descriptive statistics i: tabular and graphical methods

1 Slide

Chapter 2Chapter 2Descriptive Statistics:Descriptive Statistics:

Tabular and Graphical MethodsTabular and Graphical Methods Summarizing Qualitative DataSummarizing Qualitative Data Summarizing Quantitative DataSummarizing Quantitative Data Exploratory Data AnalysisExploratory Data Analysis CrosstabulationsCrosstabulations

and Scatter Diagramsand Scatter Diagrams

2 Slide

Summarizing Qualitative DataSummarizing Qualitative Data

Frequency DistributionFrequency Distribution Relative Frequency Relative Frequency Percent Frequency DistributionPercent Frequency Distribution Bar GraphBar Graph Pie ChartPie Chart

3 Slide

Frequency DistributionFrequency Distribution

A A frequency distributionfrequency distribution is a tabular summary is a tabular summary of data showing the frequency (or number) of of data showing the frequency (or number) of items in each of several nonoverlapping items in each of several nonoverlapping classes.classes.

The objective is to The objective is to provide insightsprovide insights about the about the data that cannot be quickly obtained by data that cannot be quickly obtained by looking only at the original data.looking only at the original data.

4 Slide

Example: Marada InnExample: Marada Inn

Guests staying at Marada Inn were asked to rate the Guests staying at Marada Inn were asked to rate the quality of their accommodations as being quality of their accommodations as being excellentexcellent, , above averageabove average, , averageaverage, , below averagebelow average, or , or poorpoor. .

The Theratings provided by a sample of 20 quests are shownratings provided by a sample of 20 quests are shownbelow.below.

Below Average Average Below Average Average Above AverageAbove AverageAbove Average Above Average Above Average Above Average Above Average Above Average Above Average Below Average Above Average Below Average Below Average Below Average Average Poor Average Poor PoorPoorAbove Average ExcellentAbove Average Excellent Above Average Above Average Average Above AverageAverage Above Average AverageAverageAbove Average Average Above Average Average

5 Slide


RatingRating FrequencyFrequencyPoorPoor 2 2Below AverageBelow Average 3 3AverageAverage 5 5Above AverageAbove Average 9 9ExcellentExcellent 1 1

TotalTotal 20 20


6 Slide

Relative Frequency DistributionRelative Frequency Distribution

The The relative frequencyrelative frequency of a class is the fraction of a class is the fraction or proportion of the total number of data items or proportion of the total number of data items belonging to the class.belonging to the class.

A A relative frequency distributionrelative frequency distribution is a tabular is a tabular summary of a set of data showing the relative summary of a set of data showing the relative frequency for each class.frequency for each class.

7 Slide

Percent Frequency DistributionPercent Frequency Distribution

The The percent frequencypercent frequency of a class is the relative of a class is the relative frequency multiplied by 100.frequency multiplied by 100.

AA percent frequency distributionpercent frequency distribution is a tabular is a tabular summary of a set of data showing the percent summary of a set of data showing the percent frequency for each class.frequency for each class.

8 Slide


Relative Frequency and Percent Frequency Relative Frequency and Percent Frequency DistributionsDistributions

RelativeRelative PercentPercentRatingRating FrequencyFrequency FrequencyFrequencyPoorPoor .10 .10 1010Below AverageBelow Average .15 .15 1515AverageAverage .25 .25 2525Above AverageAbove Average .45 .45 4545ExcellentExcellent .05 .05 5 5

TotalTotal 1.00 1.00 100 100

9 Slide

Bar GraphBar Graph

A A bar graphbar graph is a graphical device for depicting is a graphical device for depicting qualitative data.qualitative data.

On the horizontal axis we specify the labels On the horizontal axis we specify the labels that are used for each of the classes.that are used for each of the classes.

A A frequencyfrequency, , relative frequencyrelative frequency, or , or percent percent frequencyfrequency scale can be used for the vertical scale can be used for the vertical axis.axis.

Using a Using a bar of fixed widthbar of fixed width drawn above each drawn above each class label, we extend the height class label, we extend the height appropriately.appropriately.

The The bars are separatedbars are separated to emphasize the fact to emphasize the fact that each class is a separate category.that each class is a separate category.

10 Slide


Bar GraphBar Graph

123456789

Poor BelowAverage

Average AboveAverage

Excellent

Freq

uenc

y

Rating

11 Slide

Pie ChartPie Chart

The The pie chartpie chart is a commonly used graphical is a commonly used graphical device for presenting relative frequency device for presenting relative frequency distributions for qualitative data.distributions for qualitative data.

First draw a First draw a circlecircle; then use the relative ; then use the relative frequencies to subdivide the circle into sectors frequencies to subdivide the circle into sectors that correspond to the relative frequency for that correspond to the relative frequency for each class.each class.

Since there are 360 degrees in a circle, a class Since there are 360 degrees in a circle, a class with a relative frequency of .25 would with a relative frequency of .25 would consume .25(360) = consume .25(360) = 90 degrees of the circle.90 degrees of the circle.

12 Slide


Pie ChartPie Chart

Average 25%

BelowAverage 15%

Poor 10%

AboveAverage 45%

Exc. 5%

Quality Ratings

13 Slide

Insights Gained from the Preceding Pie ChartInsights Gained from the Preceding Pie Chart• One-half of the customers surveyed gave One-half of the customers surveyed gave

Marada a quality rating of “above average” Marada a quality rating of “above average” or “excellent” (looking at the left side of the or “excellent” (looking at the left side of the pie). This might please the manager.pie). This might please the manager.

• For For eacheach customer who gave an “excellent” customer who gave an “excellent” rating, there were rating, there were twotwo customers who gave customers who gave a “poor” rating (looking at the top of the a “poor” rating (looking at the top of the pie). This should displease the manager.pie). This should displease the manager.


14 Slide

Summarizing Quantitative DataSummarizing Quantitative Data

Frequency DistributionFrequency Distribution Relative Frequency and Percent Frequency Relative Frequency and Percent Frequency

DistributionsDistributions Dot PlotDot Plot HistogramHistogram Cumulative DistributionsCumulative Distributions OgiveOgive

15 Slide

91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 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

Example: Hudson Auto RepairExample: Hudson Auto Repair

The manager of Hudson Auto would like to get The manager of Hudson Auto would like to get aa

better picture of the distribution of costs for better picture of the distribution of costs for engineengine

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

been taken and the costs of parts, rounded to been taken and the costs of parts, rounded to thethe

nearest dollar, are listed below.nearest dollar, are listed below.

16 Slide


Guidelines for Selecting Number of ClassesGuidelines for Selecting Number of Classes• Use between 5 and 20 classes.Use between 5 and 20 classes.• Data sets with a larger number of elements Data sets with a larger number of elements

usually require a larger number of classes.usually require a larger number of classes.• Smaller data sets usually require fewer Smaller data sets usually require fewer

classes.classes.

17 Slide


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

Largest Data Value Smallest Data ValueNumber of Classes

18 Slide


Frequency DistributionFrequency DistributionIf we choose six classes: If we choose six classes: Approximate Class Width = (109 - 52)/6 = 9.5 Approximate Class Width = (109 - 52)/6 = 9.5 1010Cost ($)Cost ($) FrequencyFrequency 50-5950-59 2 2 60-69 60-69 13 13 70-7970-79 16 16 80-8980-89 7 7 90-9990-99 7 7100-109100-109 5 5 Total 50Total 50

19 Slide

Relative Frequency and Percent Frequency Relative Frequency and Percent Frequency DistributionsDistributions

Relative PercentRelative Percent Cost ($)Cost ($) FrequencyFrequency FrequencyFrequency 50-5950-59 .04.04 4 4 60-6960-69 .26.26 2626 70-7970-79 .32.32 3232 80-8980-89 .14.14 1414 90-9990-99 .14.14 1414 100-109100-109 .10.10 1010 Total 1.00Total 1.00 100 100


20 Slide


Insights Gained from the Percent Frequency Insights Gained from the Percent Frequency DistributionDistribution• Only 4% of the parts costs are in the $50-59 Only 4% of the parts costs are in the $50-59

class.class.• 30% of the parts costs are under $70.30% of the parts costs are under $70.• The greatest percentage (32% or almost The greatest percentage (32% or almost

one-third) of the parts costs are in the $70-one-third) of the parts costs are in the $70-79 class.79 class.

• 10% of the parts costs are $100 or more.10% of the parts costs are $100 or more.

21 Slide

Dot PlotDot Plot

One of the simplest graphical summaries of One of the simplest graphical summaries of data is a data is a dot plotdot plot..

A horizontal axis shows the range of data A horizontal axis shows the range of data values.values.

Then each data value is represented by a dot Then each data value is represented by a dot placed above the axis.placed above the axis.

22 Slide


Dot PlotDot Plot

. . .. . . .

5050 6060 7070 8080 9090 100100 110110

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

Cost ($)

23 Slide

HistogramHistogram

Another common graphical presentation of Another common graphical presentation of quantitative data is a quantitative data is a histogramhistogram..

The variable of interest is placed on the The variable of interest is placed on the horizontal axis.horizontal axis.

A rectangle is drawn above each class interval A rectangle is drawn above each class interval with its height corresponding to the interval’s with its height corresponding to the interval’s frequencyfrequency, , relative frequencyrelative frequency, or , or percent percent frequencyfrequency..

Unlike a bar graph, a histogram has Unlike a bar graph, a histogram has no natural no natural separation between rectanglesseparation between rectangles of adjacent of adjacent classes.classes.

24 Slide


HistogramHistogram

PartsCost ($)

2468

1012141618

Freq

uenc

y

50 60 70 80 90 100 110

25 Slide

Cumulative DistributionsCumulative Distributions

Cumulative frequency distributionCumulative frequency distribution -- shows the -- shows the number of itemsnumber of items with values less than or with values less than or equal to the upper limit of each class.equal to the upper limit of each class.

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

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

26 Slide


Cumulative DistributionsCumulative DistributionsCumulative CumulativeCumulative Cumulative Cumulative Relative PercentCumulative Relative PercentCost ($)Cost ($) FrequencyFrequency FrequencyFrequency FrequencyFrequency << 59 59 2 2 .04 .04 4 4 << 69 69 15 15 .30 .30 30 30 << 79 79 31 31 .62 .62 62 62 << 89 89 38 38 .76 .76 76 76 << 99 99 45 45 .90 .90 90 90 << 109 109 50 50 1.00 1.00 100 100

27 Slide

OgiveOgive

An An ogiveogive is a graph of a cumulative distribution. is a graph of a cumulative distribution. The data values are shown on the horizontal The data values are shown on the horizontal

axis.axis. Shown on the vertical axis are the:Shown on the vertical axis are the:

• cumulative frequencies, orcumulative frequencies, or• cumulative relative frequencies, orcumulative relative frequencies, or• cumulative percent frequenciescumulative percent frequencies

The frequency (one of the above) of each class The frequency (one of the above) of each class is plotted as a point.is plotted as a point.

The plotted points are connected by straight The plotted points are connected by straight lines.lines.

28 Slide


OgiveOgive• Because the class limits for the parts-cost Because the class limits for the parts-cost

data are 50-59, 60-69, and so on, there data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, appear to be one-unit gaps from 59 to 60, 69 to 70, and so on.69 to 70, and so on.

• These gaps are eliminated by plotting points These gaps are eliminated by plotting points halfway between the class limits.halfway between the class limits.

• Thus, 59.5 is used for the 50-59 class, 69.5 Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on.is used for the 60-69 class, and so on.

29 Slide


Ogive with Cumulative Percent Frequencies Ogive with Cumulative Percent Frequencies

PartsPartsCost ($)Cost ($)

20

40

60

80

100

Cum

ulat

ive

Perc

ent F

requ

ency

Cum

ulat

ive

Perc

ent F

requ

ency

50 60 70 80 90 100 11050 60 70 80 90 100 110

30 Slide

Exploratory Data AnalysisExploratory Data Analysis

The techniques of The techniques of exploratory data analysisexploratory data analysis consist of simple arithmetic and easy-to-draw consist of simple arithmetic and easy-to-draw pictures that can be used to summarize data pictures that can be used to summarize data quickly.quickly.

One such technique is the One such technique is the stem-and-leaf stem-and-leaf displaydisplay..

31 Slide

Stem-and-Leaf DisplayStem-and-Leaf Display

A stem-and-leaf display shows both the A stem-and-leaf display shows both the rank rank orderorder and and shape of the distributionshape of the distribution of the data. of the data.

It is It is similar to a histogramsimilar to a histogram on its side, but it has on its side, but it has the advantage of showing the actual data values.the advantage of showing the actual data values.

The first digits of each data item are arranged to The first digits of each data item are arranged to the left of a vertical line.the left of a vertical line.

To the right of the vertical line we record the last To the right of the vertical line we record the last digit for each item in rank order.digit for each item in rank order.

Each line in the display is referred to as a Each line in the display is referred to as a stemstem.. Each digit on a stem is a Each digit on a stem is a leafleaf..

88 5 75 7 99 3 6 7 83 6 7 8

32 Slide


Leaf UnitsLeaf Units• A single digit is used to define each leaf.A single digit is used to define each leaf.• In the preceding example, the leaf unit was In the preceding example, the leaf unit was

1.1.• Leaf units may be 100, 10, 1, 0.1, and so Leaf units may be 100, 10, 1, 0.1, and so

on.on.• Where the leaf unit is not shown, it is Where the leaf unit is not shown, it is

assumed to equal 1.assumed to equal 1.

33 Slide

Example: Leaf Unit = 0.1Example: Leaf Unit = 0.1

If we have data with values such asIf we have data with values such as8.68.6 11.711.7 9.49.4 9.19.1 10.210.2 11.011.0 8.88.8

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

Leaf Unit = 0.1Leaf Unit = 0.1 8 6 88 6 8 9 1 49 1 4 10 210 2 11 0 711 0 7

34 Slide

Example: Leaf Unit = 10Example: Leaf Unit = 10

If we have data with values such asIf we have data with values such as18061806 17171717 19741974 17911791 16821682 19101910 18381838

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

Leaf Unit = 10Leaf Unit = 10 16 16 8 8 17 1 917 1 9 18 0 318 0 3 19 1 719 1 7

35 Slide



55 2 7 2 7 66 2 2 2 2 5 6 7 8 8 8 9 9 9 2 2 2 2 5 6 7 8 8 8 9 9 9 77 1 1 2 2 3 4 4 5 5 5 6 7 8 9 1 1 2 2 3 4 4 5 5 5 6 7 8 9

9 9 9 9 88 0 0 2 3 5 8 9 0 0 2 3 5 8 9 99 1 3 7 7 7 8 9 1 3 7 7 7 8 9

1010 1 4 5 5 9 1 4 5 5 9

36 Slide

Stretched Stem-and-Leaf DisplayStretched Stem-and-Leaf Display

If we believe the original stem-and-leaf display If we believe the original stem-and-leaf display has condensed the data too much, we can has condensed the data too much, we can stretch the displaystretch the display by using two more stems by using two more stems for each leading digit(s).for each leading digit(s).

Whenever a stem value is stated twice, the Whenever a stem value is stated twice, the first value corresponds to leaf values of 0-4, first value corresponds to leaf values of 0-4, and the second values corresponds to values and the second values corresponds to values of 5-9.of 5-9.

37 Slide


Stretched Stem-and-Leaf DisplayStretched Stem-and-Leaf Display 55 2 2 55 7 7 66 2 2 2 2 2 2 2 2 66 5 6 7 8 8 8 9 9 9 5 6 7 8 8 8 9 9 9 77 1 1 2 2 3 4 4 1 1 2 2 3 4 4 77 5 5 5 6 7 8 9 9 9 5 5 5 6 7 8 9 9 9 88 0 0 2 3 0 0 2 3 88 5 8 9 5 8 9 99 1 3 1 3 99 7 7 7 8 9 7 7 7 8 9

1010 1 4 1 4 1010 5 5 9 5 5 9

38 Slide

Crosstabulations and Scatter DiagramsCrosstabulations and Scatter Diagrams

Thus far we have focused on methods that are Thus far we have focused on methods that are used to summarize the data for used to summarize the data for one variable at one variable at a timea time..

Often a manager is interested in tabular and Often a manager is interested in tabular and graphical methods that will help understand graphical methods that will help understand the the relationship between two variablesrelationship between two variables..

CrosstabulationCrosstabulation and a and a scatter diagramscatter diagram are two are two methods for summarizing the data for two (or methods for summarizing the data for two (or more) variables simultaneously.more) variables simultaneously.

39 Slide

CrosstabulationCrosstabulation

CrosstabulationCrosstabulation is a tabular method for is a tabular method for summarizing the data for two variables summarizing the data for two variables simultaneously.simultaneously.

Crosstabulation can be used when:Crosstabulation can be used when:• One variable is qualitative and the other is One variable is qualitative and the other is

quantitativequantitative• Both variables are qualitativeBoth variables are qualitative• Both variables are quantitativeBoth variables are quantitative

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

40 Slide

Example: Finger Lakes HomesExample: Finger Lakes Homes

CrosstabulationCrosstabulationThe number of Finger Lakes homes sold for each The number of Finger Lakes homes sold for each style and price for the past two years is shown below.style and price for the past two years is shown below. PricePrice Home StyleHome Style

RangeRange Colonial Ranch Split A-Frame TotalColonial Ranch Split A-Frame Total

<< $99,000 18 $99,000 18 6 19 12 6 19 12 55 55

> $99,000 12 14 16 3> $99,000 12 14 16 3 4545

TotalTotal 3030 20 35 15 20 35 15 100 100

41 Slide


Insights Gained from the Preceding Insights Gained from the Preceding CrosstabulationCrosstabulation• The greatest number of homes in the The greatest number of homes in the

sample (19) are a split-level style and priced sample (19) are a split-level style and priced at less than or equal to $99,000.at less than or equal to $99,000.

• Only three homes in the sample are an A-Only three homes in the sample are an A-Frame style and priced at more than Frame style and priced at more than $99,000. $99,000.

42 Slide

Crosstabulation: Row or Column Crosstabulation: Row or Column PercentagesPercentages

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

43 Slide


Row PercentagesRow Percentages

PricePrice Home StyleHome Style RangeRange Colonial Ranch Split A-Frame Colonial Ranch Split A-Frame

TotalTotal

<< $99,000 32.73 $99,000 32.73 10.91 34.55 10.91 34.55 21.8221.82 100 100

> $99,000 26.67 31.11 35.56 6.67> $99,000 26.67 31.11 35.56 6.67 100 100

Note: row totals are actually 100.01 due to rounding.Note: row totals are actually 100.01 due to rounding.

44 Slide


Column PercentagesColumn Percentages

PricePrice Home StyleHome Style RangeRange Colonial Ranch Split A-FrameColonial Ranch Split A-Frame << $99,000 60.00 $99,000 60.00 30.00 54.29 30.00 54.29

80.0080.00 > $99,000 40.00> $99,000 40.00 70.00 45.71 70.00 45.71

20.0020.00

TotalTotal 100 100 100 100 100 100 100100

45 Slide

Scatter DiagramScatter Diagram

A scatter diagram is a graphical presentation A scatter diagram is a graphical presentation of the relationship between two of the relationship between two quantitativequantitative variables.variables.

One variable is shown on the horizontal axis One variable is shown on the horizontal axis and the other variable is shown on the vertical and the other variable is shown on the vertical axis.axis.

The general pattern of the plotted points The general pattern of the plotted points suggests the overall relationship between the suggests the overall relationship between the variables.variables.

46 Slide

Example: Panthers Football TeamExample: Panthers Football Team

Scatter DiagramScatter DiagramThe Panthers football team is interested The Panthers football team is interested

in investigating the relationship, if any, in investigating the relationship, if any, between interceptions made and points between interceptions made and points scored.scored.

xx = Number of = Number of yy = Number of = Number of InterceptionsInterceptions Points ScoredPoints Scored

11 14 14 33 24 24 22 18 18 11 17 17 33 27 27

47 Slide


Scatter DiagramScatter Diagramyy

xx

Number of InterceptionsNumber of Interceptions11 22 33Nu

mbe

r of P

oint

s Sco

red

Num

ber o

f Poi

nts S

core

d

00

5510101515202025253030

00

48 Slide


The preceding scatter diagram indicates a The preceding scatter diagram indicates a positive relationship between the number of positive relationship between the number of interceptions and the number of points scored.interceptions and the number of points scored.

Higher points scored are associated with a Higher points scored are associated with a higher number of interceptions.higher number of interceptions.

The relationship is not perfect; all plotted The relationship is not perfect; all plotted points in the scatter diagram are not on a points in the scatter diagram are not on a straight line.straight line.

49 Slide


A Positive RelationshipA Positive Relationship

xx

yy

50 Slide


A Negative RelationshipA Negative Relationship

xx

yy

51 Slide


No Apparent RelationshipNo Apparent Relationship

xx

yy

52 Slide

Tabular and Graphical ProceduresTabular and Graphical ProceduresDataData

Qualitative DataQualitative Data Quantitative DataQuantitative Data

TabularTabularMethodsMethods

TabularTabularMethodsMethods

GraphicalGraphical MethodsMethods

GraphicalGraphical MethodsMethods

•FrequencyFrequency DistributionDistribution•Rel. Freq. Dist.Rel. Freq. Dist.•% Freq. Dist.% Freq. Dist.•CrosstabulationCrosstabulation

•Bar GraphBar Graph•Pie ChartPie Chart

•FrequencyFrequency DistributionDistribution•Rel. Freq. Dist.Rel. Freq. Dist.•Cum. Freq. Dist.Cum. Freq. Dist.•Cum. Rel. Freq.Cum. Rel. Freq. Distribution Distribution •Stem-and-LeafStem-and-Leaf DisplayDisplay•CrosstabulationCrosstabulation

•Dot PlotDot Plot•HistogramHistogram•OgiveOgive•ScatterScatter DiagramDiagram

53 Slide

End of Chapter 2End of Chapter 2

descriptive statistics i: tabular and graphical methods

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