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 • Analyze Numerical Data UsingSummary Measures

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

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

Presenting Qualitative Data

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:|||| |||||||| ||||

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

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°

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

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

Bar Graph Solution* Market Share (%) Browser

Pie Chart Solution* Market Share

Pareto Diagram Solution* Market Share (%) Browser

Presenting Quantitative Data

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

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

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

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

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

Numerical Data Properties

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

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

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

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

Central Tendency

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)

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

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

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

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

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

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

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

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.

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

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

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

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

Shape

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

Variation

Statistics for Business and Economics