stats chap03 bluman ppt

8/16/2019 Stats Chap03 bluman ppt

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Chapter 3

Data Description

1© McGraw-Hill, Bluman, 5

th

ed, Chapter3


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Chapter 3 Overview

Introduction

3- Measures o! Central "endenc#

3-$ Measures o! %ariation

3-3 Measures o! &osition

3-' ()plorator# Data *nal#sis

2Bluman, Chapter 3


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Chapter 3 O+ectives

.ummari/e data usin0 measures o!central tendenc#

$ Descri+e data usin0 measures o!

variation3 Identi!# the position o! a data value in a

data set

' 1se +o)plots and !ive-num+ersummaries to discover various aspectso! data

3Bluman, Chapter 3


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Introduction

Traditional Statistics

Average

Variation

Position

4Bluman, Chapter 3


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3 Measures o! Central "endenc#

* statisticstatistic is a characteristic or measureo+tained +# usin0 the data values !rom asample

* parameter parameter is a characteristic ormeasure o+tained +# usin0 all the data

values !or a speci!ic population

5Bluman, Chapter 3


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Measures o! Central "endenc#4

Mean "he meanmean is the uotient o! the sum o!the values and the total num+er o! values

"he s#m+ol is used !or sample mean

6or a population, the Gree7 letter μ 8mu9is used !or the mean

X 1 2 3 n

X X X X X X

n n

+ + + += =

∑L

1 2 3 N X X X X X

N N µ

+ + + += =

∑L

8Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-()ample 3-

&a0e :;<

9Bluman, Chapter 3


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()ample 3-4 Da#s O!! per =ear

"he data represent the num+er o! da#s o!! per#ear !or a sample o! individuals selected !romnine di!!erent countries 6ind the mean

$;, $


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Rounding Rule: Mean

"he mean should +e rounded to one moredecimal place than occurs in the raw data

"he mean, in most cases, is not an actualdata value

11Bluman, Chapter 3


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Mean !or Grouped Data "he mean !or 0rouped data is calculated

+# multipl#in0 the !reuencies and

midpoints o! the classes

m f X X

n

×=

∑

12Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-()ample 3-3

&a0e :;>

13Bluman, Chapter 3


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()ample 3-34 Miles ?un

Class Boundaries 6reuenc#

55 - ;5;5 - 5555 - $;5$;5 - $55

$55 - 3;53;5 - 355355 - ';5

$35

'3$

Below is a !reuenc# distri+ution o! milesrun per wee7 6ind the mean

Σ f @ $;

14Bluman, Chapter 3


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Median "he medianmedian is the midpoint o! the data

arra# "he s#m+ol !or the median is MD

"he median will +e one o! the data valuesi! there is an odd num+er o! values

"he median will +e the avera0e o! twodata values i! there is an even num+er o!values

16Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-()ample 3-'

&a0e :;

17Bluman, Chapter 3


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()ample 3-'4 Hotel ?ooms

"he num+er o! rooms in the seven hotels indowntown &itts+ur0h is >3, 3;;,


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Chapter 3

Data Description

.ection 3-()ample 3-<

&a0e :;

19Bluman, Chapter 3


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()ample 3-


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Mode "he modemode is the value that occurs most

o!ten in a data set

It is sometimes said to +e the most t#picalcase

"here ma# +e no mode, one mode8unimodal9, two modes 8+imodal9, orman# modes 8multimodal9

21Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-()ample 3-

&a0e :

22Bluman, Chapter 3


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()ample 3-4 6 .i0nin0 Bonuses

6ind the mode o! the si0nin0 +onuses o!ei0ht 6 pla#ers !or a speci!ic #ear "he+onuses in millions o! dollars are

A;, ';, 3'5, ;, 3, ;, $', ;

=ou ma# !ind it easier to sort !irst

;, ;, ;, 3, $', ';, A;, 3'5

.elect the value that occurs the most

23Bluman, Chapter 3

"he mode is ; million dollars


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()ample 3-;4 Coal (mplo#ees in &*

6ind the mode !or the num+er o! coal emplo#ees percount# !or ; selected counties in southwestern&enns#lvania

;, >3, ;3, A', $;, A, >, ;3, >5$

o value occurs more than once

25Bluman, Chapter 3

"here is no mode


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()ample 3-4 icensed uclear

?eactors"he data show the num+er o! licensed nuclearreactors in the 1nited .tates !or a recent 5-#earperiod 6ind the mode

;' ;' ;' ;' ;' ;> ; ; ; ;; $ ;

;' and ; +oth occur the most "he data setis said to +e +imodal

27Bluman, Chapter 3

"he modes are ;' and ;

;' ;' ;' ;' ;' ;> ; ; ; ;; $ ;


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()ample 3-$4 Miles ?un per 2ee76ind the modal class !or the !reuenc# distri+utiono! miles that $; runners ran in one wee7

29Bluman, Chapter 3

"he modal class is

$;5 E $55

Class Frequency

55 E ;5

;5 E 55 $

55 E $;5 3

$;5 E $55 5

$55 E 3;5 '

3;5 E 355 3

355 E ';5 $

"he mode, the midpointo! the modal class, is

$3 miles per wee7


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Midran0e "he midrangemidrange is the avera0e o! the

lowest and hi0hest values in a data set

2

Lowest Highest MR

+=

30Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-()ample 3-5

&a0e :'

31Bluman, Chapter 3


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()ample 3-54 2ater-ine Brea7s

In the last two winter seasons, the cit# o!Brownsville, Minnesota, reported thesenum+ers o! water-line +rea7s per month6ind the midran0e

$, 3,


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2ei0hted Mean 6ind the weighted meanweighted mean o! a varia+le +#

multipl#in0 each value +# its

correspondin0 wei0ht and dividin0 the sumo! the products +# the sum o! the wei0hts

1 1 2 2

1 2

n n

n

wX w X w X w X X w w w w

+ + += =+ + + ∑∑

LL

33Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-()ample 3->

&a0e :5

34Bluman, Chapter 3


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()ample 3->4 Grade &oint *vera0e * student received the !ollowin0 0rades 6ind the correspondin0 G&*

35Bluman, Chapter 3

"he 0rade point avera0e is $>

wX

w X = ∑∑

Course Credits, w Grade, X

(n0lish Composition 3 * 8' points9

Introduction to &s#cholo0# 3 C 8$ points9Biolo0# ' B 83 points9

&h#sical (ducation $ D 8 point9

322.712

3 4 3 2 4 3 2 1

3 3 4 2 =× + × + × + ×

= =+ + +


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&roperties o! the Mean

1ses all data values %aries less than the median or mode

1sed in computin0 other statistics, such as

the variance 1niue, usuall# not one o! the data values

Cannot +e used with open-ended classes

*!!ected +# e)tremel# hi0h or low values,called outliers

36Bluman, Chapter 3


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&roperties o! the Median

Gives the midpoint 1sed when it is necessar# to !ind out

whether the data values !all into the upper

hal! or lower hal! o! the distri+ution Can +e used !or an open-ended

distri+ution

*!!ected less than the mean +# e)tremel#hi0h or e)tremel# low values

37Bluman, Chapter 3


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&roperties o! the Mode

1sed when the most t#pical case isdesired

(asiest avera0e to compute

Can +e used with nominal data ot alwa#s uniue or ma# not e)ist

38Bluman, Chapter 3


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&roperties o! the Midran0e

(as# to compute Gives the midpoint

*!!ected +# e)tremel# hi0h or low values in

a data set

39Bluman, Chapter 3


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Distri+utions

40Bluman, Chapter 3


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3-$ Measures o! %ariation

How Can 2e Measure %aria+ilit#?an0e

%ariance.tandard Deviation

Coe!!icient o! %ariation

Che+#shevFs "heorem

(mpirical ?ule 8ormal9

41Bluman, Chapter 3


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Measures o! %ariation4 ?an0e

"he rangerange is the di!!erence +etween thehi0hest and lowest values in a data set

R Highest Lowest = −

42Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-$()ample 3-A

&a0e :$3$5

43Bluman, Chapter 3


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()ample 3-A4 Outdoor &aint

"wo e)perimental +rands o! outdoor paint aretested to see how lon0 each will last +e!ore!adin0 .i) cans o! each +rand constitute asmall population "he results 8in months9 are

shown 6ind the mean and ran0e o! each 0roup

44Bluman, Chapter 3

Brand A Brand B

; 35


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()ample 3-A4 Outdoor &aint

45Bluman, Chapter 3

Brand A Brand B

; 35


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Measures o! %ariation4 %ariance

.tandard Deviation "he variancevariance is the avera0e o! the

suares o! the distance each value is

!rom the mean "he standard deviationstandard deviation is the suare

root o! the variance

"he standard deviation is a measure o!how spread out #our data are

46Bluman, Chapter 3


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1ses o! the %ariance and .tandard

Deviation "o determine the spread o! the data

"o determine the consistenc# o! a

varia+le "o determine the num+er o! data values

that !all within a speci!ied interval in a

distri+ution 8Che+#shevFs "heorem9 1sed in in!erential statistics

47Bluman, Chapter 3


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Measures o! %ariation4

%ariance .tandard Deviation8&opulation "heoretical Model9 "he population variancepopulation variance is

"he population standard deviationpopulation standard deviation is

( )2

2 X N

µ σ −= ∑

( )2

X

N

µ σ

−=

∑

48Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-$()ample 3-$

&a0e :$5

49Bluman, Chapter 3


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()ample 3-$4 Outdoor &aint6ind the variance and standard deviation !or thedata set !or Brand * paint ;,


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%ariance .tandard Deviation8.ample "heoretical Model9 "he sample variancesample variance is

"he sample standard deviationsample standard deviation is

( )2

2

1

X X sn

−=−

∑

( ) 2

1

X X s

n

−=

−

∑

51Bluman, Chapter 3


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%ariance .tandard Deviation8.ample Computational Model9 Is mathematicall# euivalent to the

theoretical !ormula .aves time when calculatin0 +# hand

Does not use the mean Is more accurate when the mean has

+een rounded

52Bluman, Chapter 3


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%ariance .tandard Deviation8.ample Computational Model9 "he sample variancesample variance is

"he sample standard deviationsample standard deviation is

53Bluman, Chapter 3

( )

( )

2 2

2

1

−=

−

∑ ∑ X X n s

n n

2 s s=


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Chapter 3

Data Description

.ection 3-$()ample 3-$3

&a0e :$

54Bluman, Chapter 3


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5A'

()ample 3-$34 (uropean *uto .ales6ind the variance and standard deviation !or the

amount o! (uropean auto sales !or a sample o! <#ears "he data are in millions o! dollars

$, , $;, $A, 3', '3

55Bluman, Chapter 3

X X2

$$;$A3''3

$5'''


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Coe!!icient o! %ariation"he coefficient of variationcoefficient of variation is thestandard deviation divided +# the

mean, e)pressed as a percenta0e

1se CVAR to compare standarddeviations when the units are di!!erent

100% s

CVAR X

= ×

56Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-$()ample 3-$5

&a0e :3$

57Bluman, Chapter 3


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()ample 3-$54 .ales o! *utomo+iles

"he mean o! the num+er o! sales o! cars over a3-month period is A>, and the standard deviationis 5 "he mean o! the commissions is J5$$5,and the standard deviation is J>>3 Compare

the variations o! the two

58Bluman, Chapter 3

Commissions are more varia+le than sales

5100% 5.7% ales

87CVar = × =

773100% 14.8% !"mmissi"ns5225CVar = × =


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?an0e ?ule o! "hum+"he Range Rule of hum!Range Rule of hum! appro)imates the standard deviation

as

when the distri+ution is unimodal andappro)imatel# s#mmetric

4

Range s ≈

59Bluman, Chapter 3


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?an0e ?ule o! "hum+1se to appro)imate the lowestvalue and to appro)imate the

hi0hest value in a data set

60Bluman, Chapter 3

2 X s−2 X s+

#$amle: 10& 12 X Range= =

12 34

s ≈ = ( )( )

10 2 3 410 2 3 16

LOW HIGH

≈ − =≈ + =


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"he proportion o! values !rom an# data set that!all within k standard deviations o! the mean will+e at least -k $, where k is a num+er 0reaterthan 8k is not necessaril# an inte0er9

Measures o! %ariation4Che+#shevFs "heorem

61Bluman, Chapter 3

: o!standard

deviations, k

Minimum &roportionwithin k standard

deviations

Minimum &ercenta0ewithin k standard

deviations

$ -'@3' >5K

3 -@A AAAK

' -5K


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Measures o! %ariation4Che+#shevFs "heorem

62Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-$()ample 3-$>

&a0e :35

63Bluman, Chapter 3


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()ample 3-$>4 &rices o! Homes"he mean price o! houses in a certain

nei0h+orhood is J5;,;;;, and the standarddeviation is J;,;;; 6ind the price ran0e !orwhich at least >5K o! the houses will sell

Che+#shevFs "heorem states that at least >5K o!a data set will !all within $ standard deviations o!the mean

5;,;;; E $8;,;;;9 @ 3;,;;;

5;,;;; L $8;,;;;9 @ >;,;;;

64Bluman, Chapter 3

*t least >5K o! all homes sold in the area will have aprice ran0e !rom J3;,;;; and J>5,;;;


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Chapter 3

Data Description

.ection 3-$()ample 3-$A

&a0e :35

65Bluman, Chapter 3


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"he percenta0e o! values !rom a data set that!all within 7 standard deviations o! the mean ina normal 8+ell-shaped9 distri+ution is listed

+elow: o! standarddeviations, 7

&roportion within 7 standarddeviations

K

Measures o! %ariation4(mpirical ?ule 8ormal9

67Bluman, Chapter 3


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Measures o! %ariation4(mpirical ?ule 8ormal9

68Bluman, Chapter 3


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3-3 Measures o! &osition

-score&ercentile

Nuartile

Outlier

69Bluman, Chapter 3


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Measures o! &osition4 -score

* "#score"#score or standard scorestandard score !or a valueis o+tained +# su+tractin0 the mean !romthe value and dividin0 the result +# thestandard deviation

* /-score represents the num+er o!standard deviations a value is a+ove or+elow the mean

X X s−= X

µ

σ

−=

70Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-3()ample 3-$

&a0e :'$

71Bluman, Chapter 3


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()ample 3-$4 "est .cores * student scored


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Measures o! &osition4 &ercentiles

$ercentiles$ercentiles separate the data set into ;;eual 0roups

* percentile ran7 !or a datum representsthe percenta0e o! data values +elow the

datum( ), "- al(es /el" 0.5

100%*"*al , "- al(es

X !er"enti#e

+= ×

73Bluman, Chapter 3

100

n $" ×=


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Measures o! &osition4 ()ample o!

a &ercentile Graph

74Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-3()ample 3-3$

&a0e :'>

75Bluman, Chapter 3

( l 3 3$ " t .


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()ample 3-3$4 "est .cores * teacher 0ives a $;-point test to ; students

6ind the percentile ran7 o! a score o! $A, 5, $,


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Chapter 3

Data Description

.ection 3-3()ample 3-3'

&a0e :'A

77Bluman, Chapter 3


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M ! & iti


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Measures o! &osition4Nuartiles and Deciles %eciles%eciles separate the data set into ;

eual 0roups D110& D440

&uartiles&uartiles separate the data set into 'eual 0roups 125& 2MD& 375

2 median"&)i

1 median"&23 median2&)i

"he 'nterquartile Range'nterquartile Range, R 3 1

79Bluman, Chapter 3


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Chapter 3

Data Description

.ection 3-3()ample 3-3<

&a0e :5;

80Bluman, Chapter 3

( l 3 3< N til


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()ample 3-3


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Measures o! &osition4Outliers *n outlier outlier is an e)tremel# hi0h or low

data value when compared with the resto! the data values

* data value less than 1 1.5R or0reater than 3 ; 1.5R can +econsidered an outlier

82Bluman, Chapter 3


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3' ()plorator# Data *nal#sis

"he Five#(um!er )ummaryFive#(um!er )ummary iscomposed o! the !ollowin0 num+ers4"& 1& MD& 3& )i

"he 6ive-um+er .ummar# can +e0raphicall# represented usin0 a

Bo*plotBo*plot

83Bluman, Chapter 3


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&rocedure "a+le

Constructin0 Bo)plots

6ind the !ive-num+er summar#

$ Draw a hori/ontal a)is with a scale that includes

the ma)imum and minimum data values3 Draw a +o) with vertical sides throu0h '( and

', and draw a vertical line thou0h the median

' Draw a line !rom the minimum data value to thele!t side o! the +o) and a line !rom the ma)imumdata value to the ri0ht side o! the +o)

84Bluman, Chapter $


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Chapter 3

Data Description

.ection 3-'()ample 3-3A

&a0e :


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()ample 3-3A4 Meteorites"he num+er o! meteorites !ound in ; 1. states

is shown Construct a +o)plot !or the data A, '>, A, A, 3A, -A35- A35

stats chap03 bluman ppt

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