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Slides Prepared by JOHN S. LOUCKS St. Edward’s University © 2003 South-Western/Thomson LearningTM Slide 1 Chapter 3 Descriptive Statistics: Numerical Methods, Part A Measures of Location Measures of Variability x © 2003 South-Western/Thomson LearningTM Slide 2 Measures of Location Mean Median Mode Percentiles Quartiles © 2003 South-Western/Thomson LearningTM Slide 3 Example: Apartment Rents Given below is a sample of monthly rent values ($) for one-bedroom apartments. The data is a sample of 70 apartments in a particular city. The data are presented in ascending order. 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 © 2003 South-Western/Thomson LearningTM Slide 4 Mean The mean of a data set is the average of all the data values. If the data are from a sample, the mean is denoted by x. xi x n If the data are from a population, the mean is denoted by m (mu). xi N © 2003 South-Western/Thomson LearningTM Slide 5 Example: Apartment Rents Mean xi 34, 356 x 490.80 n 70 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 © 2003 South-Western/Thomson LearningTM Slide 6 Median The median is the measure of location most often reported for annual income and property value data. A few extremely large incomes or property values can inflate the mean. © 2003 South-Western/Thomson LearningTM Slide 7 Median The median of a data set is the value in the middle when the data items are arranged in ascending order. For an odd number of observations, the median is the middle value. For an even number of observations, the median is the average of the two middle values. © 2003 South-Western/Thomson LearningTM Slide 8 Example: Apartment Rents Median Median = 50th percentile i = (p/100)n = (50/100)70 = 35.5 Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 © 2003 South-Western/Thomson LearningTM Slide 9 Mode The mode of a data set is the value that occurs with greatest frequency. The greatest frequency can occur at two or more different values. If the data have exactly two modes, the data are bimodal. If the data have more than two modes, the data are multimodal. © 2003 South-Western/Thomson LearningTM Slide 10 Example: Apartment Rents Mode 450 occurred most frequently (7 times) Mode = 450 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 © 2003 South-Western/Thomson LearningTM Slide 11 Using Excel to Compute the Mean, Median, and Mode Formula Worksheet 1 2 3 4 5 6 A Apartment 1 2 3 4 5 B C D E Monthly Rent ($) 525 Mean =AVERAGE(B2:B71) 440 Median =MEDIAN(B2:B71) 450 Mode =MODE(B2:B71) 615 480 Note: Rows 7-71 are not shown. © 2003 South-Western/Thomson LearningTM Slide 12 Using Excel to Compute the Mean, Median, and Mode Value Worksheet 1 2 3 4 5 6 A Apartment 1 2 3 4 5 B C D Monthly Rent ($) 525 Mean 440 Median 450 Mode 615 480 E 490.80 475.00 450.00 Note: Rows 7-71 are not shown. © 2003 South-Western/Thomson LearningTM Slide 13 Percentiles A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. Admission test scores for colleges and universities are frequently reported in terms of percentiles. © 2003 South-Western/Thomson LearningTM Slide 14 Percentiles The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more. • Arrange the data in ascending order. • Compute index i, the position of the pth percentile. i = (p/100)n • If i is not an integer, round up. The pth percentile is the value in the ith position. • If i is an integer, the pth percentile is the average of the values in positions i and i+1. © 2003 South-Western/Thomson LearningTM Slide 15 Example: Apartment Rents 90th Percentile i = (p/100)n = (90/100)70 = 63 Averaging the 63rd and 64th data values: 90th Percentile = (580 + 590)/2 = 585 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 © 2003 South-Western/Thomson LearningTM Slide 16 Quartiles Quartiles are specific percentiles First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile © 2003 South-Western/Thomson LearningTM Slide 17 Example: Apartment Rents Third Quartile Third quartile = 75th percentile i = (p/100)n = (75/100)70 = 52.5 = 53 Third quartile = 525 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 © 2003 South-Western/Thomson LearningTM Slide 18 Using Excel to Compute Percentiles and Quartiles 1 2 3 4 5 6 Unsorted Monthly Rent ($) A B C D Apart- Monthly ment Rent ($) 1 525 2 440 3 450 4 615 5 480 Note: Rows 7-71 are not shown. E F © 2003 South-Western/Thomson LearningTM Slide 19 Using Excel to Compute Percentiles and Quartiles Sorting Data Step 1 Select any cell containing data in column B Step 2 Select the Data pull-down menu Step 3 Choose the Sort option Step 4 When the Sort dialog box appears: In the Sort by box, make sure that Monthly Rent ($) appears and that Ascending is selected In the My list has box, make sure that Header row is selected Click OK © 2003 South-Western/Thomson LearningTM Slide 20 Using Excel to Compute Percentiles and Quartiles 1 2 3 4 5 6 Sorted Monthly Rent ($) A B C D Apart- Monthly ment Rent ($) 1 425 2 430 3 430 4 435 5 435 Note: Rows 7-71 are not shown. E F © 2003 South-Western/Thomson LearningTM Slide 21 Using Excel to Compute Percentiles and Quartiles 1 2 3 4 5 6 Formula Worksheet for 90th Percentile’s Index A B C D E F Apart- Monthly Number of ment Rent ($) Observations Percentile Index i 1 425 70 90 =(E2/100)*D2 2 430 3 430 4 435 5 435 Note: Rows 7-71 are not shown. © 2003 South-Western/Thomson LearningTM Slide 22 Using Excel to Compute Percentiles and Quartiles 1 2 3 4 5 6 Value Worksheet for 90th Percentile’s Index A B C D E Apart- Monthly Number of ment Rent ($) Observations Percentile 1 425 70 90 2 430 3 430 4 435 5 435 Note: Rows 7-71 are not shown. F Index i 63.00 © 2003 South-Western/Thomson LearningTM Slide 23 Using Excel to Compute Percentiles and Quartiles 1 2 3 4 5 6 Value Worksheet for 3rd Quartile’s Index A B C D E Apart- Monthly Number of ment Rent ($) Observations Percentile 1 425 70 75 2 430 3 430 4 435 5 435 Note: Rows 7-71 are not shown. F Index i 52.50 © 2003 South-Western/Thomson LearningTM Slide 24 Measures of Variability It is often desirable to consider measures of variability (dispersion), as well as measures of location. For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each. © 2003 South-Western/Thomson LearningTM Slide 25 Measures of Variability Range Interquartile Range Variance Standard Deviation Coefficient of Variation © 2003 South-Western/Thomson LearningTM Slide 26 Range The range of a data set is the difference between the largest and smallest data values. It is the simplest measure of variability. It is very sensitive to the smallest and largest data values. © 2003 South-Western/Thomson LearningTM Slide 27 Example: Apartment Rents Range Range = largest value - smallest value Range = 615 - 425 = 190 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 © 2003 South-Western/Thomson LearningTM Slide 28 Interquartile Range The interquartile range of a data set is the difference between the third quartile and the first quartile. It is the range for the middle 50% of the data. It overcomes the sensitivity to extreme data values. © 2003 South-Western/Thomson LearningTM Slide 29 Example: Apartment Rents Interquartile Range 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 - Q1 = 525 - 445 = 80 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 © 2003 South-Western/Thomson LearningTM Slide 30 Variance The variance is a measure of variability that utilizes all the data. It is based on the difference between the value of each observation (xi) and the mean (x for a sample, for a population). © 2003 South-Western/Thomson LearningTM Slide 31 Variance The variance is the average of the squared differences between each data value and the mean. If the data set is a sample, the variance is denoted by s2. s2 2 ( x x ) i n 1 If the data set is a population, the variance is denoted by 2. 2 ( x ) i 2 N © 2003 South-Western/Thomson LearningTM Slide 32 Standard Deviation The standard deviation of a data set is the positive square root of the variance. It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean. If the data set is a sample, the standard deviation is denoted s. s s2 If the data set is a population, the standard deviation is denoted (sigma). 2 © 2003 South-Western/Thomson LearningTM Slide 33 Coefficient of Variation The coefficient of variation indicates how large the standard deviation is in relation to the mean. If the data set is a sample, the coefficient of variation is computed as follows: s (100) x If the data set is a population, the coefficient of variation is computed as follows: (100) © 2003 South-Western/Thomson LearningTM Slide 34 Example: Apartment Rents Variance s n 1 2 ( xi x ) 2 2 , 996.16 Standard Deviation s s2 2996. 47 54. 74 Coefficient of Variation s 54. 74 100 100 11.15 x 490.80 © 2003 South-Western/Thomson LearningTM Slide 35 Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation Formula Worksheet 1 2 3 4 5 6 7 A B C D E Apart- Monthly ment Rent ($) 1 525 Mean =AVERAGE(B2:B71) 2 440 Median =MEDIAN(B2:B71) 3 450 Mode =MODE(B2:B71) 4 615 Variance =VAR(B2:B71) 5 480 Std. Dev. =STDEV(B2:B71) 6 510 C.V. =E6/E2*100 Note: Rows 8-71 are not shown. © 2003 South-Western/Thomson LearningTM Slide 36 Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation Value Worksheet 1 2 3 4 5 6 7 A B C D Apart- Monthly ment Rent ($) 1 525 Mean 2 440 Median 3 450 Mode 4 615 Variance 5 480 Std. Dev. 6 510 C.V. E 490.80 475.00 450.00 2996.16 54.74 11.15 Note: Rows 8-71 are not shown. © 2003 South-Western/Thomson LearningTM Slide 37 Using Excel’s Descriptive Statistics Tool Step 1 Select the Tools pull-down menu Step 2 Choose the Data Analysis option Step 3 Choose Descriptive Statistics from the list of Analysis Tools … continued © 2003 South-Western/Thomson LearningTM Slide 38 Using Excel’s Descriptive Statistics Tool Step 4 When the Descriptive Statistics dialog box appears: Enter B1:B71 in the Input Range box Select Grouped By Columns Select Labels in First Row Select Output Range Enter D1 in the Output Range box Select Summary Statistics Click OK © 2003 South-Western/Thomson LearningTM Slide 39 Using Excel’s Descriptive Statistics Tool 1 2 3 4 5 6 7 8 Value Worksheet (Partial) A B C D E Apart- Monthly ment Rent ($) Monthly Rent ($) 1 525 2 440 Mean 490.8 3 450 Standard Error 6.542348114 4 615 Median 475 5 480 Mode 450 6 510 Standard Deviation 54.73721146 7 575 Sample Variance 2996.162319 Note: Rows 9-71 are not shown. © 2003 South-Western/Thomson LearningTM Slide 40 Using Excel’s Descriptive Statistics Tool Value Worksheet (Partial) 9 10 11 12 13 14 15 16 A 8 9 10 11 12 13 14 15 B 430 440 450 470 485 515 575 430 C D Kurtosis Skewness Range Minimum Maximum Sum Count E -0.334093298 0.924330473 190 425 615 34356 70 Note: Rows 1-8 and 17-71 are not shown. © 2003 South-Western/Thomson LearningTM Slide 41 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 South-Western/Thomson LearningTM Slide 42 Measures of Relative Location and Detecting Outliers z-Scores Chebyshev’s Theorem Empirical Rule Detecting Outliers © 2003 South-Western/Thomson LearningTM Slide 43 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 z-score 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 z-score of zero. © 2003 South-Western/Thomson LearningTM Slide 44 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 © 2003 South-Western/Thomson LearningTM Slide 45 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. © 2003 South-Western/Thomson LearningTM Slide 46 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 © 2003 South-Western/Thomson LearningTM Slide 47 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 © 2003 South-Western/Thomson LearningTM Slide 48 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 South-Western/Thomson LearningTM Slide 49 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 South-Western/Thomson LearningTM Slide 50 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 South-Western/Thomson LearningTM Slide 51 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 655.02 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 % 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 © 2003 South-Western/Thomson LearningTM Slide 52 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 ! © 2003 South-Western/Thomson LearningTM Slide 53 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 © 2003 South-Western/Thomson LearningTM Slide 54 Exploratory Data Analysis Five-Number Summary Box Plot © 2003 South-Western/Thomson LearningTM Slide 55 Five-Number Summary Smallest Value First Quartile Median Third Quartile Largest Value © 2003 South-Western/Thomson LearningTM Slide 56 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 © 2003 South-Western/Thomson LearningTM Slide 57 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. Limits are located (not drawn) using the interquartile range (IQR). • The lower limit is located 1.5(IQR) below Q1. • The upper limit is located 1.5(IQR) above Q3. • Data outside these limits are considered outliers. … continued © 2003 South-Western/Thomson LearningTM Slide 58 Box Plot (Continued) Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits. The locations of each outlier is shown with the symbol * . © 2003 South-Western/Thomson LearningTM Slide 59 Example: Apartment Rents Box Plot Lower Limit: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5 Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5 There are no outliers. 37 5 40 0 42 5 45 0 47 5 50 0 52 5 550 575 600 625 © 2003 South-Western/Thomson LearningTM Slide 60 Measures of Association between Two Variables Covariance Correlation Coefficient © 2003 South-Western/Thomson LearningTM Slide 61 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. © 2003 South-Western/Thomson LearningTM Slide 62 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 © 2003 South-Western/Thomson LearningTM Slide 63 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 x y xy . © 2003 South-Western/Thomson LearningTM Slide 64 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) © 2003 South-Western/Thomson LearningTM Slide 65 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 E Pop. Covariance Samp. Correlation -5.9 -0.9631 © 2003 South-Western/Thomson LearningTM Slide 66 The Weighted Mean and Working with Grouped Data Weighted Mean Mean for Grouped Data Variance for Grouped Data Standard Deviation for Grouped Data © 2003 South-Western/Thomson LearningTM Slide 67 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. © 2003 South-Western/Thomson LearningTM Slide 68 Weighted Mean x = wi xi wi where: xi = value of observation i wi = weight for observation i © 2003 South-Western/Thomson LearningTM Slide 69 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. © 2003 South-Western/Thomson LearningTM Slide 70 Mean for Grouped Data Sample Data fM x f i fM i i i Population Data i N where: fi = frequency of class i Mi = midpoint of class i © 2003 South-Western/Thomson LearningTM Slide 71 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 © 2003 South-Western/Thomson LearningTM Slide 72 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 iMi 3436.0 7641.5 5634.0 3916.0 3566.5 2118.0 1099.0 2278.0 1179.0 3657.0 34525.0 34, 525 493. 21 70 This approximation differs by $2.41 from the actual sample mean of $490.80. x © 2003 South-Western/Thomson LearningTM Slide 73 Variance for Grouped Data Sample Data 2 f ( M x ) i i s2 n 1 Population Data 2 f ( M ) i i 2 N © 2003 South-Western/Thomson LearningTM Slide 74 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. © 2003 South-Western/Thomson LearningTM Slide 75 End of Chapter 3 © 2003 South-Western/Thomson LearningTM Slide 76

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