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Business Statistics (BUSA 3101)Business Statistics (BUSA 3101)

Dr.Dr. LariLari H.H. [email protected]@clayton.edu

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Chapter 4 (Part A)Chapter 4 (Part A)

Descriptive Statistics: Numerical MeasuresDescriptive Statistics: Numerical Measures

Measures of LocationMeasures of Location

Measures of VariabilityMeasures of VariabilityNumerical Data

Properties

Mean

Median

Mode

Midrange

Midhinge

CentralTendency

Range

InterquartileRange

Variance

Standard Deviation

Coeff. of Variation

Variation

Skew

Kurtosis

Shape

Numerical DataProperties

Mean

Median

Mode

Midrange

Midhinge

CentralTendency

Range

InterquartileRange

Variance

Standard Deviation

Coeff. of Variation

Variation

Skew

Kurtosis

Shape

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33SlideSlide

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 calledthey are called sample statisticssample statistics..

If the measures are computedIf the measures are computed

for data from a population,for data from a population,they are calledthey are called population parameterspopulation parameters..

A sample statistic is referred toA sample statistic is referred toas theas the point estimatorpoint estimator

of theof the

corresponding population parameter.corresponding population parameter.For example,For example,

thethe

sample mean is asample mean is a

point estimator of the population mean.point estimator of the population mean.

MeanMean

MedianMedian

ModeMode

PercentilesPercentiles

QuartilesQuartiles

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MeanMean

TheThe

meanmean

of a data set is the average of all the dataof a data set is the average of all the data

values.values.

As we said, the sample mean is the point estimatorAs we said, the sample mean is the point estimator

of the population meanof the population mean ..xx

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Sample MeanSample Mean xx

Number of

observationsin the sample

Number ofNumber of

observationsobservationsin the samplein the sample

Sum of the valuesof the n

observations

Sum of the valuesSum of the valuesof theof the nn

observationsobservations

ixx

n

ix

x

n

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Population MeanPopulation Mean

Number of

observations inthe population

Number ofNumber of

observations inobservations inthe populationthe population

Sum of the valuesof the N

observations

Sum of the valuesSum of the valuesof theof the NN

observationsobservations

ix

N

ix

N

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Seventy efficiency apartmentsSeventy efficiency apartments

were randomly sampled inwere randomly sampled ina 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:Example:

Apartment RentsApartment 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 Example ContinuedExample Continued

Monthly Rent for 70 ApartmentsMonthly Rent for 70 Apartments

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34,356490.80

70ixx

n 34,356 490.80

70ixx

n

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 Example ContinuedExample Continued


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11--

Every set of intervalEvery set of interval--level and ratiolevel and ratio--level data has alevel data has amean.mean.

22-- All the values are included in computing the mean.All the values are included in computing the mean.33--

A set of data has a unique mean.A set of data has a unique mean.

44--

The mean is affected by unusually large or small dataThe mean is affected by unusually large or small data

values.values.55--

The arithmetic mean is the only measure of centralThe arithmetic mean is the only measure of centraltendency where thetendency where the sum of the deviations of each valuesum of the deviations of each value

from the mean is zero.from the mean is zero.

Properties of the Arithmetic MeanProperties of the Arithmetic MeanProperties of the Arithmetic Mean

( )X X 0See next

Slide forAn example

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Illustration of Item

Number 5 on Previous Slide Illustration of ItemIllustration of Item

NumberNumber

55 on Previous Slideon Previous Slide

Consider the set of values: 3, 8, and 4. TheConsider the set of values: 3, 8, and 4. The meanmean

is 5.is 5.

So (3So (3 --5) + (85) + (8 --

5) + (45) + (4 --

5) =5) = --2 + 32 + 3 --

1 = 0.1 = 0.

Symbolically we write:Symbolically we write:

( )X X 0

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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 often

reported for annual income and property value data.reported for annual income and property value data.

TheThe

medianmedian

of a data set is the value in the middleof a data set is the value in the middle

when the data items are arranged in ascending orderwhen the data items are arranged in ascending order..

Positioning Point n 1

2

Positioning Point n 1

2

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MedianMedian

1212 1414 1919 2626 27271818 2727

For anFor 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 anFor 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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Median:Median: ExampleExample

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

TheThe modemode

of a data set is the value that occurs withof a data set is the value that occurs with

greatest frequencygreatest 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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Mode:Mode: ExampleExample

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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Mode: Another ExampleMode:Mode: Another ExampleAnother Example

No ModeNo Mode

Raw Data:Raw Data: 10.310.3

4.94.9

8.98.9

11.711.7

6.36.3

7.77.7

One ModeOne Mode Raw Data:Raw Data: 6.06.0

4.94.9

6.0 8.96.0 8.9

6.36.3 4.94.9

4.94.9

More Than 1 ModeMore Than 1 Mode

Raw Data:Raw Data: 2121 2828 2828 4141 4343 4343

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Use Excel to ComputeUse Excel to Compute the Mean, Median, and Modethe Mean, Median, and Mode

of the Following Data and Explain the Answers:of the Following Data and Explain the Answers:

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

STUDENTS

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PercentilesPercentiles

A percentileA percentile

provides information about how theprovides information about how the

data are spread over the intervaldata are spread over the interval from the smallestfrom 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.

You are familiar withYou are familiar with percentilepercentile

score of nationalscore of national

educational tests such as ACT, and SAT, whicheducational tests such as ACT, and SAT, which tell youtell youwhere you stand in comparison with others.where you stand in comparison with others.

For example, if you are in the 83th percentile, then 83%For example, if you are in the 83th percentile, then 83%

of the testof the test--takers scored below you and you are in the toptakers scored below you and you are in the top17% of the test takers.17% of the test takers.

P tilP til

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2121SlideSlide

TheTheppthth

percentilepercentile

of a data set is a value such that atof a data set is a value such that at

leastleastpp

percent of the items take on thispercent of the items take on this value or lessvalue or less

and at leastand at least (100(100 --

pp) percent) percent

of the items take on thisof the items take on this

value or morevalue or more..

PercentilesPercentiles DefinitionDefinition

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2222SlideSlide

Steps for Finding PercentilesSteps for Finding Percentiles

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

Compute indexCompute index

ii, the, the

positionposition

of theof theppthth

percentile.percentile.

ii

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

IfIf

ii

is notis not

an integer,an integer, round upround up. The. Thepp thth


is the value in theis the value in the ii

thth

position.position.

IfIf ii is an integer, theis an integer, thepp thth percentile is the averagepercentile is the averageof the values in positionsof the values in positions ii

andand ii

+1.+1.

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2323SlideSlide

8080thth

Percentile:Percentile: ExampleExample

ii

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

= (80/100)70 = 56= (80/100)70 = 56

Averaging the 56Averaging the 56thth

and 57and 57thth

data values:data values:

80th Percentile = (535 + 549)/2 = 54280th Percentile = (535 + 549)/2 = 542

Note: Data is in ascending order.Note: Data is in ascending order.

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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8080thth

Percentile:Percentile: Example ContinuedExample Continued

At least 80%At least 80%of the itemsof the items

take on a valuetake on a value

of 542 or less.of 542 or less.

At least 20%At least 20%of the itemsof the items

take on a valuetake on a value

of 542 or more.of 542 or more.56/70 = .8 or 80%56/70 = .8 or 80% 14/70 = .2 or 20%14/70 = .2 or 20%

425 430 430 435 435 435 435 435 440 440440 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 510510 515 525 525 525 535 549 550 570 570

575 575 580 590 600 600 600 600 615 615

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A B C D E

1Apart-ment

MonthlyRent ($) 80th Percentile

2 1 525 =PERCENTILE(B2:B71,.8)

3 2 440

4 3 450 5 4 615

6 5 480

Use Excel to Find 80Use Excel to Find 80thth

PercentilePercentile

Excel Formula WorksheetExcel Formula Worksheet

Note: Rows 7Note: Rows 7--71 are not shown.71 are not shown.

It is not necessaryIt is not necessary

to put the datato put the data

in ascending order.in ascending order.

8080thth


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8080thth

PercentilePercentile

Excel Value WorksheetExcel Value Worksheet

A B C D E

1Apart-ment

MonthlyRent ($) 80th Percentile

2 1 525 537.8

3 2 440

4 3 450 5 4 615

6 5 480


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EXAMPLEXAMPL

Given the following data, use Excel toGiven the following data, use Excel to

find the 25find the 25thth percentile:percentile:

357 550357 550654 290654 290

763 700763 700621 789621 789

900 605900 605

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QuartilesQuartiles

Quartiles are specific percentiles.Quartiles are specific percentiles.

First Quartile =First Quartile = 25th Percentile25th Percentile

Second Quartile =Second Quartile = 50th Percentile50th Percentile == MedianMedian Third Quartile =Third Quartile = 75th Percentile75th Percentile

Unless the sample size is large, percentiles may not makeUnless the sample size is large, percentiles may not makesense, since percentiles divide the data into 100 groups.sense, since percentiles divide the data into 100 groups.

In smaller samples, we might divide the data into fourIn smaller samples, we might divide the data into fourgroupsgroups ((quartilesquartiles).).

Since almost any sample can beSince almost any sample can be

divided into four groups, the quartiles are importantdivided into four groups, the quartiles are importantdescriptive statistics to explain.descriptive statistics to explain.

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A B C D E

1Apart-ment

MonthlyRent ($) Third Quarti le

2 1 525 =QUARTILE(B2:B71,3)

3 2 440

4 3 450 5 4 615

6 5 480

Excel Formula WorksheetExcel Formula Worksheet


It is not necessaryIt is not necessaryto put the datato put the data

in ascending order.in ascending order.

Third QuartileThird Quartile

33rdrd

quartilequartile

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Excel Value WorksheetExcel Value Worksheet

Third QuartileThird Quartile

A B C D E

1Apart-ment

MonthlyRent ($) Third Quarti le

2 1 525 522.5

3 2 440

4 3 450 5 4 615

6 5 480


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find the second quartile:find the second quartile:

357 550357 550654 290654 290

763 700763 700621 789621 789

900 605900 605

EXAMPLEXAMPL

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Measures of VariabilityMeasures of Variability (Dispersion)(Dispersion)

It is often desirable to consider measures ofIt is often desirable to consider measures of variabilityvariability(dispersion),(dispersion),

as well as measures of location.as well as measures of location.

For example, in choosing supplierFor example, in choosing supplier AA or supplieror supplier BB wewemight consider not only themight consider not only the average delivery timeaverage delivery time forforeach, but also theeach, but also the variability in delivery timevariability in delivery time for each.for each.

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3333SlideSlide

Measures of VariabilityMeasures of Variability (Dispersion)(Dispersion)

RangeRange

Interquartile Range or MidspreadInterquartile Range or Midspread

VarianceVariance

Standard DeviationStandard Deviation

Coefficient of VariationCoefficient of Variation

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RangeRange

TheThe rangerange

of a data set is the difference between theof a data set is the difference between the

largest and smallest data values.largest and smallest data values.

It is theIt is the simplest measuresimplest measure

of variability.of variability.

It isIt is very sensitivevery sensitive

to the smallest and largest datato the smallest and largest data

values.values.

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Range:Range: ExampleExample

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 valueRange = largest value --

smallest valuesmallest value

Range = 615Range = 615 --

425 = 190425 = 190


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Interquartile RangeInterquartile Range oror MidspreadMidspread TheThe interquartile rangeinterquartile range

of a data set is the differenceof a data set is the difference

between thebetween the third quartilethird quartile and theand thefirst quartilefirst quartile..

It is the range for theIt is the range for the middle 50%middle 50%

of the data.of the data.

It overcomes the sensitivity to extreme data valuesIt overcomes the sensitivity to extreme data valuesit isit isnot effected by the extreme values.not effected by the extreme values.

Interquartile Range Q Q3 1Interquartile Range Q Q3 1

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Interquartile Range:Interquartile Range: ExampleExample

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) = 5251st Quartile (1st Quartile (QQ1) = 4451) = 445

Interquartile Range =Interquartile Range = QQ33 --

QQ1 = 5251 = 525 --

445 = 80445 = 80


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find the Interquartile Range :find the Interquartile Range :

357 550357 550654 290654 290

763 700763 700621 789621 789

900 605900 605

EXAMPLEXAMPL

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TheThe variancevariance

is a measure of variability that utilizesis a measure of variability that utilizes

all 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,

for a population).for a population).xx

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VarianceVariance

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

The variance is theThe 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 a

samplesample

for afor a

populationpopulation

2

2

( )x

N

i

22

( )x

N

i

s

xi

x

n

22

1

( )

s

xi

x

n

22

1

( )

S d d D

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4141SlideSlide


TheThe standard deviationstandard deviation

of a data set is the positiveof a data set is the positive

square root of the variance.square root of the variance.

It is measured in theIt is measured in the same units as the datasame units as the data,,

makingmaking

it more easily interpreted than the variance.it more easily interpreted than the variance.

d dS d d D i i

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

for afor a

samplesample

for afor a

populationpopulation


2

2

2

2

C ffi i t f V i tiCoefficient of Variation

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4343SlideSlide

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


100 %s

x

100 %

s

x

TheThe

coefficient of variationcoefficient of variation

indicates how large theindicates how large the

standard deviation is in relation to the mean.standard deviation is in relation to the mean.

for afor asamplesample

for afor apopulationpopulation

100 %

100 %

C ffi i t f V i tiCoefficient of Variation (C ti d)(Continued)

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Measure ofMeasure of relativerelative

dispersiondispersion

Always a %Always a %

CV is the standard deviation expressed as percent ofCV is the standard deviation expressed as percent ofthe meanthe mean

Used to compare two or more groupsUsed to compare two or more groups

Weakness: CV is undefined if the mean is zero or ifWeakness: CV is undefined if the mean is zero or ifdata are negative.data are negative.

Thus, CV is used only for variables whose values areThus, CV is used only for variables whose values are

X>=0X>=0

Coefficient of VariationCoefficient of Variation (Continued)(Continued)

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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

Example ContinuedExample Continued


Given the following monthly rent prices for 70 apartments, findGiven the following monthly rent prices for 70 apartments, find

variance, standard deviation, and the coefficient of variation.:variance, standard deviation, and the coefficient of variation.: useuse

equations & Excelequations & Excel

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4646SlideSlide

54.74100 % 100 % 11.15%

490.80

s

x

54.74100 % 100 % 11.15%

490.80

s

x

22 ( ) 2,996.16

1ix xs

n

22 ( ) 2,996.16

1ix xs

n

2

2996.47 54.74s s 2

2996.47 54.74s s

the standardthe standarddeviation isdeviation is

about 11% ofabout 11% ofof the meanof the mean

VarianceVariance



SolutionsSolutions

Note thatNote that CV is the standard deviation expressed as percent ofCV is the standard deviation expressed as percent ofthe mean.the mean.

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4848SlideSlide


find the followings:find the followings:

357 550357 550654 290654 290

763 700763 700621 789621 789

900 605900 605

EXAMPLEXAMPL

EXAMPLEEXAMPLE

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4949SlideSlide

EXAMPLEEXAMPLE

Given theGiven thefollowing data:following data:

357 550357 550

654 290654 290

763 700763 700

621 789621 789

900 605900 605

Use Excel to find:

A.The mean

B. The modeC.The medianD.The 75th percentile

E.The first and the thirdquartileF.The rangeG.The interquartile range ormidspreadH. The standard deviationI.The coefficient of variation

Use Excel to find:Use Excel to find:

A.The mean

B. The modeC.The medianD.The 75th percentile

E.The first and the thirdquartileF.The rangeG.The interquartile range ormidspreadH. The standard deviationI.The coefficient of variationIf you need help with

this, see next slides.

If you need help withIf you need help with

this, see next slides.this, see next slides.

A P bl U i E lA P bl U i E l

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A Problem Using ExcelA Problem Using Excel

A private researchA private researchorganization studyingorganization studying

families in variousfamilies in various

countries reportedcountries reportedthe following data forthe following data for

the amount of time 4the amount of time 4--

year old childrenyear old childrenspent alone with theirspent alone with their

fathers each day.fathers each day.

Country Time with Dad(minutes)

Belgium 30

Canada 44

China 54

Finland 50

Germany 36

Nigeria 42Sweden 46

U.S.A. 42

A Problem Using ExcelA Problem Using Excel

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Use Excel, answer the following questions and explainyour answers (round all numbers into two decimalplaces):

A. The mean B. The mode

C. The median

D. The 75th percentile E. The first and the third quartile

F. The range

G. The interquartile range or midspread

H. The standard deviation I. The coefficient of variation

gg(Continued)(Continued)

Note:Note: All results are rounded to two decimal places.All results are rounded to two decimal places.

Using SWStatUsing SWStat++

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gg((Creating Data AreaCreating Data Area))

Data AreaData Area


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gg(Choose Statistics; Ungrouped Data; Choose Measures)(Choose Statistics; Ungrouped Data; Choose Measures)

Using SWStat+

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g(Numerical Data, Summary Measures (Sample); Calculate)

Using SWStatUsing SWStat++ (Results)(Results)

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Using SWStatUsing SWStat+ (Results)(Results)


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gg(Numerical Data; Percentile; Calculate)(Numerical Data; Percentile; Calculate)

Using SWStatUsing SWStat++ (Results)(Results)

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Using SWStatUsing SWStat (Results)(Results)

End of Chapter 4End of Chapter 4 Part APart A

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End of Chapter 4,End of Chapter 4, Part APart A

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chap4-a