Chapter 3Important Characteristics of dataDescribing Central TendencySlide 4MeanExample 3.1Example 3.2 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.The MedianSlide 9Example 3.3Example 3.2 RevisitedModeSlide 13Percentiles and QuartilesPercentilesSlide 16Slide 17QuartilesExample 2.4 RevisitedMeasures of VariationExample 3.3 RevisitedSlide 22VarianceStandard DeviationExample 3.1 Revisited (recall =277.4)Example 3.1 Revisited (continued)Example 3.4Z scoreExample 3.1 RevisitedChebyshev’s RuleSlide 31Empirical Rule for Normal PopulationsOutliersExample (Outliers)The EndDr. Constance Lightner- Fayetteville State University1Chapter 3Descriptive Statistics Part IIDescribing Central TendencyMeasures of VariationDr. Constance Lightner- Fayetteville State University2Important Characteristics of dataCenter: A value that indicates where the middle of the data set is located.Variation: A measure of the amount that the data values vary among themselvesDistribution: The nature or shape of the distribution of data (such as bell-shaped, uniform, or skewed)Outliers: Sample values that lie very far away from the vast majority of the other samplesDr. Constance Lightner- Fayetteville State University3Describing Central TendencyMean, , is the average or expected value Median, Md, is the middle point of the ordered measurementsMode, Mo, is the most frequent valuePercentiles and QuartilesDr. Constance Lightner- Fayetteville State University4The sample size, i.e. the number of items in the sample, is denoted n.The population size, i.e. the total number of items in the entire population, is denoted N.Basic SymbolsDr. Constance Lightner- Fayetteville State University5MeanIf the data are from a population, the mean is denoted by  (mu).If the data are from a sample, the mean is denoted by . The sample mean is a point estimate of the population mean .xNixNixxxnixxniDr. Constance Lightner- Fayetteville State University6Example 3.1Suppose we compiled a sample of the weights of 5 professional football players255, 216, 346, 300, 270555432151xxxxxxxii  255 + 216 + 346 + 300 + 270513875277 4.Dr. Constance Lightner- Fayetteville State University7Example 3.2Given below is a sample of monthly rent values ($) for one-iven below is a sample of monthly rent values ($) for one-bedroom apartments. The data is a sample of 70 bedroom apartments. The data is a sample of 70 apartments in a particular city. The data are presented in apartments in a particular city. The data are presented in ascending order.ascending order. 425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615xxni 34 35670490 80,.xxni 34 35670490 80,.Anderson, Sweeney, and WilliamsDr. Constance Lightner- Fayetteville State University8The Median The median is a value such that at least 50% of all measurements are less than or equal to it and at least 50% of all measurements are greater than or equal to it .The median is the measure of location most often reported for annual income and property value data.This measure is used instead of the mean since a few extremely large incomes or property values can inflate the mean.Dr. Constance Lightner- Fayetteville State University9The median Md is found as follows: 1. Arrange values in ascending order (smallest to largest).2. If the number of measurements is odd, the median is the middle value.3. If the number of measurements is even, the median is the average of the two middle values.Dr. Constance Lightner- Fayetteville State University10Example 3.3Suppose the following represent a sample of salaries of 13 Internist(x$1000)127 132 138 141 144 146 152 154 165 171 177 192 241Since n = 13 (odd,) then the median is the middlemost or 7th measurement, Md=152Dr. Constance Lightner- Fayetteville State University11Example 3.2 Revisited425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615Median = (475 + 475)/2 = 475Dr. Constance Lightner- Fayetteville State University12ModeThe 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.Mode is an important measure of location for qualitative data (can not compute median and mean for qualitative data)The mode, Mo , is the measurement that occurs most frequently.Dr. Constance Lightner- Fayetteville State University13Mode450 occurred most frequently (7 times) Mode = 450425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615Dr. Constance Lightner- Fayetteville State University14Percentiles and QuartilesA 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.Dr. Constance