1 1 Slide © 2003 Thomson/South-Western

2 2 Slide © 2003 Thomson/South-Western Chapter 3 Descriptive Statistics: Numerical Methods Part B n Measures of Relative Location and Detecting Outliers n Exploratory Data Analysis n Measures of Association Between Two Variables n The Weighted Mean and Working with Grouped Data Working with Grouped Data     % % x x

3 3 Slide © 2003 Thomson/South-Western Measures of Relative Location and Detecting Outliers n z-Scores n Chebyshev’s Theorem n Empirical Rule n Detecting Outliers

4 4 Slide © 2003 Thomson/South-Western z-Scores n The z-score is often called the standardized value. n It denotes the number of standard deviations a data value x i is from the mean. n A data value less than the sample mean will have a z- score less than zero. n A data value greater than the sample mean will have a z-score greater than zero. n A data value equal to the sample mean will have a z- score of zero.

5 5 Slide © 2003 Thomson/South-Western n z-Score of Smallest Value (425) Standardized Values for Apartment Rents Example: Apartment Rents

6 6 Slide © 2003 Thomson/South-Western 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 any value greater than 1. At least 75% of the items must be within 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 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 At least 94% of the items must be within z = 4 standard deviations of the mean. 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 any value greater than 1. At least 75% of the items must be within 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 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 At least 94% of the items must be within z = 4 standard deviations of the mean.

7 7 Slide © 2003 Thomson/South-Western Example: Apartment Rents n Chebyshev’s Theorem Let z = 1.5 with = and s = Let z = 1.5 with = and s = At least (1 - 1/(1.5) 2 ) = = 0.56 or 56% of the rent values must be between of the rent values must be between - z ( s ) = (54.74) = z ( s ) = (54.74) = 409 and and + z ( s ) = (54.74) = z ( s ) = (54.74) = 573

8 8 Slide © 2003 Thomson/South-Western n Chebyshev’s Theorem (continued) Actually, 86% of the rent values Actually, 86% of the rent values are between 409 and 573. are between 409 and 573. Example: Apartment Rents

9 9 Slide © 2003 Thomson/South-Western Empirical Rule For data having a bell-shaped distribution: For data having a bell-shaped distribution: Approximately 68% of the data values will be within one standard deviation of the mean. Approximately 68% of the data values will be within one standard deviation of the mean.

10 Slide © 2003 Thomson/South-Western Empirical Rule For data having a bell-shaped distribution: Approximately 95% of the data values will be within two standard deviations of the mean. Approximately 95% of the data values will be within two standard deviations of the mean.

11 Slide © 2003 Thomson/South-Western 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. Almost all (99.7%) of the items will be within three standard deviations of the mean.

12 Slide © 2003 Thomson/South-Western Example: Apartment Rents n Empirical Rule Interval % in Interval Interval % in Interval Within +/- 1 s to /70 = 69% Within +/- 2 s to /70 = 97% Within +/- 3 s to /70 = 100%

13 Slide © 2003 Thomson/South-Western Detecting Outliers n An outlier is an unusually small or unusually large value in a data set. n A data value with a z-score less than -3 or greater than +3 might be considered an outlier. n It might be: an incorrectly recorded data value an incorrectly recorded data value a data value that was incorrectly included in the data set a data value that was incorrectly included in the data set a correctly recorded data value that belongs in the data set a correctly recorded data value that belongs in the data set

14 Slide © 2003 Thomson/South-Western Example: Apartment Rents n Detecting Outliers The most extreme z-scores are and Using | z | > 3 as the criterion for an outlier, there are no outliers in this data set. Standardized Values for Apartment Rents

15 Slide © 2003 Thomson/South-Western End of Chapter 3, Part B