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Statistics for Business and Economics

Statistics for Business and Economics. Chapter 2 Methods for Describing Sets of Data. Learning Objectives. Describe Qualitative Data Graphically Describe Quantitative Data Graphically Explain Numerical Data Properties Describe Summary Measures

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Statistics for Business and Economics

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  1. Statistics for Business and Economics Chapter 2 Methods for Describing Sets of Data

  2. Learning Objectives • Describe Qualitative Data Graphically • Describe Quantitative Data Graphically • Explain Numerical Data Properties • Describe Summary Measures • Analyze Numerical Data UsingSummary Measures

  3. Thinking Challenge 36% Our market share far exceeds all competitors! - VP 34% 32% 30% X Y Us

  4. Data Presentation QualitativeData QuantitativeData SummaryTable Stem-&-LeafDisplay FrequencyDistribution ParetoDiagram BarGraph PieChart Histogram Data Presentation

  5. Presenting Qualitative Data

  6. Data Presentation QualitativeData QuantitativeData SummaryTable Stem-&-LeafDisplay FrequencyDistribution ParetoDiagram BarGraph PieChart Histogram Data Presentation

  7. Summary Table • Lists categories & number of elements in category • Obtained by tallying responses in category • May show frequencies (counts), % or both Row Is Category Tally:|||| |||||||| ||||

  8. Data Presentation QualitativeData QuantitativeData SummaryTable Stem-&-LeafDisplay FrequencyDistribution ParetoDiagram BarGraph PieChart Histogram Data Presentation

  9. Bar Graph Equal Bar Widths Bar Height Shows Frequency or % Percent Used Also Frequency Vertical Bars for Qualitative Variables Zero Point

  10. Data Presentation QualitativeData QuantitativeData SummaryTable Stem-&-LeafDisplay FrequencyDistribution ParetoDiagram BarGraph PieChart Histogram Data Presentation

  11. Pie Chart • Shows breakdown of total quantity into categories • Useful for showing relative differences • Angle size • (360°)(percent) Majors Mgmt. Econ. 25% 10% 36° Acct. 65% (360°) (10%) = 36°

  12. Data Presentation QualitativeData QuantitativeData SummaryTable Stem-&-LeafDisplay FrequencyDistribution ParetoDiagram BarGraph PieChart Histogram Data Presentation

  13. Pareto Diagram Like a bar graph, but with the categories arranged by height in descending order from left to right. Equal Bar Widths Bar Height Shows Frequency or % Percent Used Also Frequency Vertical Bars for Qualitative Variables Zero Point

  14. You’re an analyst for IRI. You want to show the market shares held by Web browsers in 2006. Construct abargraph, pie chart, & Paretodiagram to describe the data. Thinking Challenge

  15. Bar Graph Solution* Market Share (%) Browser

  16. Pie Chart Solution* Market Share

  17. Pareto Diagram Solution* Market Share (%) Browser

  18. Presenting Quantitative Data

  19. Data Presentation QualitativeData QuantitativeData SummaryTable Stem-&-LeafDisplay FrequencyDistribution ParetoDiagram BarGraph PieChart Histogram Data Presentation

  20. Stem-and-Leaf Display 1. Divide each observation into stem value and leaf value • Stem value defines class • Leaf value defines frequency (count) 2 144677 26 3 028 4 1 2. Data: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41

  21. Data Presentation QualitativeData QuantitativeData SummaryTable Stem-&-LeafDisplay FrequencyDistribution ParetoDiagram BarGraph PieChart Histogram Data Presentation

  22. Frequency Distribution Table Steps • Determine range • Select number of classes • Usually between 5 & 15 inclusive • Compute class intervals (width) • Determine class boundaries (limits) • Compute class midpoints • Count observations & assign to classes

  23. Frequency Distribution Table Example Raw Data:24, 26, 24, 21, 272730, 41, 32,38 Class Midpoint Frequency 15.5 – 25.5 20.5 3 Width 25.5 – 35.5 30.5 5 35.5 – 45.5 40.5 2 (Lower + Upper Boundaries) / 2 Boundaries

  24. Relative Frequency & % Distribution Tables Relative Frequency Distribution Percentage Distribution Class Prop. Class % 15.5 – 25.5 .3 15.5 – 25.5 30.0 25.5 – 35.5 .5 25.5 – 35.5 50.0 35.5 – 45.5 .2 35.5 – 45.5 20.0

  25. Data Presentation QualitativeData QuantitativeData SummaryTable Stem-&-LeafDisplay FrequencyDistribution ParetoDiagram BarGraph PieChart Histogram Data Presentation

  26. Histogram Class Freq. Count 15.5 – 25.5 3 25.5 – 35.5 5 5 35.5 – 45.5 2 4 Frequency Relative Frequency Percent 3 Bars Touch 2 1 0 0 15.5 25.5 35.5 45.5 55.5 Lower Boundary

  27. Numerical Data Properties

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

  29. X 2 2 S  Standard Notation Measure Sample Population Mean  StandardDeviation S  Variance Size n N

  30. Numerical Data Properties Central Tendency (Location) Variation (Dispersion) Shape

  31. Numerical DataProperties & Measures Numerical Data Properties RelativeStanding Central Variation Tendency Mean Range Percentiles Interquartile Range Median Z–scores Mode Variance Standard Deviation

  32. Central Tendency

  33. Numerical DataProperties & Measures Numerical Data Properties RelativeStanding Central Variation Tendency Mean Range Percentiles Interquartile Range Median Z–scores Mode Variance Standard Deviation

  34. n  X i X  X  …  X 1 2 n i  1 X   n n Mean • Measure of central tendency • Most common measure • Acts as ‘balance point’ • Affected by extreme values (‘outliers’) • Formula (sample mean)

  35. Mean Example Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 n  X i X  X  X  X  X  X 1 2 3 4 5 6 i  1 X   n 6 10 . 3  4 . 9  8 . 9  11 . 7  6 . 3  7 . 7  6  8 . 30

  36. Numerical DataProperties & Measures Numerical Data Properties RelativeStanding Central Variation Tendency Mean Range Percentiles Median Interquartile Range Z–scores Mode Variance Standard Deviation

  37. n  1 Positioning Point  2 Median • Measure of central tendency • Middle value in ordered sequence • If n is odd, middle value of sequence • If n is even, average of 2 middle values • Position of median in sequence • Not affected by extreme values

  38. Median Example Odd-Sized Sample • Raw Data: 24.1 22.6 21.5 23.7 22.6 • Ordered: 21.5 22.6 22.6 23.7 24.1 • Position: 1 2 3 4 5 n  1 5  1 Positioning Point    3 . 0 2 2 Median  22 . 6

  39. Median Example Even-Sized Sample • Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 • Ordered: 4.9 6.3 7.78.9 10.3 11.7 • Position: 1 2 34 5 6 n  1 6  1 Positioning Point    3 . 5 2 2 7 . 7  8 . 9 Median   8 . 30 2

  40. Numerical DataProperties & Measures Numerical Data Properties RelativeStanding Central Variation Tendency Mean Range Percentiles Interquartile Range Median Z–scores Mode Variance Standard Deviation

  41. Mode • Measure of central tendency • Value that occurs most often • Not affected by extreme values • May be no mode or several modes • May be used for quantitative or qualitative data

  42. Mode Example • No ModeRaw Data: 10.3 4.9 8.9 11.7 6.3 7.7 • One ModeRaw Data: 6.3 4.9 8.9 6.3 4.9 4.9 • More Than 1 ModeRaw Data: 2128 2841 4343

  43. 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 pricesin terms of central tendency.

  44. Central Tendency Solution* Mean n  X i X  X  …  X 1 2 8 i  1 X   n 8 17  16  21  18  13  16  12  11  8  15 . 5

  45. Central Tendency Solution* Median • Raw Data: 17 16 21 18 13 16 12 11 • Ordered: 11 12 13 16 16 17 18 21 • Position: 1 2 3 4 5 6 7 8 n  1 8  1 Positioning Point    4 . 5 2 2 16  16 Median   16 2

  46. Central Tendency Solution* Mode Raw Data: 17 16 21 18 13 16 12 11 Mode = 16

  47. Summary of Central Tendency Measures Measure Formula Description Mean Balance Point  X / n i Median ( n +1) Middle Value Position 2 When Ordered Mode none Most Frequent

  48. Shape

  49. Shape • Describes how data are distributed • Measures of Shape • Skew = Symmetry Left-Skewed Symmetric Right-Skewed Mean Median Mean = Median Median Mean

  50. Variation

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