2 - 1 © 2001 Prentice-Hall, Inc. Statistics for Business and Economics Methods for Describing Sets of Data Chapter 2

2 - 2 © 2001 Prentice-Hall, Inc. Learning Objectives 1.Describe Qualitative Data Graphically 2.Describe Numerical Data Graphically 3.Create & Interpret Graphical Displays 4.Explain Numerical Data Properties 5.Describe Summary Measures 6.Analyze Numerical Data Using Summary Measures

2 - 3 © 2001 Prentice-Hall, Inc. Thinking Challenge Our market share far exceeds all competitors! - VP 30%32%34%36% Us Y X

2 - 4 © 2001 Prentice-Hall, Inc. Data Presentation

2 - 5 © 2001 Prentice-Hall, Inc. Presenting Qualitative Data

2 - 6 © 2001 Prentice-Hall, Inc. Data Presentation

2 - 7 © 2001 Prentice-Hall, Inc. Summary Table 1.Lists Categories & No. Elements in Category 2.Obtained by Tallying Responses in Category 3.May Show Frequencies (Counts), % or Both Row Is Category Tally: |||| |||| |||| |||| MajorCount Accounting130 Economics 20 Management 50 Total200

2 - 8 © 2001 Prentice-Hall, Inc. Data Presentation

2 - 9 © 2001 Prentice-Hall, Inc Acct. Econ. Mgmt. Bar Chart Horizontal Bars for Categorical Variables Bar Length Shows Frequency or % 1/2 to 1 Bar Width Equal Bar Widths Zero Point Frequency Major Percent Used Also

© 2001 Prentice-Hall, Inc. Data Presentation

© 2001 Prentice-Hall, Inc. Econ. 10% Mgmt. 25% Acct. 65% Pie Chart 1.Shows Breakdown of Total Quantity into Categories 2.Useful for Showing Relative Differences 3.Angle Size (360°)(Percent) (360°)(Percent) Majors (360°) (10%) = 36° 36°

© 2001 Prentice-Hall, Inc. Data Presentation

© 2001 Prentice-Hall, Inc Acct. Econ. Mgmt. Dot Chart Frequency Major Line Length Shows Frequency or % Equal Spacing Like Horizontal Bar Chart Percent Used Also Horizontal Lines for Categorical Variables Zero Point

© 2001 Prentice-Hall, Inc. Graphical Excellence 1.Greatest Number of Ideas 2.Shortest Time to Understand 3.Least ‘Ink’ (Fill & Lines) 4.Smallest Space Dot Chart Bar Chart Pie Chart

© 2001 Prentice-Hall, Inc. Thinking Challenge You’re an analyst for IRI. You want to show the market shares held by Windows program manufacturers in Construct a bar chart, pie chart, & dot chart to describe the data. Mfg.Mkt. Share (%) Lotus15 Microsoft60 WordPerfect10 Others15

© 2001 Prentice-Hall, Inc. Bar Chart Solution* Market Share (%) Mfg. 0%20%40%60% Lotus Microsoft Wordperf. Others

© 2001 Prentice-Hall, Inc. Pie Chart Solution* Market Share Lotus 15% Others 15% Wordperf. 10% Microsoft 60%

© 2001 Prentice-Hall, Inc. Dot Chart Solution* Market Share (%) Mfg. 0%20%40%60% Others Wordperf. Microsoft Lotus

© 2001 Prentice-Hall, Inc. Presenting Numerical Data

© 2001 Prentice-Hall, Inc. Data Presentation

© 2001 Prentice-Hall, Inc. Stem-and-Leaf Display 1.Divide Each Observation into Stem Value and Leaf Value Stem Value Defines Class Stem Value Defines Class Leaf Value Defines Frequency (Count) Leaf Value Defines Frequency (Count) 2. Data: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 26

© 2001 Prentice-Hall, Inc. Data Presentation

© 2001 Prentice-Hall, Inc. Frequency Distribution Table Raw Data: 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 ClassFrequency 15 but < but < but < 45 2

© 2001 Prentice-Hall, Inc. Frequency Distribution Table Steps 1.Determine Range 2.Select Number of Classes Usually Between 5 & 15 Inclusive Usually Between 5 & 15 Inclusive 3.Compute Class Intervals (Width) 4.Determine Class Boundaries (Limits) 5.Compute Class Midpoints 6.Count Observations & Assign to Classes

© 2001 Prentice-Hall, Inc. Frequency Distribution Table Example Raw Data: 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Boundaries (Upper + Lower Boundaries) / 2 Width ClassMidpointFrequency 15 but < but < but <

© 2001 Prentice-Hall, Inc. Relative Frequency & % Distribution Tables Percentage Distribution Relative Frequency Distribution ClassProp. 15 but < but < but < 45.2 Class% 15 but < but < but <

© 2001 Prentice-Hall, Inc. Cumulative Percentage Distribution Table Percentage Less than Lower Class Boundary Raw Data: 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Lower Class Boundary 30% + 50% 80% + 20% ClassCumulative Percentage 15 but < but < but < but <

© 2001 Prentice-Hall, Inc. Data Presentation

© 2001 Prentice-Hall, Inc Histogram Frequency Relative Frequency Percent Lower Boundary Bars Touch ClassFreq. 15 but < but < but < 45 2 Count

© 2001 Prentice-Hall, Inc. Numerical Data Properties

© 2001 Prentice-Hall, Inc. Thinking Challenge... employees cite low pay -- most workers earn only $20, President claims average pay is $70,000! $400,000 $70,000 $50,000 $30,000 $20,000

© 2001 Prentice-Hall, Inc. Standard Notation MeasureSamplePopulation Mean  X  Stand. Dev. S  Variance S 2  2 SizenN

© 2001 Prentice-Hall, Inc. Numerical Data Properties Central Tendency (Location) Variation (Dispersion) Shape

© 2001 Prentice-Hall, Inc. Numerical Data Properties & Measures Numerical Data Properties Mean Median Mode Central Tendency Range Variance Standard Deviation Variation Skew Shape Interquartile Range

© 2001 Prentice-Hall, Inc. Central Tendency

© 2001 Prentice-Hall, Inc. Numerical Data Properties & Measures Numerical Data Properties Mean Median Mode Central Tendency Range Variance Standard Deviation Variation Skew Shape Interquartile Range

© 2001 Prentice-Hall, Inc. Mean 1.Measure of Central Tendency 2.Most Common Measure 3.Acts as ‘Balance Point’ 4.Affected by Extreme Values (‘Outliers’) 5. Formula (Sample Mean) X X n XXX n i i n n     1 12 

© 2001 Prentice-Hall, Inc. Mean Example Raw Data: X X n XXXXXX i i n       

© 2001 Prentice-Hall, Inc. Numerical Data Properties & Measures Numerical Data Properties Mean Median Mode Central Tendency Range Variance Standard Deviation Variation Skew Shape Interquartile Range

© 2001 Prentice-Hall, Inc. Median 1.Measure of Central Tendency 2.Middle Value In Ordered Sequence If Odd n, Middle Value of Sequence If Odd n, Middle Value of Sequence If Even n, Average of 2 Middle Values If Even n, Average of 2 Middle Values 3. Position of Median in Sequence 4.Not Affected by Extreme Values Positionin g Point  n1 2

© 2001 Prentice-Hall, Inc. Median Example Odd-Sized Sample Raw Data: Ordered: Position:12345 Positionin g Point Median       n

© 2001 Prentice-Hall, Inc. Median Example Even-Sized Sample Raw Data: Ordered: Position: Positionin g Point Median         n

© 2001 Prentice-Hall, Inc. Numerical Data Properties & Measures Numerical Data Properties Mean Median Mode Central Tendency Range Variance Standard Deviation Variation Skew Shape Interquartile Range

© 2001 Prentice-Hall, Inc. Mode 1.Measure of Central Tendency 2.Value That Occurs Most Often 3.Not Affected by Extreme Values 4.May Be No Mode or Several Modes 5.May Be Used for Numerical & Categorical Data

© 2001 Prentice-Hall, Inc. Mode Example No Mode Raw Data: One Mode Raw Data: More Than 1 Mode Raw Data:

© 2001 Prentice-Hall, Inc. Thinking Challenge You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. Describe the stock prices in terms of central tendency.

© 2001 Prentice-Hall, Inc. Central Tendency Solution* Mean X X n XXX i i n        .

© 2001 Prentice-Hall, Inc. Central Tendency Solution* Median Raw Data: Ordered: Position: Positionin g Point Median         n

© 2001 Prentice-Hall, Inc. Central Tendency Solution* Mode Raw Data: Ordered:

© 2001 Prentice-Hall, Inc. Summary of Central Tendency Measures MeasureEquationDescription Mean  X i /n Balance Point Median(n+1) Position Position 2 Middle Value When Ordered Modenone Most Frequent

© 2001 Prentice-Hall, Inc. Variation

© 2001 Prentice-Hall, Inc. Numerical Data Properties & Measures Numerical Data Properties Mean Median Mode Central Tendency Range Variance Standard Deviation Variation Skew Shape Interquartile Range

© 2001 Prentice-Hall, Inc. Range 1.Measure of Dispersion 2.Difference Between Largest & Smallest Observations 3.Ignores How Data Are Distributed RangeXX lestsmallestarg

© 2001 Prentice-Hall, Inc. Numerical Data Properties & Measures Numerical Data Properties Mean Median Mode Central Tendency Range Interquartile Range Variance Standard Deviation Variation Skew Shape

© 2001 Prentice-Hall, Inc. Variance & Standard Deviation 1.Measures of Dispersion 2.Most Common Measures 3.Consider How Data Are Distributed 4.Show Variation About Mean (  X or  ) X = 8.3 = 8.3

© 2001 Prentice-Hall, Inc. Sample Variance Formula n - 1 in denominator! (Use N if Population Variance) S XX n XXXXXX n i i n n         ch c h c h c h 

© 2001 Prentice-Hall, Inc. Sample Standard Deviation Formula SS XX n XXXXXX n i i n n          ch chchch 

© 2001 Prentice-Hall, Inc. Variance Example Raw Data: S XX n X X n S i i n i i n           ch afafaf where 

© 2001 Prentice-Hall, Inc. Thinking Challenge You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. What are the variance and standard deviation of the stock prices?

© 2001 Prentice-Hall, Inc. Variation Solution* Sample Variance Raw Data: S XX n X X n S i i n i i n           ch a f a f a f where..... 

© 2001 Prentice-Hall, Inc. Variation Solution* Sample Standard Deviation SS XX n i i n       ch..

© 2001 Prentice-Hall, Inc. Summary of Variation Measures MeasureEquationDescription Range X largest -X smallest Total Spread Interquartile Range Q 3 -Q 1 Spread of Middle 50% Standard Deviation (Sample) XX n i    21 Dispersion about Sample Mean Standard Deviation (Population) X N iX    2 Dispersion about Population Mean Variance (Sample)  (X i -  X) 2 n Squared Dispersion about Sample Mean

© 2001 Prentice-Hall, Inc. Shape

© 2001 Prentice-Hall, Inc. Numerical Data Properties & Measures Numerical Data Properties Mean Median Mode Central Tendency Range Interquartile Range Variance Standard Deviation Variation Skew Shape

© 2001 Prentice-Hall, Inc. Shape 1.Describes How Data Are Distributed 2.Measures of Shape Skew = Symmetry Skew = Symmetry Right-SkewedLeft-SkewedSymmetric Mean =Median =Mode Mean Median Mode Mode Median Mean

© 2001 Prentice-Hall, Inc. Quartiles & Box Plots

© 2001 Prentice-Hall, Inc. Quartiles 1.Measure of Noncentral Tendency 2.Split Ordered Data into 4 Quarters 3.Position of i-th Quartile 25%25%25%25% Q1Q1Q1Q1 Q2Q2Q2Q2 Q3Q3Q3Q3 Positionin g Point of Q in i  1 4 af

© 2001 Prentice-Hall, Inc. Quartile (Q 1 ) Example Raw Data: Ordered: Position: QPosition Q 1       n afaf..

© 2001 Prentice-Hall, Inc. Quartile (Q 2 ) Example Raw Data: Ordered: Position: QPosition Q 2         n afaf....

© 2001 Prentice-Hall, Inc. Quartile (Q 3 ) Example Raw Data: Ordered: Position: QPosition Q 3       n afaf..

© 2001 Prentice-Hall, Inc. Numerical Data Properties & Measures Numerical Data Properties Mean Median Mode Central Tendency Range Interquartile Range Variance Standard Deviation Variation Skew Shape

© 2001 Prentice-Hall, Inc. Interquartile Range 1.Measure of Dispersion 2.Also Called Midspread 3.Difference Between Third & First Quartiles 4.Spread in Middle 50% 5.Not Affected by Extreme Values Interquart ile Range QQ 31

© 2001 Prentice-Hall, Inc. Thinking Challenge You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. What are the quartiles, Q 1 and Q 3, and the interquartile range?

© 2001 Prentice-Hall, Inc. Q 1 Raw Data: Ordered: Position: Quartile Solution* QPosition Q 1       n afaf..

© 2001 Prentice-Hall, Inc. Quartile Solution* Q 3 Raw Data: Ordered: Position: Q Position Position Q 3       n afaf.

© 2001 Prentice-Hall, Inc. Interquartile Range Solution* Interquartile Range Raw Data: Ordered: Position: Interquart ile Range QQ

© 2001 Prentice-Hall, Inc. Box Plot 1.Graphical Display of Data Using 5-Number Summary Median Q 3 Q 1 X largest X smallest

© 2001 Prentice-Hall, Inc. Shape & Box Plot Right-SkewedLeft-SkewedSymmetric Q 1 Median Q 3 Q 1 Median Q 3 Q 1 Median Q 3

© 2001 Prentice-Hall, Inc. Distorting the Truth with Descriptive Techniques

© 2001 Prentice-Hall, Inc. Errors in Presenting Data 1.Using ‘Chart Junk’ 2.No Relative Basis in Comparing Data Batches 3.Compressing the Vertical Axis 4.No Zero Point on the Vertical Axis

© 2001 Prentice-Hall, Inc. ‘Chart Junk’ Bad Presentation Good Presentation 1960: $ : $ : $ : $3.80 Minimum Wage $

© 2001 Prentice-Hall, Inc. No Relative Basis Good Presentation A’s by Class Bad Presentation FRSOJRSR Freq. 0% 10% 20% 30% FRSOJRSR %

© 2001 Prentice-Hall, Inc. Compressing Vertical Axis Good Presentation Quarterly Sales Bad Presentation Q1Q2Q3Q4 $ Q1Q2Q3Q4 $

© 2001 Prentice-Hall, Inc. No Zero Point on Vertical Axis Good Presentation Monthly Sales Bad Presentation JMMJSN $ JMMJSN $

© 2001 Prentice-Hall, Inc. Conclusion 1.Described Qualitative Data Graphically 2.Described Numerical Data Graphically 3.Created & Interpreted Graphical Displays 4.Explained Numerical Data Properties 5.Described Summary Measures 6.Analyzed Numerical Data Using Summary Measures

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