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35 Pages

### ESBE4-03a

Course: ORMS 5310, Fall 2008
School: Texas A&M...
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Word Count: 1039

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Prepared Slides by JOHN S. LOUCKS St. Edward' s University 2006 Thomson/SouthWestern Slide 1 Chapter 3 Descriptive Statistics: Numerical Measures Part A s s Measures of Location Measures of Variability 2006 Thomson/SouthWestern Slide 2 Measures of Location s s s s s Mean Median Mode Percentiles Quartiles If the measures are computed for data from a sample, they are...

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Prepared Slides by JOHN S. LOUCKS St. Edward' s University 2006 Thomson/SouthWestern Slide 1 Chapter 3 Descriptive Statistics: Numerical Measures Part A s s Measures of Location Measures of Variability 2006 Thomson/SouthWestern Slide 2 Measures of Location s s s s s 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. 2006 Thomson/SouthWestern Slide 3 Mean s s The mean of a data set is the average of all the data values. The sample mean is the point estimator of the population mean . x 2006 Thomson/SouthWestern Slide 4 Sample Mean x Sum of the values of the n observations i x= x n Number of observations in the sample 2006 Thomson/SouthWestern Slide 5 Population Mean Sum of the values of the N observations i = x N Number of observations in the population 2006 Thomson/SouthWestern Slide 6 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 in ascending order on the next slide. 2006 Thomson/SouthWestern Slide 7 Sample Mean 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 2006 Thomson/SouthWestern Slide 8 Sample Mean x x= 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 34, 356 = = 490.80 n 70 i 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 2006 Thomson/SouthWestern Slide 9 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. 2006 Thomson/SouthWestern Slide 10 Median For an odd number of observations: 26 18 27 12 14 27 19 12 14 18 19 26 27 27 the median is the middle value. Median = 19 7 observations in ascending order 2006 Thomson/SouthWestern Slide 11 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 2006 Thomson/SouthWestern Slide 12 Median 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 2006 Thomson/SouthWestern Slide 13 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. 2006 Thomson/SouthWestern Slide 14 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 2006 Thomson/SouthWestern Slide 15 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. 2006 Thomson/SouthWestern Slide 16 Percentiles s 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. 2006 Thomson/SouthWestern Slide 17 Percentiles 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 p th percentile is the value in the i th position. If i is an integer, the p th percentile is the average of the values in positions i and i +1. 2006 Thomson/SouthWestern Slide 18 90th Percentile i = (p/100)n = (90/100)70 = 63 Averaging 63rd the 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 Slide 19 2006 Thomson/SouthWestern 90 Percentile th "At least 90% of the items take on a value of 585 or less." 63/70 = .9 or 90% 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 "At least 10% of the items take on a value of 585 or more." 7/70 = .1 or 10% 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 20 2006 Thomson/SouthWestern Quartiles Quartiles are specific percentiles. First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile 2006 Thomson/SouthWestern Slide 21 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 Slide 22 2006 Thomson/SouthWestern 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. 2006 Thomson/SouthWestern Slide 23 Measures of Variability s s s s s Range Interquartile Range Variance Standard Deviation Coefficient of Variation 2006 Thomson/SouthWestern Slide 24 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. 2006 Thomson/SouthWestern Slide 25 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 2006 Thomson/SouthWestern Slide 26 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. 2006 Thomson/SouthWestern Slide 27 Interquartile Range 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interqua...

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Texas A&M University, Corpus Christi - ORMS - 5310
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Texas A&M University, Corpus Christi - ORMS - 5310
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