Statistics  for Management  Lesson 3 Summarizing and Describing Numerical Data
Lesson Topics <ul><li>Measures of   Central Tendency </li></ul><ul><li>Mean, Median, Mode, Midrange,  Midhinge </li></ul><...
Summary Measures  Central Tendency Mean Median Mode Midrange Quartile Midhinge Summary Measures Variation Variance Standar...
Measures of Central Tendency Central Tendency Mean Median Mode Midrange Midhinge
The Mean  (Arithmetic Average) <ul><li>It is the   Arithmetic Average   of data values: </li></ul><ul><li>The  Most Common...
The Median 0  1  2  3  4  5  6  7  8  9  10 0  1  2  3  4  5  6  7  8  9  10  12  14  Median = 5 Median = 5 <ul><li>Import...
The Mode 0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  Mode = 9 <ul><li>A Measure of Central Tendency </li></ul><ul><l...
Midrange <ul><li>A Measure of Central Tendency </li></ul><ul><li>Average of Smallest and Largest   </li></ul><ul><li>Obser...
Quartiles <ul><li>Not a Measure of Central Tendency </li></ul><ul><li>Split Ordered Data into 4 Quarters </li></ul><ul><li...
Midhinge <ul><li>A Measure of Central Tendency </li></ul><ul><li>The Middle point of 1st and 3rd Quarters </li></ul><ul><l...
Summary Measures  Central Tendency Mean Median Mode Midrange Quartile Midhinge Summary Measures Variation Variance Standar...
Measures of Variation Variation Variance Standard Deviation Coefficient of Variation Population Variance Sample  Variance ...
<ul><li>Measure of Variation </li></ul><ul><li>Difference Between Largest & Smallest   </li></ul><ul><li>Observations: </l...
<ul><li>Measure of Variation </li></ul><ul><li>Also Known as   Midspread:   </li></ul><ul><li>Spread in the Middle 50% </l...
<ul><li>Important Measure of Variation  </li></ul><ul><li>Shows Variation About the Mean: </li></ul><ul><li>For the Popula...
<ul><li>Most Important Measure of Variation  </li></ul><ul><li>Shows Variation About the Mean: </li></ul><ul><li>For the P...
Sample Standard Deviation For the  Sample  : use   n - 1   in the denominator. Data:   10  12  14  15  17  18  18  24 s = ...
Comparing Standard Deviations s  =  =  4.2426 =  3.9686 Value for the Standard Deviation is   larger   for data considered...
Comparing Standard Deviations Mean = 15.5 s  =  3.338   11  12  13  14  15  16  17  18  19  20  21 11  12  13  14  15  16 ...
Coefficient of Variation <ul><li>Measure of   Relative Variation </li></ul><ul><li>Always a   % </li></ul><ul><li>Shows Va...
Comparing Coefficient of Variation <ul><li>Stock A:   Average Price last year  =   $50 </li></ul><ul><li>  Standard Deviat...
Shape <ul><li>Describes How Data Are Distributed </li></ul><ul><li>Measures of Shape:  </li></ul><ul><li>Symmetric or skew...
Box-and-Whisker Plot <ul><li>Graphical Display of Data Using 50%-Number Summary </li></ul>Median 4 6 8 10 12 Q 3 Q 1 X lar...
Distribution Shape &  Box-and-Whisker Plots Right-Skewed Left-Skewed Symmetric Q 1 Median Q 3 Q 1 Median Q 3 Q 1 Median Q 3
Lesson Summary <ul><li>Discussed Measures of   Central Tendency </li></ul><ul><li>Mean, Median,  Mode, Midrange, Midhinge ...
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  1. 1. Statistics for Management Lesson 3 Summarizing and Describing Numerical Data
  2. 2. Lesson Topics <ul><li>Measures of Central Tendency </li></ul><ul><li>Mean, Median, Mode, Midrange, Midhinge </li></ul><ul><li>Quartile </li></ul><ul><li>Measures of Variation </li></ul><ul><li>The Range, Interquartile Range, Variance and </li></ul><ul><li>Standard Deviation, Coefficient of variation </li></ul><ul><li>Shape </li></ul><ul><li>Symmetric, Skewed, using Box-and-Whisker </li></ul><ul><li>Plots </li></ul>
  3. 3. Summary Measures Central Tendency Mean Median Mode Midrange Quartile Midhinge Summary Measures Variation Variance Standard Deviation Coefficient of Variation Range
  4. 4. Measures of Central Tendency Central Tendency Mean Median Mode Midrange Midhinge
  5. 5. The Mean (Arithmetic Average) <ul><li>It is the Arithmetic Average of data values: </li></ul><ul><li>The Most Common Measure of Central Tendency </li></ul><ul><li>Affected by Extreme Values (Outliers) </li></ul>0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5 Mean = 6 Sample Mean
  6. 6. The Median 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 Median = 5 <ul><li>Important Measure of Central Tendency </li></ul><ul><li>In an ordered array, the median is the </li></ul><ul><li>“ middle” number. </li></ul><ul><ul><li>If n is odd , the median is the middle number . </li></ul></ul><ul><ul><li>If n is even , the median is the average of the 2 </li></ul></ul><ul><ul><li>middle numbers. </li></ul></ul><ul><li>Not Affected by Extreme Values </li></ul>
  7. 7. The Mode 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 <ul><li>A Measure of Central Tendency </li></ul><ul><li>Value that Occurs Most Often </li></ul><ul><li>Not Affected by Extreme Values </li></ul><ul><li>There May Not be a Mode </li></ul><ul><li>There May be Several Modes </li></ul><ul><li>Used for Either Numerical or Categorical Data </li></ul>0 1 2 3 4 5 6 No Mode
  8. 8. Midrange <ul><li>A Measure of Central Tendency </li></ul><ul><li>Average of Smallest and Largest </li></ul><ul><li>Observation: </li></ul><ul><li>Affected by Extreme Value </li></ul>Midrange 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Midrange = 5 Midrange = 5
  9. 9. Quartiles <ul><li>Not a Measure of Central Tendency </li></ul><ul><li>Split Ordered Data into 4 Quarters </li></ul><ul><li>Position of i-th Quartile: position of point </li></ul>25% 25% 25% 25% Q 1 Q 2 Q 3 Q i(n+1) i  4 Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Position of Q 1 = 2.50 Q 1 =12.5 = 1•(9 + 1) 4
  10. 10. Midhinge <ul><li>A Measure of Central Tendency </li></ul><ul><li>The Middle point of 1st and 3rd Quarters </li></ul><ul><li>Not Affected by Extreme Values </li></ul>Midhinge = Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Midhinge =
  11. 11. Summary Measures Central Tendency Mean Median Mode Midrange Quartile Midhinge Summary Measures Variation Variance Standard Deviation Coefficient of Variation Range
  12. 12. Measures of Variation Variation Variance Standard Deviation Coefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range Interquartile Range
  13. 13. <ul><li>Measure of Variation </li></ul><ul><li>Difference Between Largest & Smallest </li></ul><ul><li>Observations: </li></ul><ul><li>Range = </li></ul><ul><li>Ignores How Data Are Distributed: </li></ul>The Range 7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 Range = 12 - 7 = 5
  14. 14. <ul><li>Measure of Variation </li></ul><ul><li>Also Known as Midspread: </li></ul><ul><li>Spread in the Middle 50% </li></ul><ul><li>Difference Between Third & First </li></ul><ul><li>Quartiles: Interquartile Range = </li></ul><ul><li>Not Affected by Extreme Values </li></ul>Interquartile Range Data in Ordered Array: 11 12 13 16 16 17 17 18 21 = 17.5 - 12.5 = 5
  15. 15. <ul><li>Important Measure of Variation </li></ul><ul><li>Shows Variation About the Mean: </li></ul><ul><li>For the Population: </li></ul><ul><li>For the Sample: </li></ul>Variance For the Population : use N in the denominator. For the Sample : use n - 1 in the denominator.
  16. 16. <ul><li>Most Important Measure of Variation </li></ul><ul><li>Shows Variation About the Mean: </li></ul><ul><li>For the Population: </li></ul><ul><li>For the Sample: </li></ul>Standard Deviation For the Population: use N in the denominator. For the Sample : use n - 1 in the denominator .
  17. 17. Sample Standard Deviation For the Sample : use n - 1 in the denominator. Data: 10 12 14 15 17 18 18 24 s = n = 8 Mean =16 = 4.2426 s
  18. 18. Comparing Standard Deviations s = = 4.2426 = 3.9686 Value for the Standard Deviation is larger for data considered as a Sample . Data : 10 12 14 15 17 18 18 24 N= 8 Mean =16
  19. 19. Comparing Standard Deviations Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 4.57 Data C
  20. 20. Coefficient of Variation <ul><li>Measure of Relative Variation </li></ul><ul><li>Always a % </li></ul><ul><li>Shows Variation Relative to Mean </li></ul><ul><li>Used to Compare 2 or More Groups </li></ul><ul><li>Formula ( for Sample): </li></ul>
  21. 21. Comparing Coefficient of Variation <ul><li>Stock A: Average Price last year = $50 </li></ul><ul><li> Standard Deviation = $5 </li></ul><ul><li>Stock B: Average Price last year = $100 </li></ul><ul><li> Standard Deviation = $5 </li></ul>Coefficient of Variation: Stock A: CV = 10% Stock B: CV = 5%
  22. 22. Shape <ul><li>Describes How Data Are Distributed </li></ul><ul><li>Measures of Shape: </li></ul><ul><li>Symmetric or skewed </li></ul>Right-Skewed Left-Skewed Symmetric Mean = Median = Mode Mean Median Mode Median Mean Mode
  23. 23. Box-and-Whisker Plot <ul><li>Graphical Display of Data Using 50%-Number Summary </li></ul>Median 4 6 8 10 12 Q 3 Q 1 X largest X smallest
  24. 24. Distribution Shape & Box-and-Whisker Plots Right-Skewed Left-Skewed Symmetric Q 1 Median Q 3 Q 1 Median Q 3 Q 1 Median Q 3
  25. 25. Lesson Summary <ul><li>Discussed Measures of Central Tendency </li></ul><ul><li>Mean, Median, Mode, Midrange, Midhinge </li></ul><ul><li>Quartiles </li></ul><ul><li>Addressed Measures of Variation </li></ul><ul><li>The Range, Interquartile Range, Variance, </li></ul><ul><li> Standard Deviation, Coefficient of Variation </li></ul><ul><li>Determined Shape of Distributions </li></ul><ul><li>Symmetric, Skewed, Box-and-Whisker Plot </li></ul>Mean = Median = Mode Mean Median Mode Mode Median Mean
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