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Slides by JOHN LOUCKS St. Edward’s University © 2009 Thomson South-Western. All Rights Reserved Slide 1 Chapter 3, Part A Descriptive Statistics: Numerical Measures Measures of Location Measures of Variability © 2009 Thomson South-Western. All Rights Reserved 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. © 2009 Thomson South-Western. All Rights Reserved Slide 3 Mean 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. © 2009 Thomson South-Western. All Rights Reserved Slide 4 Sample Mean x x x Sum of the values of the n observations i n Number of observations in the sample © 2009 Thomson South-Western. All Rights Reserved Slide 5 Population Mean m m x Sum of the values of the N observations i N Number of observations in the population © 2009 Thomson South-Western. All Rights Reserved 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 on the next slide. © 2009 Thomson South-Western. All Rights Reserved Slide 7 Sample Mean Apartment Rent Sample Data 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 © 2009 Thomson South-Western. All Rights Reserved 600 445 460 480 500 550 435 440 450 465 570 500 480 430 615 450 480 465 480 510 440 Slide 8 Sample Mean x x 34, 356 490.80 n 70 445 440 465 450 600 570 510 615 440 450 470 485 515 575 430 440 525 490 580 450 490 590 525 450 472 470 445 435 i 435 425 450 475 490 525 600 600 445 460 475 500 535 435 460 575 435 500 549 475 445 © 2009 Thomson South-Western. All Rights Reserved 600 445 460 480 500 550 435 440 450 465 570 500 480 430 615 450 480 465 480 510 440 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. © 2009 Thomson South-Western. All Rights Reserved Slide 10 Median For an odd number of observations: 26 18 27 12 14 27 12 14 18 26 27 27 19 19 7 observations in ascending order the median is the middle value. Median = 19 © 2009 Thomson South-Western. All Rights Reserved Slide 11 Median For an even number of observations: 26 18 27 12 14 27 30 19 8 observations 12 14 18 19 26 27 27 30 in ascending order the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5 © 2009 Thomson South-Western. All Rights Reserved 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 Note: Data is in ascending order. © 2009 Thomson South-Western. All Rights Reserved 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. © 2009 Thomson South-Western. All Rights Reserved 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 Note: Data is in ascending order. © 2009 Thomson South-Western. All Rights Reserved 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. © 2009 Thomson South-Western. All Rights Reserved Slide 16 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. © 2009 Thomson South-Western. All Rights Reserved 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 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. © 2009 Thomson South-Western. All Rights Reserved Slide 18 80th Percentile i = (p/100)n = (80/100)70 = 56 Averaging the 56th and 57th data values: 80th Percentile = (535 + 549)/2 = 542 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 Note: Data is in ascending order. © 2009 Thomson South-Western. All Rights Reserved Slide 19 80th Percentile 425 440 450 465 480 510 575 “At least 80% of the items take on a value of 542 or less.” “At least 20% of the items take on a value of 542 or more.” 56/70 = .8 or 80% 14/70 = .2 or 20% 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 © 2009 Thomson South-Western. All Rights Reserved 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 Slide 20 Quartiles Quartiles are specific percentiles First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile © 2009 Thomson South-Western. All Rights Reserved 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 Note: Data is in ascending order. © 2009 Thomson South-Western. All Rights Reserved Slide 22 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. © 2009 Thomson South-Western. All Rights Reserved Slide 23 Measures of Variability Range Interquartile Range Variance Standard Deviation Coefficient of Variation © 2009 Thomson South-Western. All Rights Reserved 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. © 2009 Thomson South-Western. All Rights Reserved 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 Note: Data is in ascending order. © 2009 Thomson South-Western. All Rights Reserved 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. © 2009 Thomson South-Western. All Rights Reserved Slide 27 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 Note: Data is in ascending order. © 2009 Thomson South-Western. All Rights Reserved Slide 28 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). © 2009 Thomson South-Western. All Rights Reserved Slide 29 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 © 2009 Thomson South-Western. All Rights Reserved Slide 30 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. © 2009 Thomson South-Western. All Rights Reserved Slide 31 Standard Deviation The standard deviation is computed as follows: s s2 2 for a sample for a population © 2009 Thomson South-Western. All Rights Reserved Slide 32 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 for a sample 100 % m for a population © 2009 Thomson South-Western. All Rights Reserved Slide 33 Variance, Standard Deviation, And Coefficient of Variation Variance s2 (x 2 x ) i n1 2, 996.16 Standard Deviation s s2 2996.16 54.74 Coefficient of Variation the standard deviation is about 11% of the mean s 54.74 100 % 11.15% 100 % x 490.80 © 2009 Thomson South-Western. All Rights Reserved Slide 34 End of Chapter 3, Part A © 2009 Thomson South-Western. All Rights Reserved Slide 35