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.. .. .. .. .. .. SLIDES BY John Loucks St. Edward’s University © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 1 Chapter 3, Part A Descriptive Statistics: Numerical Measures Measures of Location Measures of Variability © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 2 Measures of Location Mean Median Mode Percentiles Quartiles If the measures are computed for data from a sample, they are called sample statistics. If the measures are computed for data from a population, they are called population parameters. A sample statistic is referred to as the point estimator of the corresponding population parameter. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 3 Mean Perhaps the most important measure of location is the mean. The mean provides a measure of central location. The mean of a data set is the average of all the data values. The sample mean x is the point estimator of the population mean m. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 4 Sample Mean x x x Sum of the values of the n observations i n Number of observations in the sample © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 5 Population Mean m m x Sum of the values of the N observations i N Number of observations in the population © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 6 Sample Mean Example: Apartment Rents Fifty efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed below. 438 476 119 237 49 253 282 66 221 49 103 330 190 345 66 211 505 73 34 43 235 68 378 206 270 119 250 143 391 141 282 28 273 463 192 106 220 216 511 131 763 478 15 558 130 128 525 340 334 80 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 7 Sample Mean Xbar = SUM(X)/n SUM Count Average Average 12064 50 241.28 241.28 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 8 Median The median of a data set is the value in the middle when the data items are arranged in ascending order. Whenever a data set has extreme values, the median is the preferred measure of central location. 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 9 Median For an odd number of observations: 26 18 27 12 14 27 19 12 14 18 19 26 27 27 7 observations in ascending order the median is the middle value. Median = 19 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 10 Median For an even number of observations: 26 18 27 12 14 27 30 19 12 14 18 19 26 27 27 30 8 observations in ascending order the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 11 Median Example: Apartment Rents Averaging the 25 and 26th data values: Median = (216 + 220)/2 = 218 15 28 34 43 49 49 66 66 68 73 80 103 106 119 119 128 130 131 141 143 190 192 206 211 216 220 221 235 237 250 253 270 273 282 282 330 334 340 345 378 391 438 463 476 478 505 511 525 558 763 Note: Data is in ascending order. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 12 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. Caution: If the data are bimodal or multimodal, Excel’s MODE function will incorrectly identify a single mode. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 13 Mode Example: Apartment Rents 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 Note: Data is in ascending order. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 14 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 15 Measures of Variability Range Variance Standard Deviation Coefficient of Variation © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 16 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 17 Range Example: Apartment Rents 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 Note: Data is in ascending order. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 18 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, m for a population). The variance is useful in comparing the variability of two or more variables. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 19 Variance The variance is the average of the squared differences between each data value and the mean. The variance is computed as follows: 2 ( xi x ) s n 1 for a sample 2 ( xi m ) N 2 2 for a population © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 20 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 interpreted than the variance. The standard deviation is computed as follows: s s2 2 for a sample for a population © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 21 Coefficient of Variation The coefficient of variation indicates how large the standard deviation is in relation to the mean. The coefficient of variation is computed as follows: s 100 % x 100 % m for a sample for a population © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 22 Sample Mean Example: Apartment Rents Seventy efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed below. 445 440 465 450 600 570 510 615 440 450 470 485 515 575 430 440 525 490 580 450 490 590 525 450 472 470 445 435 435 425 450 475 490 525 600 600 445 460 475 500 535 435 460 575 435 500 549 475 445 600 445 460 480 500 550 435 440 450 465 570 500 480 430 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 615 450 480 465 480 510 440 Slide 23 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 ($) 1 525 Mean =AVERAGE(B2:B71) 2 440 Median =MEDIAN(B2:B71) 3 450 Mode =MODE.SNGL(B2:B71) 4 615 Variance =VAR.S(B2:B71) 5 480 Std. Dev. =STDEV.S(B2:B71) 6 510 C.V. =E6/E2*100 Note: Rows 8-71 are not shown. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 24 Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation Value Worksheet 1 2 3 4 5 6 7 A B C D Apart- Monthly ment Rent ($) 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 25 Using Excel’s Descriptive Statistics Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose Descriptive Statistics from the list of Analysis Tools Step 4 When the Descriptive Statistics dialog box appears: (see details on next slide) © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 26 Using Excel’s Descriptive Statistics Tool Excel’s Descriptive Statistics Dialog Box © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 27 Using Excel’s Descriptive Statistics Tool Excel Value Worksheet (Partial) 1 2 3 4 5 6 7 8 E D C B A Apart- Monthly Monthly Rent ($) ment Rent ($) 525 1 490.8 Mean 440 2 6.542348114 Standard Error 450 3 475 Median 615 4 450 Mode 480 5 Standard Deviation 54.73721146 510 6 2996.162319 Sample Variance 575 7 Note: Rows 9-71 are not shown. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 28 Using Excel’s Descriptive Statistics Tool Excel 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 29 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. 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 30 Using Excel’s Rank and Percentile Tool to Compute Percentiles and Quartiles Using Excel’s Percentile Function The formula Excel uses to compute the location (Lp) of the pth percentile is Lp = pn + (1 – p) Excel would compute the location of the 80th percentile for the apartment rent data as follows: L80 = (0.8)70 + (1 – 0.8) = 56 + .2 = 56.2 The 80th percentile would be 535 + .2(549 - 535) = 535 + 2.8 = 537.8 Excel interpolates over the interval from 0 to n. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 31 Using Excel’s Rank and Percentile Tool to Compute Percentiles and Quartiles Excel Formula Worksheet 1 2 3 4 5 6 80th percentile A B C D Apart- Monthly ment Rent ($) 80th Percentile 1 525 =PERCENTILE.INC(B2:B71,.8) 2 440 3 450 It is not necessary 4 615 5 480 to put the data Note: Rows 7-71 are not shown. in ascending order. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 32 Quartiles Quartiles are specific percentiles. First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 33 Third Quartile Using Excel’s QUARTILE.INC Function Excel computes the locations of the 1st, 2nd, and 3rd quartiles by first converting the quartiles to percentiles and then using the following formula to compute the location (Lp) of the pth percentile: Lp = (p/100)n + (1 – p/100) Excel would compute the location of the 3rd quartile (75th percentile) for the rent data as follows: L75 = (75/100)70 + (1 – 75/100) = 52.5 + .25 = 52.75 The 3rd quartile would be 515 + .75(525 - 515) = 515 + 7.5 = 522.5 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 34 Third Quartile Excel Formula Worksheet 1 2 3 4 5 6 A B Apart- Monthly ment Rent ($) 1 525 2 440 3 450 4 615 5 480 C 3rd quartile D Third Quartile =QUARTILE.INC(B2:B71,3) It is not necessary to put the data in ascending order. Note: Rows 7-71 are not shown. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 35 Using Excel’s QUARTILE.INC Function If the value of 1 in the QUARTILE.INC function is changed to 0, Excel computes the minimum value in the data set. If the value of 1 is changed to 4, Excel computes the maximum value in the data set. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 36 Excel’s Rank and Percentile Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose Rank and Percentile from the list of Analysis Tools Step 4 When the Rank and Percentile dialog box appears (see details on next slide) © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 37 Excel’s Rank and Percentile Tool Step 4 Complete the Rank and Percentile dialog box as follows: © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 38 Excel’s Rank and Percentile Tool Excel Value Worksheet 1 2 3 4 5 6 7 8 9 10 B Rent 525 440 450 615 480 510 575 430 440 C D Point 4 63 35 42 49 56 28 21 7 E F Rent Rank 615 1 615 1 600 3 600 3 600 3 600 3 590 7 580 8 575 9 G Percent 98.50% 98.50% 92.70% 92.70% 92.70% 92.70% 91.30% 89.80% 86.90% Note: Rows 11-71 are not shown. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 39 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 40 Interquartile Range Example: Apartment Rents 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 Note: Data is in ascending order. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 41 Trimmed Mean Another measure, sometimes used when extreme values are present, is the trimmed mean. It is obtained by deleting a percentage of the smallest and largest values from a data set and then computing the mean of the remaining values. For example, the 5% trimmed mean is obtained by removing the smallest 5% and the largest 5% of the data values and then computing the mean of the remaining values. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 42 72 91 74 67 36 38 60 44 14 3 13 17 47 44 28 38 31 33 35 36 37 37 38 39 Count 0.07 Round Up 50 3.5 4 Drop Top and Bottom 59.45 59.45 Average Static 59.45 Average Dynamic OFFSET © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 43 End of Chapter 3, Part A © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 44