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Chapter 3 Descriptive Statistics: Numerical Measures Part A. Measures of Location Central Tendency Measures Percentiles and Quartiles. Mean. 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 m. Sample Mean. - PowerPoint PPT Presentation
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Chapter 3Chapter 3 Descriptive Statistics: Numerical Descriptive Statistics: Numerical
MeasuresMeasuresPart APart A
Measures of LocationMeasures of Location
•Central Tendency MeasuresCentral Tendency Measures
•Percentiles and QuartilesPercentiles and Quartiles
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MeanMean
The The meanmean of a data set is the of a data set is the averageaverage of of all the data values.all the data values.
The sample mean is the point The sample mean is the point estimator of the population mean estimator of the population mean . .
xx
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Sample Mean Sample Mean xx
Number ofNumber ofobservationsobservationsin the samplein the sample
Number ofNumber ofobservationsobservationsin the samplein the sample
Sum of the valuesSum of the valuesof the of the nn observations observations
Sum of the valuesSum of the valuesof the of the nn observations observations
ixx
n ix
xn
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Population Mean Population Mean
Number ofNumber ofobservations inobservations inthe populationthe population
Number ofNumber ofobservations inobservations inthe populationthe population
Sum of the valuesSum of the valuesof the of the NN observations observations
Sum of the valuesSum of the valuesof the of the NN observations observations
ix
N
ix
N
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Seventy efficiency apartmentsSeventy efficiency apartments
were randomly sampled inwere randomly sampled in
a small college town. Thea small college town. The
monthly rent prices formonthly rent prices for
these apartments are listedthese apartments are listedin ascending order on the next slide. in ascending order on the next slide.
Sample MeanSample Mean
Example: Apartment RentsExample: Apartment Rents
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Sample MeanSample Mean
34,356 490.80
70ix
xn
34,356 490.80
70ix
xn
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 575 580 590 600 600 600 600 615 615
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 575 580 590 600 600 600 600 615 615
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Considerations in Using the MeanConsiderations in Using the Mean
Requires interval/ratio levelRequires interval/ratio level Influenced by extreme scoresInfluenced by extreme scores Balancing point of the distribution (+ Balancing point of the distribution (+
and -deviations cancel out)and -deviations cancel out) Minimizes the “sum of squares” (sum of Minimizes the “sum of squares” (sum of
squared deviations around mean is squared deviations around mean is smaller than around any other number)smaller than around any other number)
•Mean is our “best guess” or estimate Mean is our “best guess” or estimate – minimizes errors in prediction– minimizes errors in prediction
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MedianMedian
1212 1414 1919 2626 27271818 2727
The The medianmedian is the is the middle score or midpoint middle score or midpoint of ofa set of scores when scores are arranged in order.a set of scores when scores are arranged in order.
For an For an odd numberodd number of observations: of observations:
in ascending orderin ascending order
2626 1818 2727 1212 1414 2727 1919 7 observations7 observations
Median = 19Median = 19
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1212 1414 1919 2626 27271818 2727
MedianMedian
For an For an even numbereven number of observations: of observations:
in ascending orderin ascending order
2626 1818 2727 1212 1414 2727 3030 8 observations8 observations
the median is the average of the middle two values.the median is the average of the middle two values.
Median = (19 + 26)/2 = 22.5Median = (19 + 26)/2 = 22.5
1919
3030
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MedianMedian
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 575 580 590 600 600 600 600 615 615
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 575 580 590 600 600 600 600 615 615
Averaging the 35th and 36th data values:Averaging the 35th and 36th data values:Median = (475 + 475)/2 = 475Median = (475 + 475)/2 = 475
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Considerations in using MedianConsiderations in using Median
Not sensitive to extreme scores – good for Not sensitive to extreme scores – good for skewed distributionsskewed distributions
Examples: annual income and property valuesExamples: annual income and property values
Requires ordinal level or higherRequires ordinal level or higher
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ModeMode
The The modemode of a data set is the value that occurs of a data set is the value that occurs with greatest frequency.with greatest frequency. The greatest frequency can occur at two or moreThe greatest frequency can occur at two or more different values.different values. If the data have exactly two modes, the data areIf the data have exactly two modes, the data are bimodalbimodal.. If the data have more than two modes, they areIf the data have more than two modes, they are multimodalmultimodal..
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ModeMode
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 575 580 590 600 600 600 600 615 615
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 575 580 590 600 600 600 600 615 615
450 occurred most frequently (7 times)450 occurred most frequently (7 times)
Mode = 450Mode = 450
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Considerations in Using ModeConsiderations in Using Mode
May be used at any level of May be used at any level of measurementmeasurement
May not imply “majority “ or “most”May not imply “majority “ or “most”
““Most common” score or value may not Most common” score or value may not be representative of most casesbe representative of most cases
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PercentilesPercentiles
Admission test scores for colleges and universitiesAdmission test scores for colleges and universities are frequently reported in terms of percentiles.are frequently reported in terms of percentiles.
The The ppth percentileth percentile of a data set is a value such of a data set is a value such that at least that at least pp percent of the items take on this percent of the items take on this value or less and at least (100 - value or less and at least (100 - pp) percent of ) percent of the items take on this value or more.the items take on this value or more.
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PercentilesPercentiles
Arrange the data in ascending order.Arrange the data in ascending order.
Compute index Compute index ii, the, the position position of the of the ppth percentile.th percentile.
ii = ( = (pp/100)/100)nn
If If ii is is not an integernot an integer, round , round upup. The . The pp th percentileth percentile is the is the value in the value in the ii th positionth position..
If If ii is an is an integerinteger, the , the pp th percentile is the th percentile is the averageaverage of the values in positionsof the values in positions i i and and ii +1.+1.
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9090thth Percentile Percentile
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 575 580 590 600 600 600 600 615 615
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 575 580 590 600 600 600 600 615 615
ii = ( = (pp/100)/100)nn = (90/100)70 = 63 = (90/100)70 = 63Averaging the 63rd and 64th data values:Averaging the 63rd and 64th data values:
90th Percentile = (580 + 590)/2 = 58590th Percentile = (580 + 590)/2 = 585
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9090thth Percentile Percentile
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 575 580 590 600 600 600 600 615 615
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 575 580 590 600 600 600 600 615 615
““At least 90%At least 90% of the itemsof the items
take on a valuetake on a value of 585 or less.”of 585 or less.”
““At least 10%At least 10% of the itemsof the items
take on a valuetake on a value of 585 or more.”of 585 or more.”
63/70 = .9 or 90%63/70 = .9 or 90% 7/70 = .1 or 10%7/70 = .1 or 10%
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QuartilesQuartiles
Quartiles are specific percentiles.Quartiles are specific percentiles.
First Quartile = 25th PercentileFirst Quartile = 25th Percentile
Second Quartile = 50th Percentile = MedianSecond Quartile = 50th Percentile = Median
Third Quartile = 75th PercentileThird Quartile = 75th Percentile
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Third QuartileThird Quartile
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 575 580 590 600 600 600 600 615 615
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 575 580 590 600 600 600 600 615 615
Third quartile = 75th percentileThird quartile = 75th percentile
i i = (= (pp/100)/100)nn = (75/100)70 = 52.5 = 53 = (75/100)70 = 52.5 = 53Third quartile = 525Third quartile = 525
