02_FrequencyHistogram

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STAT 201Introduction to Business Statistics

Class 2: Describing Data by Graphs, Charts,Tables


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Takeaways from Class 1

Statisticsis the study of the collection, organization,

analysis, interpretation, and presentation of data gives managers a better understanding of the

business environment

enables them to make more informed and betterdecisions

Descriptive Statistics: Summaries of data which may

be graphical, tabular, or numerical Inferential Statistics:Procedures that help draw

conclusions about a set of data from a subset of thatdata

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Population:the set of all items or individuals of

interest A parameteris a summary measure computed to

describe a characteristic of the population

Sample:a subset of the population

A statisticis a summary measure computed to

describe a characteristic of the sample drawn from

the population

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Types of Data:

Qualitative (Categorical):Data grouped by specificcategories (e.g., eye color, marital status), generally

Nominal Scale: only labels

Ordinal Scale: can be ordered

Quantitative (Numerical):Data grouped bynumerical values (e.g., number of children, weight),generally

Interval Scale: meaningful intervals in addition to ordinalscale

Ratio Scale: meaningful ratios in addition to interval scale(there is a zero value)

Time Series Data:Collected over several timeperiods

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Todays Focus

Introducing tabular and graphical methods

commonly used to summarize both categorical andquantitative data.

Tabular and graphical summaries of data are foundin:

Annual reports

Newspaper articles

Research studies

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Sources: The Economist, RealClearPolitics.com


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Summarizing Categorical Data

Frequency Distribution:

A frequency distribution is a tabular summary of datashowing the number (frequency) of items of severalnon-overlapping classes.

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Example: Soft Drink Purchases*

Data from a sample of 50 soft drink purchases

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Source: Modern Business Statistics by Anderson, Sweeney, Williams

Coke Classic Sprite Pepsi

Diet Coke Coke Classic Coke Classic

Pepsi Diet Coke Coke Classic


Coke Classic Diet Coke Pepsi

Coke Classic Coke Classic Dr. Pepper

Dr. Pepper Sprite Coke ClassicDiet Coke Pepsi Diet Coke

Pepsi Coke Classic Pepsi


Coke Classic Coke Classic Pepsi

Dr. Pepper Pepsi Pepsi

Sprite Coke Classic Coke ClassicCoke Classic Sprite Dr. Pepper

Diet Coke Dr. Pepper Pepsi

Coke Classic Pepsi Sprite

Coke Classic Diet Coke


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To develop a frequency distribution, count the

number of times each item type appears in data.

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Soft Drink

Frequenc

y

Coke ClassicDiet Coke

Dr. Pepper

Pepsi

Sprite


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11

Soft Drink

Frequenc

y

Coke Classic 19Diet Coke

Dr. Pepper

Pepsi

Sprite


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12

















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13

Soft Drink

Frequenc

y

Coke Classic 19Diet Coke 8

Dr. Pepper

Pepsi

Sprite


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14

















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15

Soft Drink

Frequenc

y


Dr. Pepper 5

Pepsi 13

Sprite 5


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What can we say by looking at this data? Who is the

market leader? Coke Classic is the market leader

Pepsi is second

Diet Coke is third

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Soft Drink

Frequenc

yCoke Classic 19

Diet Coke 8

Dr. Pepper 5

Pepsi 13

Sprite 5


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The summary provides more insight than the rawdata!

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Soft Drink

Frequenc

y


Dr. Pepper 5Pepsi 13

Sprite 5



Pepsi Diet Coke Coke ClassicDiet Coke Coke Classic Coke Classic



Dr. Pepper Sprite Coke Classic

Diet Coke Pepsi Diet Coke





Sprite Coke Classic Coke Classic

Coke Classic Sprite Dr. Pepper


Coke Classic Pepsi SpriteCoke Classic Diet Coke


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Relative frequency of a class equals the fraction or

proportion of items belonging to a class.

For a data with n observations:

Relative frequency of a class = Frequency of class / n

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Relative frequency of a class equals the fraction or

proportion of items belonging to a class.

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Soft Drink Frequency Relative Frequency

Percent

Frequency

Coke Classic 19 0.38 38

Diet Coke 8 0.16 16

Dr. Pepper 5 0.10 10

Pepsi 13 0.26 26

Sprite 5 0.10 10

Total 50 1.00 100


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Summarizing Quantitative Data

A frequency distribution is a tabular summary of datashowing the number (frequency) of items of severalnon-overlappingclasses.

Three steps necessary to define the classes for afrequency distribution with quantitative data:

Determine the number of non-overlapping classes

Determine the width of each class Determine the class limits

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Too many classes:

May yield a very jagged distribution with gaps from emptyclasses

Can give a poor indication of how frequency varies acrossclasses

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0

0.5

1

1.5

2

2.5

3

3.5

4 812

16

20

24

28

32

36

40

44

48

52

56

60

More

Temperature

Frequency


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Too few classes:

May compress variation too much and yield a blockydistribution

Can obscure important patterns of variation

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0

2

4

6

8

10

12

0 30 60 More

Temperature

Frequency


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Number of classes:

Number of Data Points Number of

Classes

under 50 5 - 750100 6 - 10

100250 7 - 12over 250 10 - 20

Class widths can typically be reduced as the number of

observations increases

Distributions with numerous observations are more likelyto be smooth and have gaps filled since data are plentiful

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Width of classes:

If possible, use the same width for each class.

Range of data = Largest data pointSmallest datapoint

Approximate class width = Range / (Number of

classes)

Generally round to a convenient number

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Class Limits:

Class limits must be chosen so that each data itembelongs to one and only one class.

Lower class limit is the smallest possible data valueassigned to the class. Upper class limit is the largestpossible data value assigned to the class.

Class midpoint: The value halfway between thelower and upper class limits.

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Important Considerations for Selecting Classes:

Must be mutually exclusive

Must be all-inclusive

Categories (classes) should be of equal width

Avoid empty categories

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Example: Credit Card Balances

(See Class 02ExampleCredit Card Balances.xls)

Minimum Data value: $99.00

Maximum Date value: $1493.00

Range: $1493.00 - $99.00 = $1394.00

Approximate Class Size:

Approx Class Size = Range/(# of

classes)=1394/9=154.89 For convenience and better representation, will pick

200.00

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Determining the class limits:

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Class Lower Limit Class Upper Limit0 199.99

200 399.99

400 599.99600 799.99800 999.99

1000 1199.991200 1399.99

1400 1599.99

Omitted the 9thclass $1600 and under $1799.99 as nodata falls in the class (maximum data value is $1493).So, will use 8 classes in total.


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Excel

ArrayFormula !


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Using PhStat2

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Using PhStat2

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Histogram:A histogram is constructed by placing thevariable of interest on the horizontal axis and thefrequency, relative frequency, or percent frequencyon the vertical axis.

Rectangles with bases determined by the class limitson the horizontal axis and heights corresponding tofrequency, relative frequency, or percent frequency.

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Histogram:

Adjacent rectangles of a histogram touch oneanother. (Unlike a bar graph, no separation betweenthe rectangles.)

Histograms provide information about the shape or

form of a distribution.

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Histogram:

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0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

0

10

20

30

40

50

60

70

80

90

199.99 399.99 599.99 799.99 999.99 1199.99 1399.99 1599.99

Frequency

Upper Class Limit

Histogram

Frequency

Cumulative %


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Cumulative Distributions (Ogives): shows thenumber of data items with values less than or equalto the upper class limit of each class.

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02_FrequencyHistogram

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