Lecture2-1-Descriptive Statistcs-Numerical Measures PArt I

Lecture2-1-Descriptive Statistcs-Numerical Measures PArt I...

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Unformatted text preview: EM 521 Applied Statistics Descriptive Statistics: Numerical Measures - I AS&W – Chapter 3 1 Descriptive Statistics: Numerical Measures Several numerical measures that provide additional alternatives for summarizing data: • Measures of Location • Measures of Variability 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. 3 Mean Provides a measure of central location for the data 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 . 4 Sample Mean x For variable x x x Sum of the values of the n observations i n Number of observations in the sample Sample mean is a point estimator of the population mean. 5 Population Mean For variable x x Sum of the values of the N observations i N Number of observations in the population 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. 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 8 440 450 465 480 510 570 615 Sample Mean x 34,356 x= = = i 70 n 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 490.80 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 9 440 450 465 480 510 570 615 Median The median of a data set is the value in the middle when the data items are arranged in ascending ord Whenever a data set has extreme values, the med 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. 10 Median For an odd number of observations: 26 18 27 12 14 27 19 7 observations 12 14 18 19 26 27 27 in ascending order the median is the middle value. Median = 19 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 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 13 440 450 465 480 510 570 615 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. Important measure of location for qualitative data. 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 15 440 450 465 480 510 570 615 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. 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. 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. 18 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 19 440 450 465 480 510 570 615 90th Percentile “At least 90% of the items take on a value of 585 or less.” “At least 10% of the items take on a value of 585 or more.” 63/70 = .9 or 90% 7/70 = .1 or 10% 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 20 440 450 465 480 510 570 615 Quartiles Quartiles are specific percentiles. First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile 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 22 440 450 465 480 510 570 615 Measures of Variability It is often desirable to consider measures of variabil (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 f each, but also the variability in delivery time for eac 23 Measures of Variability Range Interquartile Range Variance Standard Deviation Coefficient of Variation 24 Range The range of a data set is the difference between th largest and smallest data values. It is the simplest measure of variability. It is very sensitive to the smallest and largest data values. 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 26 440 450 465 480 510 570 615 Interquartile Range The interquartile range of a data set is the differenc 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 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 28 440 450 465 480 510 570 615 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 ( for a sample x for a population). 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 2 ( x x ) i s2 n 1 for a sample 2 ( x ) i 2 N for a population 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. 31 Standard Deviation The standard deviation is computed as follows: s s2 2 for a sample for a population 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 100 % for a sample for a population 33 Variance, Standard Deviation, And Coefficient of Variation Variance 2 ( x x ) i s2 2,996.16 n 1 Standard Deviation the standard s s 2996.47 54.74 deviation is about 11% Coefficient of Variation of of the s 54.74 100 % 100 % 11.15% mean x 490.80 2 2 34 ...
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