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© 2003 Thomson/South-Western Slide 1 Chapter 3 Descriptive Statistics: 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 x © 2003 Thomson/South-Western Slide 2 Measures of Relative Location and Detecting Outliers z-Scores Chebyshev’s Theorem Empirical Rule Detecting Outliers © 2003 Thomson/South-Western 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. © 2003 Thomson/South-Western 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 © 2003 Thomson/South-Western -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/z2) of the items in any data set will be within z standard deviations of the mean, where z is any value greater than 1. • At least 75% of the items must be within z = 2 standard deviations of the mean. • At least 89% of the items must be within z = 3 standard deviations of the mean. • At least 94% of the items must be within z = 4 standard deviations of the mean. © 2003 Thomson/South-Western Slide 6 Example: Apartment Rents Chebyshev’s Theorem Let z = 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 - z(s) = 490.80 - 1.5(54.74) = 409 and x + z(s) = 490.80 + 1.5(54.74) = 573 © 2003 Thomson/South-Western 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 © 2003 Thomson/South-Western 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 Empirical Rule For data having a bell-shaped distribution: • Approximately 68% of the data values will be within one standard deviation of the mean. © 2003 Thomson/South-Western Slide 9 Empirical Rule For data having a bell-shaped distribution: • Approximately 95% of the data values will be within two standard deviations of the mean. © 2003 Thomson/South-Western Slide 10 Empirical Rule For data having a bell-shaped distribution: • Almost all (99.7%) of the items will be within three standard deviations of the mean. © 2003 Thomson/South-Western Slide 11 Example: Apartment Rents 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 © 2003 Thomson/South-Western 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 12 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 • a data value that was incorrectly included in the data set • a correctly recorded data value that belongs in the data set © 2003 Thomson/South-Western Slide 13 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 © 2003 Thomson/South-Western -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 14 End of Chapter 3, Part B © 2003 Thomson/South-Western Slide 15