot placed above the axis.

.

. .. . . .

. .. .. .. .. . .

. . . ..... .......... .. . .. . . ... . .. .

5060708090100110

Cost ($)

- Dot Plot

- Another common graphical presentation of quantitative data is a histogram.
- The variable of interest is placed on the horizontal axis.
- A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency.
- Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes.

- Histogram

18

16

14

12

Frequency

10

8

6

4

2

Parts

Cost ($)

50 60 70 80 90 100 110

- Cumulative frequency distribution -- shows the number of items with values less than or equal to the upper limit of each class.
- Cumulative relative frequency distribution -- shows the proportion of items with values less than or equal to the upper limit of each class.
- Cumulative percent frequency distribution -- shows the percentage of items with values less than or equal to the upper limit of each class.

- Cumulative Distributions
Cumulative Cumulative

Cumulative Relative Percent

Cost ($)FrequencyFrequencyFrequency

< 59 2 .04 4

< 69 15 .30 30

< 79 31 .62 62

< 89 38 .76 76

< 99 45 .90 90

< 109 50 1.00 100

- An ogive is a graph of a cumulative distribution.
- The data values are shown on the horizontal axis.
- Shown on the vertical axis are the:
- cumulative frequencies, or
- cumulative relative frequencies, or
- cumulative percent frequencies

- The frequency (one of the above) of each class is plotted as a point.
- The plotted points are connected by straight lines.

- Ogive
- Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, 69 to 70, and so on.
- These gaps are eliminated by plotting points halfway between the class limits.
- Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on.

- Ogive with Cumulative Percent Frequencies

100

80

60

Cumulative Percent Frequency

40

20

Parts

Cost ($)

50 60 70 80 90 100 110

- The techniques of exploratory data analysis consist of simple arithmetic and easy-to-draw pictures that can be used to summarize data quickly.
- One such technique is the stem-and-leaf display.

- A stem-and-leaf display shows both the rank order and shape of the distribution of the data.
- It is similar to a histogram on its side, but it has the advantage of showing the actual data values.
- The first digits of each data item are arranged to the left of a vertical line.
- To the right of the vertical line we record the last digit for each item in rank order.
- Each line in the display is referred to as a stem.
- Each digit on a stem is a leaf.
85 7

93 6 7 8

- Leaf Units
- A single digit is used to define each leaf.
- In the preceding example, the leaf unit was 1.
- Leaf units may be 100, 10, 1, 0.1, and so on.
- Where the leaf unit is not shown, it is assumed to equal 1.

If we have data with values such as

8.611.79.49.110.211.08.8

a stem-and-leaf display of these data will be

Leaf Unit = 0.1

8 6 8

9 1 4

10 2

11 0 7

If we have data with values such as

1806171719741791168219101838

a stem-and-leaf display of these data will be

Leaf Unit = 10

16 8

17 1 9

18 0 3

19 1 7

- Stem-and-Leaf Display
5 2 7

6 2 2 2 2 5 6 7 8 8 8 9 9 9

7 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9

8 0 0 2 3 5 8 9

9 1 3 7 7 7 8 9

10 1 4 5 5 9

- If we believe the original stem-and-leaf display has condensed the data too much, we can stretch the display by using two more stems for each leading digit(s).
- Whenever a stem value is stated twice, the first value corresponds to leaf values of 0-4, and the second values corresponds to values of 5-9.

- Stretched Stem-and-Leaf Display
5 2

5 7

6 2 2 2 2

6 5 6 7 8 8 8 9 9 9

7 1 1 2 2 3 4 4

7 5 5 5 6 7 8 9 9 9

8 0 0 2 3

8 5 8 9

9 1 3

9 7 7 7 8 9

10 1 4

10 5 5 9

- Thus far we have focused on methods that are used to summarize the data for one variable at a time.
- Often a manager is interested in tabular and graphical methods that will help understand the relationship between two variables.
- Crosstabulation and a scatter diagram are two methods for summarizing the data for two (or more) variables simultaneously.

- Crosstabulation is a tabular method for summarizing the data for two variables simultaneously.
- Crosstabulation can be used when:
- One variable is qualitative and the other is quantitative
- Both variables are qualitative
- Both variables are quantitative

- The left and top margin labels define the classes for the two variables.

- Crosstabulation
The number of Finger Lakes homes sold for each style and price for the past two years is shown below.

PriceHome Style

RangeColonial Ranch Split A-Frame Total

< $99,000 18 6 19 12 55

> $99,000 12 14 16 3 45

Total30 20 35 15 100

- Insights Gained from the Preceding Crosstabulation
- The greatest number of homes in the sample (19) are a split-level style and priced at less than or equal to $99,000.
- Only three homes in the sample are an A-Frame style and priced at more than $99,000.

- Converting the entries in the table into row percentages or column percentages can provide additional insight about the relationship between the two variables.

- Row Percentages
PriceHome Style

RangeColonial Ranch Split A-Frame Total

< $99,000 32.73 10.91 34.55 21.82 100

> $99,000 26.67 31.11 35.56 6.67 100

Note: row totals are actually 100.01 due to rounding.

- Column Percentages
PriceHome Style

RangeColonial Ranch Split A-Frame

< $99,000 60.00 30.00 54.29 80.00

> $99,000 40.00 70.00 45.71 20.00

Total 100 100 100 100

- A scatter diagram is a graphical presentation of the relationship between two quantitative variables.
- One variable is shown on the horizontal axis and the other variable is shown on the vertical axis.
- The general pattern of the plotted points suggests the overall relationship between the variables.

- Scatter Diagram
The Panthers football team is interested in investigating the relationship, if any, between interceptions made and points scored.

x = Number of y = Number of

InterceptionsPoints Scored

1 14

3 24

2 18

1 17

3 27

- Scatter Diagram

y

30

25

20

Number of Points Scored

15

10

5

x

0

1

0

2

3

Number of Interceptions

- The preceding scatter diagram indicates a positive relationship between the number of interceptions and the number of points scored.
- Higher points scored are associated with a higher number of interceptions.
- The relationship is not perfect; all plotted points in the scatter diagram are not on a straight line.

y

x

- A Positive Relationship

y

x

- A Negative Relationship

y

x

- No Apparent Relationship

Data

Qualitative Data

Quantitative Data

Tabular

Methods

Graphical

Methods

Tabular

Methods

Graphical

Methods

- Frequency
- Distribution
- Rel. Freq. Dist.
- % Freq. Dist.
- Crosstabulation

- Bar Graph
- Pie Chart

- Dot Plot
- Histogram
- Ogive
- Scatter
- Diagram

- Frequency
- Distribution
- Rel. Freq. Dist.
- Cum. Freq. Dist.
- Cum. Rel. Freq.
- Distribution
- Stem-and-Leaf
- Display
- Crosstabulation