1. 1 1 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. SLIDES. BY John Loucks St. Edward’s University......................

2. 2 2 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 3, Part B Descriptive Statistics: Numerical Measures n Measures of Distribution Shape, Relative Location, and Detecting Outliers n Five-Number Summaries and Box Plots n Measures of Association Between Two Variables n Data Dashboards: Adding Numerical Measures to Improve Effectiveness

3. 3 3 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Measures of Distribution Shape, Relative Location, and Detecting Outliers n Distribution Shape n z-Scores n Chebyshev’s Theorem n Empirical Rule n Detecting Outliers

4. 4 4 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Distribution Shape: Skewness n An important measure of the shape of a distribution is called skewness. n The formula for the skewness of sample data is n Skewness can be easily computed using statistical software.

5. 5 5 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Distribution Shape: Skewness n Symmetric (not skewed) Relative Frequency.05.10.15.20.25.30.35 0 0 Skewness = 0 Skewness = 0 Skewness is zero. Skewness is zero. Mean and median are equal. Mean and median are equal.

6. 6 6 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Relative Frequency.05.10.15.20.25.30.35 0 0 Distribution Shape: Skewness n Moderately Skewed Left Skewness = .31 Skewness = .31 Skewness is negative. Skewness is negative. Mean will usually be less than the median. Mean will usually be less than the median.

7. 7 7 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Distribution Shape: Skewness n Moderately Skewed Right Relative Frequency.05.10.15.20.25.30.35 0 0 Skewness =.31 Skewness =.31 Skewness is positive. Skewness is positive. Mean will usually be more than the median. Mean will usually be more than the median.

8. 8 8 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Distribution Shape: Skewness n Highly Skewed Right Relative Frequency.05.10.15.20.25.30.35 0 0 Skewness = 1.25 Skewness = 1.25 Skewness is positive (often above 1.0). Skewness is positive (often above 1.0). Mean will usually be more than the median. Mean will usually be more than the median.

9. 9 9 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seventy efficiency apartments were randomly Seventy efficiency apartments were randomly sampled in a college town. The monthly rent prices for the apartments are listed below in ascending order. Distribution Shape: Skewness n Example: Apartment Rents

10. 10 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Relative Frequency.05.10.15.20.25.30.35 0 0 Skewness =.92 Skewness =.92 Distribution Shape: Skewness n Example: Apartment Rents

11. 11 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The z-score is often called the standardized value. The z-score is often called the standardized value. It denotes the number of standard deviations a data It denotes the number of standard deviations a data value x i is from the mean. value x i is from the mean. It denotes the number of standard deviations a data It denotes the number of standard deviations a data value x i is from the mean. value x i is from the mean. z-Scores Excel’s STANDARDIZE function can be used to Excel’s STANDARDIZE function can be used to compute the z-score. compute the z-score. Excel’s STANDARDIZE function can be used to Excel’s STANDARDIZE function can be used to compute the z-score. compute the z-score.

12. 12 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. z-Scores A data value less than the sample mean will have a A data value less than the sample mean will have a z-score less than zero. z-score less than zero. A data value greater than the sample mean will have A data value greater than the sample mean will have a z-score greater than zero. a z-score greater than zero. A data value equal to the sample mean will have a A data value equal to the sample mean will have a z-score of zero. z-score of zero. An observation’s z-score is a measure of the relative An observation’s z-score is a measure of the relative location of the observation in a data set. location of the observation in a data set.

13. 13 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. z-Score of Smallest Value (525) z-Score of Smallest Value (525) z-Scores Example: Apartment Rents Example: Apartment Rents Standardized Values for Apartment Rents

14. 14 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chebyshev’s Theorem At least (1 - 1/ z 2 ) of the items in any data set will be At least (1 - 1/ z 2 ) of the items in any data set will be within z standard deviations of the mean, where z is within z standard deviations of the mean, where z is any value greater than 1. any value greater than 1. At least (1 - 1/ z 2 ) of the items in any data set will be At least (1 - 1/ z 2 ) of the items in any data set will be within z standard deviations of the mean, where z is within z standard deviations of the mean, where z is any value greater than 1. any value greater than 1. Chebyshev’s theorem requires z > 1, but z need not Chebyshev’s theorem requires z > 1, but z need not be an integer. be an integer. Chebyshev’s theorem requires z > 1, but z need not Chebyshev’s theorem requires z > 1, but z need not be an integer. be an integer.

15. 15 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean. 75%75% z = 2 standard deviations z = 2 standard deviations Chebyshev’s Theorem At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean. 89%89% z = 3 standard deviations z = 3 standard deviations At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean. 94%94% z = 4 standard deviations z = 4 standard deviations

16. 16 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chebyshev’s Theorem Let z = 1.5 with = 590.80 and s = 54.74 At least (1  1/(1.5) 2 ) = 1  0.44 = 0.56 or 56% of the rent values must be between - z ( s ) = 590.80  1.5(54.74) = 509 - z ( s ) = 590.80  1.5(54.74) = 509and + z ( s ) = 590.80 + 1.5(54.74) = 673 + z ( s ) = 590.80 + 1.5(54.74) = 673 (Actually, 86% of the rent values are between 509 and 673.) are between 509 and 673.) Example: Apartment Rents Example: Apartment Rents

17. 17 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Empirical Rule When the data are believed to approximate a When the data are believed to approximate a bell-shaped distribution … bell-shaped distribution … The empirical rule is based on the normal The empirical rule is based on the normal distribution, which is covered in Chapter 6. distribution, which is covered in Chapter 6. The empirical rule is based on the normal The empirical rule is based on the normal distribution, which is covered in Chapter 6. distribution, which is covered in Chapter 6. The empirical rule can be used to determine the The empirical rule can be used to determine the percentage of data values that must be within a percentage of data values that must be within a specified number of standard deviations of the specified number of standard deviations of the mean. mean. The empirical rule can be used to determine the The empirical rule can be used to determine the percentage of data values that must be within a percentage of data values that must be within a specified number of standard deviations of the specified number of standard deviations of the mean. mean.

18. 18 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Empirical Rule For data having a bell-shaped distribution: For data having a bell-shaped distribution: of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. 68.26%68.26% +/- 1 standard deviation of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. 95.44%95.44% +/- 2 standard deviations of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. 99.72%99.72% +/- 3 standard deviations

19. 19 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Empirical Rule x  – 3   – 1   – 2   + 1   + 2   + 3  68.26% 95.44% 99.72%

20. 20 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Detecting Outliers An outlier is an unusually small or unusually large An outlier is an unusually small or unusually large value in a data set. value in a data set. A data value with a z-score less than -3 or greater A data value with a z-score less than -3 or greater than +3 might be considered an outlier. than +3 might be considered an outlier. It might be: It might be: an incorrectly recorded data value an incorrectly recorded data value a data value that was incorrectly included in the a data value that was incorrectly included in the data set data set a correctly recorded data value that belongs in a correctly recorded data value that belongs in the data set the data set

21. 21 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Detecting Outliers The most extreme z-scores are -1.20 and 2.27 The most extreme z-scores are -1.20 and 2.27 Using | z | > 3 as the criterion for an outlier, there Using | z | > 3 as the criterion for an outlier, there are no outliers in this data set. are no outliers in this data set. Standardized Values for Apartment Rents Example: Apartment Rents Example: Apartment Rents

22. 22 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Five-Number Summaries and Box Plots Summary statistics and easy-to-draw graphs can be Summary statistics and easy-to-draw graphs can be used to quickly summarize large quantities of data. used to quickly summarize large quantities of data. Summary statistics and easy-to-draw graphs can be Summary statistics and easy-to-draw graphs can be used to quickly summarize large quantities of data. used to quickly summarize large quantities of data. Two tools that accomplish this are five-number Two tools that accomplish this are five-number summaries and box plots. summaries and box plots. Two tools that accomplish this are five-number Two tools that accomplish this are five-number summaries and box plots. summaries and box plots.

23. 23 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Five-Number Summary 11 Smallest Value Smallest Value First Quartile First Quartile Median Median Third Quartile Third Quartile Largest Value Largest Value 22 33 44 55

24. 24 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Five-Number Summary Lowest Value = 525 First Quartile = 545 Median = 575 Third Quartile = 625 Largest Value = 715 Example: Apartment Rents Example: Apartment Rents

25. 25 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Box Plot A box plot is a graphical summary of data that is A box plot is a graphical summary of data that is based on a five-number summary. based on a five-number summary. A box plot is a graphical summary of data that is A box plot is a graphical summary of data that is based on a five-number summary. based on a five-number summary. A key to the development of a box plot is the A key to the development of a box plot is the computation of the median and the quartiles Q 1 and computation of the median and the quartiles Q 1 and Q 3. Q 3. A key to the development of a box plot is the A key to the development of a box plot is the computation of the median and the quartiles Q 1 and computation of the median and the quartiles Q 1 and Q 3. Q 3. Box plots provide another way to identify outliers. Box plots provide another way to identify outliers.

26. 26 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 500 525 550 575 600 625 650 675 700 725 A box is drawn with its ends located at the first and A box is drawn with its ends located at the first and third quartiles. third quartiles. Box Plot A vertical line is drawn in the box at the location of A vertical line is drawn in the box at the location of the median (second quartile). the median (second quartile). Q1 = 545 Q3 = 625 Q2 = 575 Example: Apartment Rents Example: Apartment Rents

27. 27 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Box Plot Limits are located (not drawn) using the interquartile range (IQR). Limits are located (not drawn) using the interquartile range (IQR). Data outside these limits are considered outliers. Data outside these limits are considered outliers. The locations of each outlier is shown with the symbol *. The locations of each outlier is shown with the symbol *.continued

28. 28 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Box Plot Lower Limit: Q1 - 1.5(IQR) = 545 - 1.5(80) = 425 Upper Limit: Q3 + 1.5(IQR) = 625 + 1.5(80) = 745 The lower limit is located 1.5(IQR) below Q 1. The lower limit is located 1.5(IQR) below Q 1. The upper limit is located 1.5(IQR) above Q 3. The upper limit is located 1.5(IQR) above Q 3. There are no outliers (values less than 425 or There are no outliers (values less than 425 or greater than 745) in the apartment rent data. greater than 745) in the apartment rent data. Example: Apartment Rents Example: Apartment Rents

29. 29 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Box Plot Whiskers (dashed lines) are drawn from the ends Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values of the box to the smallest and largest data values inside the limits. inside the limits. 500 525 550 575 600 625 650 675 700 725 Smallest value inside limits = 525 Largest value inside limits = 715 Example: Apartment Rents Example: Apartment Rents

30. 30 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Measures of Association Between Two Variables Thus far we have examined numerical methods used Thus far we have examined numerical methods used to summarize the data for one variable at a time. to summarize the data for one variable at a time. Thus far we have examined numerical methods used Thus far we have examined numerical methods used to summarize the data for one variable at a time. to summarize the data for one variable at a time. Often a manager or decision maker is interested in Often a manager or decision maker is interested in the relationship between two variables. the relationship between two variables. Often a manager or decision maker is interested in Often a manager or decision maker is interested in the relationship between two variables. the relationship between two variables. Two descriptive measures of the relationship Two descriptive measures of the relationship between two variables are covariance and correlation between two variables are covariance and correlation coefficient. coefficient. Two descriptive measures of the relationship Two descriptive measures of the relationship between two variables are covariance and correlation between two variables are covariance and correlation coefficient. coefficient.

31. 31 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Covariance Positive values indicate a positive relationship. Positive values indicate a positive relationship. Negative values indicate a negative relationship. Negative values indicate a negative relationship. The covariance is a measure of the linear association The covariance is a measure of the linear association between two variables. between two variables. The covariance is a measure of the linear association The covariance is a measure of the linear association between two variables. between two variables.

32. 32 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Covariance The covariance is computed as follows: The covariance is computed as follows: forsamples forpopulations

33. 33 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Correlation Coefficient Just because two variables are highly correlated, it Just because two variables are highly correlated, it does not mean that one variable is the cause of the does not mean that one variable is the cause of the other. other. Just because two variables are highly correlated, it Just because two variables are highly correlated, it does not mean that one variable is the cause of the does not mean that one variable is the cause of the other. other. Correlation is a measure of linear association and not Correlation is a measure of linear association and not necessarily causation. necessarily causation. Correlation is a measure of linear association and not Correlation is a measure of linear association and not necessarily causation. necessarily causation.

34. 34 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The correlation coefficient is computed as follows: The correlation coefficient is computed as follows: forsamplesforpopulations Correlation Coefficient

35. 35 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Correlation Coefficient Values near +1 indicate a strong positive linear Values near +1 indicate a strong positive linear relationship. relationship. Values near +1 indicate a strong positive linear Values near +1 indicate a strong positive linear relationship. relationship. Values near -1 indicate a strong negative linear Values near -1 indicate a strong negative linear relationship. relationship. Values near -1 indicate a strong negative linear Values near -1 indicate a strong negative linear relationship. relationship. The coefficient can take on values between -1 and +1. The coefficient can take on values between -1 and +1. The closer the correlation is to zero, the weaker the The closer the correlation is to zero, the weaker the relationship. relationship. The closer the correlation is to zero, the weaker the The closer the correlation is to zero, the weaker the relationship. relationship.

36. 36 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. A golfer is interested in investigating the A golfer is interested in investigating the relationship, if any, between driving distance and 18-hole score. 277.6 259.5 269.1 267.0 255.6 272.9 69 71 70 70 71 69 Average Driving Distance (yds.) Average 18-Hole Score Covariance and Correlation Coefficient Example: Golfing Study Example: Golfing Study

37. 37 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Covariance and Correlation Coefficient 277.6259.5269.1267.0255.6272.9 697170707169 xy 10.65 10.65 -7.45 -7.45 2.15 2.15 0.05 0.05-11.35 5.95 5.95 1.0 1.0 0 0 -10.65 -10.65 -7.45 -7.45 0 0-11.35 -5.95 -5.95 Average Std. Dev. 267.070.0-35.40 8.2192.8944 Total Example: Golfing Study Example: Golfing Study

38. 38 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sample Covariance Sample Covariance Sample Correlation Coefficient Sample Correlation Coefficient Covariance and Correlation Coefficient Example: Golfing Study Example: Golfing Study

39. 39 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Dashboards: Adding Numerical Measures to Improve Effectiveness The addition of numerical measures, such as the mean The addition of numerical measures, such as the mean and standard deviation of KPIs, to a data dashboard and standard deviation of KPIs, to a data dashboard is often critical. is often critical. Drilling down refers to functionality in interactive Drilling down refers to functionality in interactive dashboards that allows the user to access information dashboards that allows the user to access information and analyses at increasingly detailed level. and analyses at increasingly detailed level. Dashboards are often interactive. Dashboards are often interactive. Data dashboards are not limited to graphical displays. Data dashboards are not limited to graphical displays.

40. 40 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Dashboards: Adding Numerical Measures to Improve Effectiveness

41. 41 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 3, Part B