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Anderson u Sweeney u Williams CONTEMPORARY BUSINESS STATISTICS WITH MICROSOFT EXCEL u Slides Prepared by JOHN LOUCKS u © 2001 South-Western /Thomson Learning Slide 1 Chapter 3 Descriptive Statistics II: Numerical Methods - Part B Measures of Relative Location and Detecting Outliers Exploratory Data Analysis Measures of Association between Two Variables The Weighted Mean and Working with Grouped Data Slide 2 Measures of Relative Location and Detecting Outliers z-Scores Chebyshev’s Theorem The Empirical Rule Detecting Outliers Slide 3 z-Scores The z-score is often called the standardized value. It denotes the number of standard deviations a data value xi is from the mean. xi x zi s A data value less than the sample mean will have a zscore less than zero. A data value greater than the sample mean will have a z-score greater than zero. A data value equal to the sample mean will have a zscore of zero. Slide 4 Example: Apartment Rents z-Score of Smallest Value (425) xi x 425 490.80 z 1. 20 s 54. 74 Standardized Values for Apartment Rents -1.20 -0.93 -0.75 -0.47 -0.20 0.35 1.54 -1.11 -0.93 -0.75 -0.38 -0.11 0.44 1.54 -1.11 -0.93 -0.75 -0.38 -0.01 0.62 1.63 -1.02 -0.84 -0.75 -0.34 -0.01 0.62 1.81 -1.02 -0.84 -0.75 -0.29 -0.01 0.62 1.99 -1.02 -0.84 -0.56 -0.29 0.17 0.81 1.99 -1.02 -0.84 -0.56 -0.29 0.17 1.06 1.99 -1.02 -0.84 -0.56 -0.20 0.17 1.08 1.99 -0.93 -0.75 -0.47 -0.20 0.17 1.45 2.27 -0.93 -0.75 -0.47 -0.20 0.35 1.45 2.27 Slide 5 Chebyshev’s Theorem At least (1 - 1/k2) of the items in any data set will be within k standard deviations of the mean, where k is any value greater than 1. • At least 75% of the items must be within k = 2 standard deviations of the mean. • At least 89% of the items must be within k = 3 standard deviations of the mean. • At least 94% of the items must be within k = 4 standard deviations of the mean. Slide 6 Example: Apartment Rents Chebyshev’s Theorem Let k = 1.5 with x = 490.80 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 x - k(s) = 490.80 - 1.5(54.74) = 409 and x + k(s) = 490.80 + 1.5(54.74) = 573 Slide 7 Example: Apartment Rents Chebyshev’s Theorem (continued) Actually, 86% of the rent values are between 409 and 573. 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 Slide 8 The Empirical Rule For data having a bell-shaped distribution: • Approximately 68% of the data values will be within one standard deviation of the mean. • Approximately 95% of the data values will be within two standard deviations of the mean. • Almost all of the items (99.7%) will be within three standard deviations of the mean. Slide 9 Example: Apartment Rents The Empirical Rule Within +/- 1s Within +/- 2s Within +/- 3s 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 Interval 436.06 to 545.54 381.32 to 600.28 326.58 to 655.02 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 % in Interval 48/70 = 69% 68/70 = 97% 70/70 = 100% 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 Slide 10 Detecting Outliers An outlier is an unusually small or unusually large value in a data set. A data value with a z-score less than -3 or greater than +3 might be considered an outlier. It might be an incorrectly recorded data value. It might be a data value that was incorrectly included in the data set. It might be a correctly recorded data value that belongs in the data set ! Slide 11 Example: Apartment Rents Detecting Outliers The most extreme z-scores are -1.20 and 2.27. Using |z| > 3 as the criterion for an outlier, there are no outliers in this data set. Standardized Values for Apartment Rents -1.20 -0.93 -0.75 -0.47 -0.20 0.35 1.54 -1.11 -0.93 -0.75 -0.38 -0.11 0.44 1.54 -1.11 -0.93 -0.75 -0.38 -0.01 0.62 1.63 -1.02 -0.84 -0.75 -0.34 -0.01 0.62 1.81 -1.02 -0.84 -0.75 -0.29 -0.01 0.62 1.99 -1.02 -0.84 -0.56 -0.29 0.17 0.81 1.99 -1.02 -0.84 -0.56 -0.29 0.17 1.06 1.99 -1.02 -0.84 -0.56 -0.20 0.17 1.08 1.99 -0.93 -0.75 -0.47 -0.20 0.17 1.45 2.27 -0.93 -0.75 -0.47 -0.20 0.35 1.45 2.27 Slide 12 Exploratory Data Analysis Five-Number Summary Box Plot Slide 13 Five-Number Summary Smallest Value First Quartile Median Third Quartile Largest Value Slide 14 Example: Apartment Rents Five-Number Summary Lowest Value = 425 First Quartile = 450 Median = 475 Third Quartile = 525 Largest Value = 615 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 Slide 15 Box Plot A box is drawn with its ends located at the first and third quartiles. A vertical line is drawn in the box at the location of the median. Fences are located (not drawn) using the interquartile range (IQR). • The inner fences are located 1.5(IQR) below Q1 and 1.5(IQR) above Q3. • The outer fences are located 3(IQR) below Q1 and 3(IQR) above Q3. … continued Slide 16 Box Plot (Continued) Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the inner fences. The locations of mild outliers are shown with the symbol * . The locations of extreme outliers are shown with the symbol o . Slide 17 Example: Apartment Rents Box Plot Inner Fences: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5 Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5 Outer Fences: Q1 - 3(IQR) = 450 - 3(75) = 225 Q3 + 3(IQR) = 525 + 3(75) = 750 There are no mild or extreme outliers. 37 5 40 0 42 5 45 0 47 5 50 0 52 5 550 575 600 625 Slide 18 Measures of Association between Two Variables Covariance Correlation Coefficient Slide 19 Covariance The covariance is a measure of the linear association between two variables. Positive values indicate a positive relationship. Negative values indicate a negative relationship. Slide 20 Covariance If the data sets are samples, the covariance is denoted by sxy. ( xi x )( yi y ) sxy n 1 If the data sets are populations, the covariance is denoted by xy . xy ( xi x )( yi y ) N Slide 21 Correlation Coefficient The coefficient can take on values between -1 and +1. Values near -1 indicate a strong negative linear relationship. Values near +1 indicate a strong positive linear relationship. If the data sets are samples, the coefficient is rxy. rxy sxy sx s y If the data sets are populations, the coefficient is xy . xy xy x y Slide 22 Using Excel to Compute the Covariance and Correlation Coefficient 1 2 3 4 5 6 7 8 Formula Worksheet A Average Drive 277.6 259.5 269.1 267.0 255.6 272.9 B 18-Hole Score 69 71 70 70 71 69 C D E Pop. Covariance =COVAR(A2:A7,B2:B7) Samp. Correlation =CORREL(A2:A7,B2:B7) Slide 23 Using Excel to Compute the Covariance and Correlation Coefficient 1 2 3 4 5 6 7 8 Value Worksheet A Average Drive 277.6 259.5 269.1 267.0 255.6 272.9 B 18-Hole Score 69 71 70 70 71 69 C D Pop. Covariance Samp. Correlation E -5.9 -0.963073682 Slide 24 The Weighted Mean and Working with Grouped Data The Weighted Mean Mean for Grouped Data Variance for Grouped Data Standard Deviation for Grouped Data Slide 25 The Weighted Mean When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean. In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade. When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value. Slide 26 The Weighted Mean xwt = wi xi wi where: xi = value of observation i wi = weight for observation i Slide 27 Grouped Data The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data. To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class. We compute a weighted mean of the class midpoints using the class frequencies as weights. Similarly, in computing the variance and standard deviation, the class frequencies are used as weights. Slide 28 Mean for Grouped Data Sample Data fM x f i fM i i i Population Data i where: N fi = frequency of class i Mi = midpoint of class i Slide 29 Example: Apartment Rents Given below is the previous sample of monthly rents for one-bedroom apartments presented here as grouped data in the form of a frequency distribution. Rent ($) Frequency 420-439 8 440-459 17 460-479 12 480-499 8 500-519 7 520-539 4 540-559 2 560-579 4 580-599 2 600-619 6 Slide 30 Example: Apartment Rents Mean for Grouped Data Rent ($) 420-439 440-459 460-479 480-499 500-519 520-539 540-559 560-579 580-599 600-619 Total fi 8 17 12 8 7 4 2 4 2 6 70 Mi 429.5 449.5 469.5 489.5 509.5 529.5 549.5 569.5 589.5 609.5 f iM i 3436.0 7641.5 5634.0 3916.0 3566.5 2118.0 1099.0 2278.0 1179.0 3657.0 34525.0 34, 525 x 493. 21 70 This approximation differs by $2.41 from the actual sample mean of $490.80. Slide 31 Variance for Grouped Data Sample Data Population Data 2 f ( M x ) i i s2 n 1 2 f ( M ) i i 2 N Slide 32 Example: Apartment Rents Variance for Grouped Data s2 3, 017.89 Standard Deviation for Grouped Data s 3, 017.89 54. 94 This approximation differs by only $.20 from the actual standard deviation of $54.74. Slide 33 End of Chapter 3, Part B Slide 34

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