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Chap. 4: Summarizing & Describing Numerical Data

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Chap. 4: Summarizing & Describing Numerical Data Basic Business Statistics: Concepts & Applications Learning Objectives 1. Explain Numerical Data Properties 2. Describe Summary Measures Central Tendency Variation Shape 3. Analyze Numerical Data Using Summary Measures Thinking Challenge $400,000 $70,000 $50,000 ... employees cite low pay -- most workers earn only $30,000 $20,000. ... President claims average $20,000 pay is $70,000! Standard Notation Measure Sample Population Mean ⎯X μX Stand. Dev. S σX 2 2 Variance S σX Size n N Numerical Data Properties Central Tendency (Location) Variation (Dispersion) Shape Numerical Data Properties & Measures Numerical Data Properties Central Variation Shape Tendency Mean Range Skew Median Interquartile Kurtosis Range Mode Variance Midrange Standard Deviation Midhinge Coeff. of Variation 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) n ∑ Xi X1 + X2 + L + Xn X = i =1 = n n 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 X = i =1 = 1 2 3 4 5 6 n 6 10.3 + 4.9 + 8.9 + 11.7 + 6.3 + 7.7 = 6 = 8.30 Median 1. Measure of Central Tendency 2. Middle Value In Ordered Sequence If Odd n, Middle Value of Sequence If Even n, Average of 2 Middle Values 3. Position of Median in Sequence n + 1 Positioning Point = 2 4. 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 45 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 7.7 8.9 10.3 11.7 Position: 1 2 3456 n + 1 6 + 1 Positioning Point = = = 3.5 2 2 7.7 + 8.9 Median = = 8.30 2 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 Mode Example No Mode Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 One Mode Raw Data: 6.3 4.9 8.9 6.3 4.9 4.9 More Than 1 Mode Raw Data: 21 28 28 41 43 43 Midrange 1. Measure of Central Tendency 2. Middle of Smallest & Largest Observation X + X Midrange = smallest l argest 2 3. Affected by Extreme Values Midhinge 1. Measure of Central Tendency 2. Middle of 1st & 3rd Quartiles Q + Q Midhinge = 1 2 2 3. Not Affected by Extreme Values Midhinge Example Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 Ordered: 4.9 6.3 7.7 8.9 10.3 11.7 Position: 123456 Q + Q 6.3 + 10.3 Midhinge = 1 2 = = 8.3 2 2 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. Summary of Central Tendency Measures Measure Equation Description Mean Σ Xi / n Balance Point Median (n+1) Position Middle Value 2 When Ordered Mode none Most Frequent Midrange Xsmallest + Xl argest Middle of Smallest 2 & Largest Midhinge (Q1 + Q3) Middle of 1st & 2 3rd Quartile where Qi = i (n+1)/4 Numerical Data Properties & Measures Numerical Data Properties Central Variation Shape Tendency Mean Range Skew Median Interquartile Kurtosis Range Mode Variance Midrange Standard Deviation Midhinge Coeff. of Variation Range 1. Measure of Dispersion 2. Difference Between Largest & Smallest Observations Range = Xl argest − Xsmallest 3. Ignores How Data Are Distributed 7 8 9 10 7 8 9 10 Interquartile Range 1. Measure of Dispersion 2. Also Called Midspread 3. Difference Between Third & First Quartiles Interquartile Range = Q3 − Q1 4. Spread in Middle 50% 5. Not Affected by Extreme Values Variance & Standard Deviation 1. Measures of Dispersion 2. Most Common Measures 3. Consider How Data Are Distributed 4. Show Variation About Mean (⎯X or μX) ⎯X = 8.3 4 6 8 10 12 Sample Variance Formula n 2 (X − X) n - 1 in denominator! ∑ i (Use N if Population 2 S = i =1 Variance) n − 1 2 2 2 (X1 − X) + (X2 − X) + L + (Xn − X) = n − 1 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 volatility of the stock prices. Summary of Variation Measures Measure Equation Description Range Xlargest - Xsmallest Total Spread Interquartile Range Q3 - Q1 Spread of Middle 50% 2 Standard Deviation (X − X ) Dispersion about (Sample) ∑ i Sample Mean n − 1 2 Dispersion about Standard Deviation (X − μ ) (Population) ∑ i X Population Mean N 2 Variance Σ(Xi -⎯X ) Squared Dispersion (Sample) n -1 about Sample Mean Coeff. of Variation (S /⎯X )100% Relative Variation Summary of Variation Measures Measure Equation Description Range Xlargest - Xsmallest Total Spread Interquartile Range Q3 - Q1 Spread of Middle 50% 2 Standard Deviation (X − X ) Dispersion about (Sample) ∑ i Sample Mean n − 1 2 Standard Deviation (X − μ) Dispersion about (Population) ∑ i Population Mean N 2 Variance Σ(Xi - ⎯X) Squared Dispersion (Sample) n -1 about Sample Mean Coeff. of Variation (S / ⎯X)100% Relative Variation Numerical Data Properties & Measures Numerical Data Properties Central Variation Shape Tendency Mean Range Skew Median Interquartile Kurtosis Range Mode Variance Midrange Standard Deviation Midhinge Coeff. of Variation Shape 1. Describes How Data Are Distributed 2. Measures of Shape Kurtosis = How Peaked or Flat Skew = Symmetry Left-Skewed Symmetric Right-Skewed Mean Median Mode Mean = Median = Mode Mode Median Mean Box-and-Whisker Plot 1. Graphical Display of Data Using 5-Number Summary Xsmallest Q1 Median Q3 Xlargest 4 6 8 10 12 Shape & Box-and-Whisker Plot Left-Skewed Symmetric Right-Skewed Q1 Median Q3 Q1 Median Q3 Q1 Median Q3 Conclusion 1. Explained Numerical Data Properties 2. Described Summary Measures Central Tendency Variation Shape 3. Analyzed Numerical Data Using Summary Measures .
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