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Chapter 3Chapter 3Descriptive Statistics: Numerical MeasuresDescriptive Statistics: Numerical Measures

Part APart A

■■ Measures of LocationMeasures of Location

■■ Measures of VariabilityMeasures of Variability

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Measures of LocationMeasures of Location

If the measures are computedIf the measures are computedfor data from a sample,for data from a sample,

they are called they are called sample statisticssample statistics..

If the measures are computedIf the measures are computedfor data from a population,for data from a population,

they are called they are called population parameterspopulation parameters..

A sample statistic is referred toA sample statistic is referred toas the as the point estimatorpoint estimator of theof the

corresponding population parameter.corresponding population parameter.

■■ MeanMean

■■ MedianMedian

■■ ModeMode

■■ PercentilesPercentiles

■■ QuartilesQuartiles

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MeanMean

■■ The The meanmean of a data set is the average of all the data of a data set is the average of all the data values.values.

■■ The sample mean is the point estimator of the The sample mean is the point estimator of the population mean population mean µµ. .

x

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Sample Mean Sample Mean x

Number ofNumber ofobservationsobservationsin the samplein the sample

Sum of the valuesSum of the valuesof the of the nn observationsobservations

ix

xn

=∑

3

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Population Mean Population Mean µµ

Number ofNumber ofobservations inobservations inthe populationthe population

Sum of the valuesSum of the valuesof the of the NN observationsobservations

ix

Nµ =∑

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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 listed

in 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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425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

Sample MeanSample Mean

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Sample MeanSample Mean

34,356 490.80

70ix

xn

= = =∑

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

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MedianMedian

�� Whenever a data set has extreme values, the medianWhenever a data set has extreme values, the medianis the preferred measure of central location.is the preferred measure of central location.

�� A few extremely large incomes or property valuesA few extremely large incomes or property valuescan inflate the mean.can inflate the mean.

�� The median is the measure of location most oftenThe median is the measure of location most oftenreported for annual income and property value data.reported for annual income and property value data.

�� The The medianmedian of a data set is the value in the middleof a data set is the value in the middlewhen the data items are arranged in ascending order.when the data items are arranged in ascending order.

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MedianMedian

1212 1414 1919 2626 27271818 2727

�� 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

the median is the middle value.the median is the middle value.

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

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

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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ModeMode

�� The The modemode of a data set is the value that occurs withof a data set is the value that occurs withgreatest frequency.greatest frequency.

�� The greatest frequency can occur at two or moreThe greatest frequency can occur at two or moredifferent values.different values.

�� If the data have exactly two modes, the data areIf the data have exactly two modes, the data arebimodalbimodal..

�� If the data have more than two modes, the data areIf the data have more than two modes, the data aremultimodalmultimodal..

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ModeMode

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

450 occurred most frequently (7 times)450 occurred most frequently (7 times)

Mode = 450Mode = 450

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PercentilesPercentiles

�� A percentile provides information about how theA percentile provides information about how thedata are spread over the interval from the smallestdata are spread over the interval from the smallestvalue to the largest value.value to the largest value.

�� Admission test scores for colleges and universitiesAdmission test scores for colleges and universitiesare frequently reported in terms of percentiles.are frequently reported in terms of percentiles.

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■■ The The ppth percentileth percentile of a data set is a value such that at of a data set is a value such that at least least pp percent of the items take on this value or less percent of the items take on this value or less and at least (100 and at least (100 -- pp) percent of the items take on this ) percent of the items take on this value or more.value or more.

PercentilesPercentiles

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PercentilesPercentiles

Arrange the data in ascending order.Arrange the data in ascending order.

Compute index Compute index ii, the position of the , the position of the ppth percentile.th percentile.

ii = (= (pp/100)/100)nn

If If ii is not an integer, round up. The is not an integer, round up. The ppth percentileth percentileis the value in the is the value in the iith position.th position.

If If ii is an integer, the is an integer, the ppth percentile is the averageth percentile is the averageof the values in positionsof the values in positions i i and and ii+1.+1.

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9090thth PercentilePercentile

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

ii = (= (pp/100)/100)nn = (90/100)70 = 63= (90/100)70 = 63

Averaging 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 PercentilePercentile

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

“At least 90%“At least 90%of the itemsof the items

take on a valuetake on a valueof 585 or less.”of 585 or less.”

“At least 10%“At least 10%of the itemsof the items

take on a valuetake on a valueof 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

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

Third quartile = 75th percentileThird quartile = 75th percentile

i i = (= (pp/100)/100)nn = (75/100)70 = 52.5 = 53= (75/100)70 = 52.5 = 53

Third quartile = 525Third quartile = 525

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Measures of VariabilityMeasures of Variability

�� It is often desirable to consider measures of variabilityIt is often desirable to consider measures of variability(dispersion), as well as measures of location.(dispersion), as well as measures of location.

�� For example, in choosing supplier A or supplier B weFor example, in choosing supplier A or supplier B wemight consider not only the average delivery time formight consider not only the average delivery time foreach, but also the variability in delivery time for each.each, but also the variability in delivery time for each.

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Measures of VariabilityMeasures of Variability

■■ RangeRange

■■ Interquartile RangeInterquartile Range

■■ VarianceVariance

■■ Standard DeviationStandard Deviation

■■ Coefficient of VariationCoefficient of Variation

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RangeRange

�� The The rangerange of a data set is the difference between theof a data set is the difference between thelargest and smallest data values.largest and smallest data values.

�� It is the It is the simplest measuresimplest measure of variability.of variability.

�� It is It is very sensitivevery sensitive to the smallest and largest datato the smallest and largest datavalues.values.

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RangeRange

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

Range = largest value Range = largest value -- smallest valuesmallest value

Range = 615 Range = 615 -- 425 = 190425 = 190

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Interquartile RangeInterquartile Range

�� The The interquartile rangeinterquartile range of a data set is the differenceof a data set is the differencebetween the third quartile and the first quartile.between the third quartile and the first quartile.

�� It is the range for the It is the range for the middle 50%middle 50% of the data.of the data.

�� It overcomes the sensitivity to extreme data values.It overcomes the sensitivity to extreme data values.

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Interquartile RangeInterquartile Range

425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 450

450 450 450 450 450 460 460 460 465 465

465 470 470 472 475 475 475 480 480 480

480 485 490 490 490 500 500 500 500 510

510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

3rd Quartile (3rd Quartile (QQ3) = 5253) = 525

1st Quartile (1st Quartile (QQ1) = 4451) = 445

Interquartile Range = Interquartile Range = QQ3 3 -- QQ1 = 525 1 = 525 -- 445 = 80445 = 80

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The The variancevariance is a measure of variability that utilizesis a measure of variability that utilizesall the data.all the data.

VarianceVariance

It is based on the difference between the value ofIt is based on the difference between the value ofeach observation (each observation (xxii) and the mean ( for a sample,) and the mean ( for a sample,µµ for a population).for a population).

x

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VarianceVariance

The variance is computed as follows:The variance is computed as follows:

The variance is the The variance is the average of the squaredaverage of the squareddifferencesdifferences between each data value and the mean.between each data value and the mean.

for afor asamplesample

for afor apopulationpopulation

σ µ22

= −∑( )x

Nis

xi x

n2

2

1=

−∑

−( )

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Standard DeviationStandard Deviation

The The standard deviationstandard deviation of a data set is the positiveof a data set is the positivesquare root of the variance.square root of the variance.

It is measured in the It is measured in the same units as the datasame units as the data, making, makingit more easily interpreted than the variance.it more easily interpreted than the variance.

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The standard deviation is computed as follows:The standard deviation is computed as follows:

for afor asamplesample

for afor apopulationpopulation

Standard DeviationStandard Deviation

s s= 2 σ σ= 2

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The coefficient of variation is computed as follows:The coefficient of variation is computed as follows:

Coefficient of VariationCoefficient of Variation

100 %s

x ×

The The coefficient of variationcoefficient of variation indicates how large theindicates how large thestandard deviation is in relation to the mean.standard deviation is in relation to the mean.

for afor asamplesample

for afor apopulationpopulation

100 %σµ

×

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× = × =

54.74100 % 100 % 11.15%

490.80

s

x

22 ( )

2,996.161

ix xs

n

−= =

−∑

2 2996.47 54.74s s= = =

the standardthe standarddeviation isdeviation is

about 11% ofabout 11% ofof the mean of the mean

■■ VarianceVariance

■■ Standard DeviationStandard Deviation

■■ Coefficient of VariationCoefficient of Variation

Variance, Standard Deviation,Variance, Standard Deviation,And Coefficient of VariationAnd Coefficient of Variation

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End of Chapter 3, Part AEnd of Chapter 3, Part A