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Business Statistics (BUSA 3101)Business Statistics (BUSA 3101)
Dr.Dr. LariLari H.H. ArjomandArjomandlariarjomand@clayton.edulariarjomand@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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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
Monthly Rent for 70 ApartmentsMonthly Rent for 70 Apartments
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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
Monthly Rent for 70 ApartmentsMonthly Rent for 70 Apartments
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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
Monthly Rent for 70 ApartmentsMonthly Rent for 70 Apartments
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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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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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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
percentilepercentile
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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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 ($) 80th Percentile
2 1 525 =PERCENTILE(B2:B71,.8)
3 2 440
4 3 450 5 4 615
6 5 480
Use Excel to Find 80Use Excel to Find 80thth
PercentilePercentile
Excel Formula WorksheetExcel Formula Worksheet
Note: Rows 7Note: Rows 7--71 are not shown.71 are not shown.
It is not necessaryIt is not necessary
to put the datato put the data
in ascending order.in ascending order.
8080thth
percentilepercentile
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8080thth
PercentilePercentile
Excel Value WorksheetExcel Value Worksheet
A B C D E
1Apart-ment
MonthlyRent ($) 80th Percentile
2 1 525 537.8
3 2 440
4 3 450 5 4 615
6 5 480
Note: Rows 7Note: Rows 7--71 are not shown.71 are not shown.
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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 ($) Third Quarti le
2 1 525 =QUARTILE(B2:B71,3)
3 2 440
4 3 450 5 4 615
6 5 480
Excel Formula WorksheetExcel Formula Worksheet
Note: Rows 7Note: Rows 7--71 are not shown.71 are not shown.
It is not necessaryIt is not necessaryto put the datato put the data
in ascending order.in ascending order.
Third QuartileThird Quartile
33rdrd
quartilequartile
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Excel Value WorksheetExcel Value Worksheet
Third QuartileThird Quartile
A B C D E
1Apart-ment
MonthlyRent ($) Third Quarti le
2 1 525 522.5
3 2 440
4 3 450 5 4 615
6 5 480
Note: Rows 7Note: Rows 7--71 are not shown.71 are not shown.
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Given the following data, use Excel toGiven the following data, use Excel to
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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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
Monthly Rent for 70 ApartmentsMonthly Rent for 70 Apartments
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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
Monthly Rent for 70 ApartmentsMonthly Rent for 70 Apartments
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Given the following data, use Excel toGiven the following data, use Excel to
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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Standard DeviationStandard Deviation
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
Standard DeviationStandard Deviation
2
2
2
2
C ffi i t f V i tiCoefficient of Variation
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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
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
Monthly Rent for 70 ApartmentsMonthly Rent for 70 Apartments
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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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
Standard DeviationStandard Deviation
Coefficient of VariationCoefficient of Variation
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
Given the following data, use Excel toGiven the following data, use Excel to
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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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
Using SWStatUsing SWStat++
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gg(Choose Statistics; Ungrouped Data; Choose Measures)(Choose Statistics; Ungrouped Data; Choose Measures)
Using SWStat+
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5454SlideSlide
g(Numerical Data, Summary Measures (Sample); Calculate)
Using SWStatUsing SWStat++ (Results)(Results)
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5555SlideSlide
Using SWStatUsing SWStat+ (Results)(Results)
Using SWStatUsing SWStat++
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5656SlideSlide
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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