chapter 1 - power point slide staistic

8/18/2019 CHAPTER 1 - Power Point Slide staistic

1/156

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1

Chapter Introduction to

Statistics

1


2/156


Chapter Outline

• 1.1 An Overview of Statistics

• 1.2 Data Classification

• 1.3 Frequency Distributions and Their ra!hs

• 1." #ore ra!hs and Dis!lays

• 1.$ #easures of Central Tendency

• 1.% #easures of &ariation

• 1.' #easures of (osition


3/156


Objectives

The end of this cha!ter) students should be able *

•The definition of statistics

•Distin+uish between a !o!ulation and a sa,!le and

between a !ara,eter and a statistic

•Distin+uish between descri!tive statistics and

inferential statistics

•Distin+uish between qualitative data and quantitativedata


4/156


Objectives


•Constructed the frequency Distributions table)

Constructed frequency histo+ra,s) frequency !oly+ons)

relative frequency histo+ra,s and o+ives•ra!h the quantitative data and qualitative data.


5/156


What is Data?

Data

Consist of infor,ation co,in+ fro, observations) counts) ,easure,ents) or

res!onses.

• -(eo!le who eat three daily servin+s of whole +rains

have been shown to reduce their ris of/stroe by

3'0. (Source: Whole Grains Council)

• -Seventy !ercent of the 1$ .S. s!inal cordin4uries to ,inors result fro, vehicle accidents) and

%5 !ercent were not wearin+ a seatbelt. (Source: UPI)


6/156


What is Statistics?

Statistics

The science of collectin+)

or+ani6in+) analy6in+) and

inter!retin+ data in order to,ae decisions.


7/156Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 7

Data Sets

Population

The collection of all outco,es)

res!onses) ,easure,ents) or

counts that are of interest.

Sample

A subset) or !art) of the !o!ulation.



Ea!ple" #$enti%&in' Data Sets

7n a recent survey) 1$ adults in the nited States were

ased if they thou+ht there was solid evidence for +lobal

war,in+. 8i+ht hundred fifty9five of the adults said yes.

7dentify the !o!ulation and the sa,!le. Describe thedata set. (Adapted from: Pew Research Center)


9/156Copyright © 2015, 2012, and 2009 Pearson Education, Inc. (

S)luti)n" #$enti%&in' Data Sets

• The !o!ulation consists of theres!onses of all adults in the .S.

• The sa,!le consists of the

res!onses of the 1$ adults in the

.S. in the survey.

• The sa,!le is a subset of the

res!onses of all adults in the .S.

• The data set consists of 5$$ yes:sand %"$ no:s.

;es!onses of adults in

the .S.

;es!onses of

adults in survey


10/156Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1*

+,- #+ -O.,SE/0 1

a'e 23

.



ara!eter an$ Statistic

Parameter A nu,erical descri!tion of a !o!ulation

characteristic.

Averae ae of all people in the United States

Statistic

A nu,erical descri!tion of a sa,!le characteristic.

Averae ae of people from a sample

of three states


12/156


Ea!ple" Distin'uish ara!eter an$ Statistic

Decide whether the nu,erical value describes a

!o!ulation !ara,eter or a sa,!le statistic.

1. A recent survey of a sa,!le of colle+e

career centers re!orted that the avera+e

startin+ salary for !etroleu,en+ineerin+ ,a4ors is >53)121. (Source:

!ational Association of Collees and

"mplo#ers)

Solution:

Sa,!le statistic 53)121 is based

on a subset of the !o!ulation=


13/156





2. The 2152 students who acce!ted

ad,ission offers to ?orthwestern

niversity in 2@ have an avera+e SATscore of 1""2. (Source: !orthwesternUniversit#)

Solution:

(o!ulation !ara,eter


14/156





3. 7n a rando, chec of " retail stores) the

Food and Dru+ Ad,inistration found that

3"0 of the stores were not storin+ fish at the !ro!er te,!erature.

Solution:

Sa,!le statistic because the !ercent 3"0 is basedon a subset of the !o!ulation.


15/156


+,- #+ -O.,SE/0 2

a'e 24

.


16/156


ranches )% Statistics

DescriptiveStatistics 7nvolvesor+ani6in+)su,,ari6in+) and

dis!layin+ data.

e.+. Tables) charts)avera+es

Inferential Statistics 7nvolves usin+ sampledata to drawconclusions about a

population.


17/156


Ea!ple" Descriptive an$ #n%erential

Statistics

Decide which !art of the study re!resents the descri!tive branchof statistics. hat conclusions ,i+ht be drawn fro, the study

usin+ inferential statisticsB

Question:A lar+e sa,!le of ,en) a+ed "5)

was studied for 15 years. For

un,arried ,en) a!!roi,ately

'0 were alive at a+e %$. For,arried ,en) @0 were alive at

a+e %$.


18/156


19/156

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1(

+,- #+ -O.,SE/0 3

a'e 25

.


20/156

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 2*

+&pes )% Data

Qualitative Data

Consists of attributes) labels) or nonnu,erical entries.

#a4or (lace of birth 8ye color


21/156


+&pes )% Data

Quantitative data

?u,erical ,easure,ents or counts.

A+e ei+ht of a letter Te,!erature


22/156


Ea!ple" Classi%&in' Data b& +&pe

The base !rices of several vehicles are shown in the

table. hich data are qualitative data and which are

quantitative dataB (Source $ord %otor Compan#)


23/156


S)luti)n" Classi%&in' Data b& +&pe

uantitative Data


24/156


+,- #+ -O.,SE/0 1

a'e 2(

.


25/156


0reuenc& Distributi)n

Frequency Distribution

• A table that shows

classes or intervals of

data with a count of thenu,ber of entries in each

class.

• The frequency, f, of a

class is the nu,ber of

data entries in the class.

Class Frequency) f

1 $ $

% 1 5

11 1$ %

1% 2 5

21 2$ $

2% 3 "

Gower class

li,its

!!er class

li,its

Class width

% 1 H $


26/156


C)nstructin' a 0reuenc& Distributi)n

1. Decide on the nu,ber of classes.

sually between $ and 2I otherwise) it ,ay be

difficult to detect any !atterns.

2. Find the class width.

Deter,ine the ran+e of the data.

Divide the ran+e by the nu,ber of classes.

Round up to the ne&t convenient num'er


27/156



3. Find the class li,its. Jou can use the ,ini,u, data entry as the lower

li,it of the first class.

Find the re,ainin+ lower li,its


28/156



". #ae a tally ,ar for each data entry in the row of

the a!!ro!riate class.

$. Count the tally ,ars to find the total frequency f

for each class.


29/156


Ea!ple" C)nstructin' a 0reuenc&

Distributi)n

The followin+ sa,!le data set lists the !rices


30/156


S)luti)n" C)nstructin' a 0reuenc&

Distributi)n

1. ?u,ber of classes H '


31/156



Distributi)n

Gowerli,it

!!erli,it

$@

11$

1'1

22'

253

33@

3@$

Class

width H $%

3. se $@


32/156



Distributi)n

The u!!er li,it of the first

class is 11"


33/156



Distributi)n

". #ae a tally ,ar for each data entry in the row ofthe a!!ro!riate class.

$. Count the tally ,ars to find the total frequency f

for each class.Class Tally Frequency) f

$@ 11" IIII $

11$ 1' IIII III 5

1'1 22% IIII I %

22' 252 IIII $253 335 II 2

33@ 3@" I 1

3@$ "$ III 3


34/156


+,- #+ -O.,SE/0 1

a'e 62

.


35/156


Deter!inin' the i$p)int

Midpoint of a class


36/156


Deter!inin' the ,elative 0reuenc&

Relative Frequency of a class• (ortion or !ercenta+e of the data that falls in a

!articular class.

n

f

== si6eSa,!le

frequencyclass

frequencyrelative

Class Frequency) f ;elative Frequency

$@ 11" $

11$ 1' 5

1'1 22% %

$.1'

3≈

5.2'

3≈

%.2

3=

•


37/156


38/156


Epan$e$ 0reuenc& Distributi)n

Class Frequency) f #id!oint;elative

frequencyCu,ulativefrequency

$@ 11" $ 5%.$ .1' $

11$ 1' 5 1"2.$ .2' 131'1 22% % 1@5.$ .2 1@

22' 252 $ 2$".$ .1' 2"

253 335 2 31.$ .' 2%

33@ 3@" 1 3%%.$ .3 2'

3@$ "$ 3 "22.$ .1 3

M f H 3 1=∑n

f


39/156


+,- #+ -O.,SE/0 2

a'e 63

.


40/156


raphs )% 0reuenc& Distributi)ns

Frequency isto!ram

• A bar +ra!h that re!resents the frequency distribution.

• The hori6ontal scale is quantitative and ,easures the

data values.

• The vertical scale ,easures the frequencies of the

classes.

• Consecutive bars ,ust touch.

data values

f r e q u e n c y


41/156


Class )un$aries

Class boundaries

• The nu,bers that se!arate classes without for,in+

+a!s between the,.

ClassClass

EoundariesFrequency)

f

$@ 11" $

11$ 1' 5

1'1 22% %

• The distance fro, the u!!er

li,it of the first class to the

lower li,it of the second

class is 11$ 11" H 1.

• Nalf this distance is .$.

• First class lower boundary H $@ .$ H $5.$• First class u!!er boundary H 11" L .$ H 11".$

$5.$ 11".$


42/156


Class )un$aries

ClassClass

boundariesFrequency)

f

$@ 11" $5.$ 11".$ $

11$ 1' 11".$ 1'.$ 5

1'1 22% 1'.$ 22%.$ %

22' 252 22%.$ 252.$ $

253 335 252.$ 335.$ 2

33@ 3@" 335.$ 3@".$ 13@$ "$ 3@".$ "$.$ 3


43/156


Ea!ple" 0reuenc& ist)'ra!

Construct a frequency histo+ra, for the +lobal

!ositionin+ syste,


44/156


S)luti)n" 0reuenc& ist)'ra!

usin' i$p)ints


45/156


S)luti)n" 0reuenc& ist)'ra!

usin' class b)un$aries

Jou can see that ,ore than half of the (S navi+ators are

!riced below >22%.$.


46/156


47/156



Frequency Poly!on

• A line +ra!h that e,!hasi6es the continuous chan+e in

frequencies.

data values

f r e q u e n c y


48/156


Ea!ple" 0reuenc& )l&')n

Construct a frequency !oly+on for the (S navi+ators

frequency distribution.

Class #id!oint Frequency) f

$@ 11" 5%.$ $

11$ 1' 1"2.$ 5

1'1 22% 1@5.$ %

22' 252 2$".$ $

253 335 31.$ 2

33@ 3@" 3%%.$ 1

3@$ "$ "22.$ 3


49/156


S)luti)n" 0reuenc& )l&')n

Jou can see that the frequency of (S navi+ators increases

u! to >1"2.$ and then decreases.

The +ra!h should

be+in and end on the

hori6ontal ais) so

etend the left side to

one class width before

the first class,id!oint and etend

the ri+ht side to one

class width after the

last class ,id!oint.


50/156



Relative Frequency isto!ram

• Nas the sa,e sha!e and the sa,e hori6ontal scale as

the corres!ondin+ frequency histo+ra,.

• The vertical scale ,easures the relative frequencies)not frequencies.

data values

r e l a t i

v e

f r e q u e

n c y


51/156


52/156


Ea!ple" ,elative 0reuenc& ist)'ra!

Construct a relative frequency histo+ra, for the (S

navi+ators frequency distribution.

Class

Class

boundaries

Frequency)

f

;elative

frequency $@ 11" $5.$ 11".$ 5%.$ .1'

11$ 1' 11".$ 1'.$ 1"2.$ .2'

1'1 22% 1'.$ 22%.$ 1@5.$ .2

22' 252 22%.$ 252.$ 2$".$ .1'

253 335 252.$ 335.$ 31.$ .'

33@ 3@" 335.$ 3@".$ 3%%.$ .3

3@$ "$ 3@".$ "$.$ "22.$ .1


53/156


54/156



Cumulative Frequency "rap# or $!ive

• A line +ra!h that dis!lays the cu,ulative frequency of

each class at its u!!er class boundary.

• The u!!er boundaries are ,ared on the hori6ontalais.

• The cu,ulative frequencies are ,ared on the vertical

ais.

data values

c u , u l a t i v e

f r e q u e n c y


55/156


+,- #+ -O.,SE/0 5

a'e 66

.


56/156


C)nstructin' an O'ive

1. Construct a frequency distribution that includes

cu,ulative frequencies as one of the colu,ns.

2. S!ecify the hori6ontal and vertical scales.

The hori6ontal scale consists of the u!!er class boundaries.

The vertical scale ,easures cu,ulative

frequencies.

3. (lot !oints that re!resent the u!!er class boundaries

and their corres!ondin+ cu,ulative frequencies.


57/156


58/156


59/156


S)luti)n" O'ive

%.$ 15.$ 3.$ "2.$ $".$ %%.$ '5.$ @.$

Fro, the o+ive) you can see that about 2$ (S navi+ators cost

>3 or less. The +reatest increase occurs between >11".$ and

>1'.$.


60/156


+,- #+ -O.,SE/0 6

a'e 67

.


61/156


raphin' 9uantitative Data Sets

Stem%and%leaf plot

• 8ach nu,ber is se!arated into a stem and a leaf .

• Si,ilar to a histo+ra,.

• Still contains ori+inal data values.

Data* 21) 2$) 2$) &') 2') 25)

3) 3%) 3%) "$

&'

2 1 $ $ % ' 5

3 % %

" $

.


62/156

S l ti C t ti St $ / %


63/156


S)luti)n" C)nstructin' a Ste!:an$:/ea%

l)t

• The data entries +o fro, a low of '5 to a hi+h of 1$@.• se the ri+ht,ost di+it as the leaf.

For instance)

'5 H ' 5 and 1$@ H 1$ @• Gist the ste,s) ' to 1$) to the left of a vertical line.• For each data entry) list a leaf to the ri+ht of its ste,.

1$$ 1$@ 1"" 12@ 1$ 1"$ 12% 11% 13 11" 122 112 112 1"2 12%

1$% 115 15 122 121 1@ 1" 12% 11@ 113 11' 115 1@ 1@ 11@

13@ 13@ 122 '5 133 12% 123 1"$ 121 13" 12" 11@ 132 133 12"

12@ 112 12% 1"5 1"'


64/156


65/156


+,- #+ -O.,SE/0 1

a'e 76

.


66/156


raphin' 9uantitative Data Sets

Dot plot

• 8ach data entry is !lotted) usin+ a !oint) above a

hori6ontal ais

Data* 21) 2$) 2$) &') 2') 25) 3) 3%) 3%) "$

&'

2 21 22 23 2" 2$ 2% 2' 25 2@ 3 31 32 33 3" 3$ 3% 3' 35 3@ " "1 "2 "3 "" "$

.


67/156


Ea!ple" C)nstructin' a D)t l)t

se a dot !lot or+ani6e the tet ,essa+in+ data.

• So that each data entry is included in the dot !lot) the hori6ontal ais should include

nu,bers between ' and 1%.• To re!resent a data entry) !lot a !oint above the entryPs !osition on the ais.• 7f an entry is re!eated) !lot another !oint above the !revious !oint.

1$$ 1$@ 1"" 12@ 1$ 1"$ 12% 11% 13 11" 122 112 112 1"2 12%

1$% 115 15 122 121 1@ 1" 12% 11@ 113 11' 115 1@ 1@ 11@

13@ 13@ 122 '5 133 12% 123 1"$ 121 13" 12" 11@ 132 133 12"

12@ 112 12% 1"5 1"'

.


68/156


69/156


+,- #+ -O.,SE/0 3

a'e 77

.


70/156


raphin' 9ualitative Data Sets

Pie C#art

• A circle is divided into sectors that re!resent

cate+ories.

• The area of each sector is !ro!ortional to thefrequency of each cate+ory.


71/156


Ea!ple" C)nstructin' a ie Chart

The nu,bers of earned de+rees conferred


72/156


S)luti)n" C)nstructin' a ie Chart• Find the relative frequency

Ty!e of de+ree Frequency) f ;elative frequency

Associate:s'25

Eachelor:s 1$2$

#aster:s%"

First !rofessional @

Doctoral%

3'

'25.2"

3'≈

1$2$ .$13'

≈

%".2

3'≈

@ .33'

≈

.

%.2

3'≈


73/156


S)luti)n" C)nstructin' a ie Chart

• Construct the !ie chart usin+ the central an+le thatcorres!onds to each cate+ory.

To find the central an+le) ,ulti!ly 3%Q by the

cate+oryPs relative frequency. For ea,!le) the central an+le for cars is

3%


74/156



Ty!e of de+ree Frequency) f ;elative

frequency Central an+le

Associate:s '25 .2"

Eachelor:s 1$2$ .$1

#aster:s %" .2

First !rofessional @ .3

Doctoral % .2 3%Q


75/156



Ty!e of de+ree;elative

frequencyCentralan+le

Associate:s .2" 5%Q

Eachelor:s .$1 15"Q#aster:s .2 '2Q

First !rofessional .3 11Q

Doctoral .2 'Q

Fro, the !ie chart) you can see that ,ost fatalities in ,otor

vehicle crashes were those involvin+ the occu!ants of cars.

.


76/156


+,- #+ -O.,SE/0 4

a'e 78

.


77/156


raphin' 9ualitative Data Sets

Pareto C#art

• A vertical bar +ra!h in which the hei+ht of each bar

re!resents frequency or relative frequency.

• The bars are !ositioned in order of decreasin+ hei+ht)with the tallest bar !ositioned at the left.

Cate+ories

F r e q u e n

c y

.


78/156


raphin' aire$ Data Sets

Paired Data Sets

• 8ach entry in one data set corres!onds to one entry in

a second data set.

• ra!h usin+ a scatter plot( The ordered !airs are +ra!hed as

!oints in a coordinate !lane.

sed to show the relationshi! between two quantitative variables.

&

#

.


79/156


+,- #+ -O.,SE/0 5 ; 6

a'e 7( ; 8*

.


80/156


81/156


Objectives


•Find the ,ean) ,edian) and ,ode of a !o!ulation and

of a sa,!le


82/156


easures )% Central +en$enc&

Measure of central tendency

• A value that re!resents a ty!ical) or central) entry of a

data set.

• #ost co,,on ,easures of central tendency* #ean

#edian

#ode

.


83/156


easure )% Central +en$enc&" ean

Mean


84/156


85/156


86/156


+,- #+ -O.,SE/0 1

a'e 87

.


87/156

E l 0i $i th $i


88/156


Ea!ple" 0in$in' the e$ian

The !rices


89/156


S)luti)n" 0in$in' the e$ian

5'2 "32 3@' "2' 355 '52 3@'

• First order the data.

355 3@' 3@' "2' "32 '52 5'2

• There are seven entries "2'.

.

E l 0i $i th $i


90/156

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. (*

Ea!ple" 0in$in' the e$ian

The fli+ht !riced at >"32 is no lon+er available. hat isthe ,edian !rice of the re,ainin+ fli+htsB

5'2 3@' "2' 355 '52 3@'

.

S l ti 0i $i th $i


91/156

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. (1

S)luti)n" 0in$in' the e$ian

5'2 3@' "2' 355 '52 3@'

• First order the data.

355 3@' 3@' "2' '52 5'2

• There are si entries "12.

3@' "2'#edian "122

+

= =

.


92/156


+,- #+ -O.,SE/0 2 ; 3

a'e 88

.

% C t l + $ $


93/156


easure )% Central +en$enc&" )$e

Mode

• The data entry that occurs with the +reatest frequency.

• 7f no entry is re!eated the data set has no ,ode.

• 7f two entries occur with the sa,e +reatest frequency)each entry is a ,ode


94/156


Ea!ple" 0in$in' the )$e

The !rices


95/156


S)luti)n" 0in$in' the )$e

5'2 "32 3@' "2' 355 '52 3@'

• Orderin+ the data hel!s to find the ,ode.

355 3@' 3@' "2' "32 '52 5'2

• The entry of 3@' occurs twice) whereas the other

data entries occur only once.

The ,ode of the fli+ht !rices is >3@'.

.



96/156



At a !olitical debate a sa,!le of audience ,e,bers wasased to na,e the !olitical !arty to which they belon+.

Their res!onses are shown in the table. hat is the

,ode of the res!onsesB

Political Party Frequency, f

De,ocrat 3"

;e!ublican $%

Other 21

Did not res!ond @

.



97/156



Political Party Frequency, f

De,ocrat 3"

;e!ublican $%

Other 21Did not res!ond @

The ,ode is ;e!ublican


98/156


+,- #+ -O.,SE/0 4

a'e 8(

.

Ea!ple" C)!parin' the ean= e$ian=


99/156

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. ((

p p '

an$ )$e

Find the ,ean) ,edian) and ,ode of the sa,!le a+es ofa class shown. hich ,easure of central tendency best

describes a ty!ical entry of this data setB Are there any

outliersB

)!es in a class

2 2 2 2 2 2 21

21 21 21 22 22 22 23

23 23 23 2" 2" %$

.

S)luti)n" C)!parin' the ean= e$ian=


100/156

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1**

p '

an$ )$e

#ean* 2 2 ... 2" %$ 23.5 years2

& &

n

Σ + + + += = ≈

#edian* 21 22 21.$ years

2

+=

2 years


101/156

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1*1

'

an$ )$e

#ean R 23.5 years #edian H 21.$ years #ode H 2 years

• The ,ean taes every entry into account) but is

influenced by the outlier of %$.• The ,edian also taes every entry into account) and

it is not affected by the outlier.• 7n this case the ,ode eists) but it doesnPt a!!ear to

re!resent a ty!ical entry.

.

S)luti)n" C)!parin' the ean= e$ian=


102/156


an$ )$e

So,eti,es a +ra!hical co,!arison can hel! you decide which ,easure of centraltendency best re!resents a data set.

7n this case) it a!!ears that the median best describes the data set.

.


103/156

ean )% r)upe$ Data


104/156


ean )% r)upe$ Data

Mean of a Frequency Distribution

• A!!roi,ated by

where & and f are the ,id!oints and frequencies of a

class) res!ectively

< = & f & n f

n

Σ ×= = Σ

.

0in$in' the ean )% a 0reuenc&

Distrib ti)n


105/156


Distributi)n

In Words In S#m'ols

< = & f &

n

Σ ×=


106/156


Distributi)n

se the frequency distribution to a!!roi,ate the ,eannu,ber of ,inutes that a sa,!le of 7nternet subscribers

s!ent online durin+ their ,ost recent session.

Class #id!oint Frequency) f ' 15 12.$ %

1@ 3 2".$ 1

31 "2 3%.$ 13

"3 $" "5.$ 5$$ %% %.$ $

%' '5 '2.$ %

'@ @ 5".$ 2

.

S)luti)n" 0in$ the ean )% a 0reuenc&


107/156


Distributi)n

Class #id!oint) & Frequency) f


108/156


+,- #+ -O.,SE/0 8

a'e (2

.

+he Shape )% Distributi)ns


109/156

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1*(


S&!!etric Distributi)n

• # $ertica" "ine can %e dra&n through the idd"e o! a graph o! the

distri%ution and the resu"ting ha"$es are appro'iate"y irror iages.

.



110/156



S>ee$ /e%t Distributi)n ne'ativel& s>ee$

• The (tai") o! the graph e"ongates ore to the "e!t.

• The ean is to the "e!t o! the edian.

.



111/156



S>ee$ ,i'ht Distributi)n p)sitivel& s>ee$

• The (tai") o! the graph e"ongates ore to the right.

• The ean is to the right o! the edian.

.

,an'e


112/156


,an'e

Ran!e

• The difference between the ,ai,u, and ,ini,u,

data entries in the set.

• The data ,ust be quantitative.• ;an+e H


113/156


Ea!ple" 0in$in' the ,an'e

A cor!oration hired 1 +raduates. The startin+ salariesfor each +raduate are shown. Find the ran+e of the

startin+ salaries.

Startin+ salaries


114/156


S)luti)n" 0in$in' the ,an'e

• Orderin+ the data hel!s to find the least and +reatestsalaries.

3' 35 3@ "1 "1 "1 "2 "" "$ "'

• ;an+e H


115/156


+,- #+ -O.,SE/0 1

a'e 1*2

.

Deviati)n= @ariance= an$ Stan$ar$

D i ti


116/156


Deviati)n

Deviation

• The difference between the data entry) &) and the

,ean of the data set.

• (o!ulation data set* Deviation of & H &

• Sa,!le data set*

Deviation of & H & &

.

Ea!ple" 0in$in' the Deviati)n


117/156


Ea!ple" 0in$in' the Deviati)n

A cor!oration hired 1 +raduates. The startin+ salariesfor each +raduate are shown. Find the deviation of the

startin+ salaries.

Startin+ salaries


118/156


S)luti)n" 0in$in' the Deviati)n

• Deter,ine thedeviation for each data

entry.

Salary *+---s., x Deviation: x / 0

"1 "1 "1.$ H .$

35 35 "1.$ H 3.$

3@ 3@ "1.$ H 2.$

"$ "$ "1.$ H 3.$"' "' "1.$ H $.$

"1 "1 "1.$ H .$

"" "" "1.$ H 2.$

"1 "1 "1.$ H .$3' 3' "1.$ H ".$

"2 "2 "1.$ H .$

M & H "1$ M


119/156


Deviati)n

Population 1ariance

•

Population Standard Deviation

•

22 < = &

!

µ σ

Σ −=

Su, of squares) SS &

22 < = &

!

µ

σ σ

Σ −

= =

.

0in$in' the )pulati)n @ariance ;

St $ $ D i ti


120/156


Stan$ar$ Deviati)n

In Words In S#m'ols

1. Find the ,ean of the

!o!ulation data set.

2. Find deviation of eachentry.

3. Square each deviation.

". Add to +et the su, ofsquares.

&

! µ

Σ=

&


121/156


Stan$ar$ Deviati)n

$. Divide by ! to +et the

population variance.

%. Find the square root to +etthe population standard

deviation.

22 < = &

!

µ σ

Σ −=

2< = &

!

µ σ

Σ −=

In Words In S#m'ols

.

Ea!ple" 0in$in' the )pulati)n

St $ $ D i ti


122/156


Stan$ar$ Deviati)n

A cor!oration hired 1 +raduates. The startin+ salariesfor each +raduate are shown. Find the !o!ulation

variance and standard deviation of the startin+ salaries.

Startin+ salaries


123/156


Stan$ar$ Deviati)n

• Deter,ine SS &

• ! H 1

Salary, x Deviation: x / 0 Squares: * x / 0.&

"1 "1 "1.$ H .$


124/156


Stan$ar$ Deviati)n

Population 1ariance

•

Population Standard Deviation

•

22 < = 55.$ 5.@

1

&

!

µ σ

Σ −= = ≈

2

5.5$ 3.σ σ = = ≈

The !o!ulation standard deviation is about 3.) or >3.

.


125/156


+,- #+ -O.,SE/0 2

a'e 1*4

.

Deviati)n= @ariance= an$ Stan$ar$

Deviati)n


126/156


Deviati)n

Sample 1ariance

•

Sample Standard Deviation

•

22 < =

1

& & s

n

Σ −=

−

22 < =

1

& &

s s n

Σ −

= = −

.

0in$in' the Sa!ple @ariance ; Stan$ar$

Deviati)n


127/156


Deviati)n

In Words In S#m'ols

1. Find the ,ean of the

sa,!le data set.

2. Find deviation of eachentry.

3. Square each deviation.

". Add to +et the su, ofsquares.

& &

n

Σ=

2< = &SS & &= Σ −

2< = & &−

& &−

.

0in$in' the Sa!ple @ariance ; Stan$ar$

Deviati)n


128/156


Deviati)n

$. Divide by n 1 to +et the

sample variance.

%. Find the square root to +etthe sample standard

deviation.

In Words In S#m'ols2

2 < =

1

& & s

n

Σ −=

−

2< =

1

& & s

n

Σ −=

−

.

Ea!ple" 0in$in' the Sa!ple Stan$ar$

Deviati)n


129/156


Deviati)n

The startin+ salaries are for the Chica+o branches of acor!oration. The cor!oration has several other branches)

and you !lan to use the startin+ salaries of the Chica+o

branches to esti,ate the startin+ salaries for the lar+er

!o!ulation. Find the sample standard deviation of the

startin+ salaries.

Startin+ salaries


130/156

S)luti)n" 0in$in' the Sa!ple Stan$ar$

Deviati)n


131/156


Deviati)n

Sample 1ariance

•

Sample Standard Deviation

•

22 < = 55.$ @.5

1 1 1

& & s

n

Σ −= = ≈

− −

2 55.$

3.1@ s s= = ≈

The sa,!le standard deviation is about 3.1) or >31.

.


132/156


+,- #+ -O.,SE/0 3

a'e 1*6

.

#nterpretin' Stan$ar$ Deviati)n


133/156


• Standard deviation is a ,easure of the ty!ical a,ountan entry deviates fro, the ,ean.

• The ,ore the entries are s!read out) the +reater the

standard deviation.

.

#nterpretin' Stan$ar$ Deviati)n"

E!pirical ,ule 68 A (5 A (( 7 ,ule


134/156


E!pirical ,ule 68 A (5 A ((B7 ,ule

For data with a


135/156


E!pirical ,ule 68 A (5 A ((B7 ,ule

3 & s− & s+ 2 & s+ 3 & s+ & s− &2 & s−

%50 within 1

standard deviation

3"0 3"0

@@.'0 within 3 standard deviations

2.3$0 2.3$0

@$0 within 2 standard deviations

13.$0 13.$0

.

Stan$ar$ Deviati)n %)r r)upe$ Data


136/156


Sample standard deviation for a frequency distribution

•

• hen a frequency distribution has classes) esti,ate the sa,!le ,ean an d standard deviation by usin+ the ,id!oint of each class.

2< =

1

& & f s

n

Σ −=

−

where nH M f


137/156


%)r r)upe$ Data

Jou collect a rando, sa,!le of the

nu,ber of children !er household in

a re+ion. Find the sa,!le ,ean and

the sa,!le standard deviation of thedata set.

7umber of C#ildren in5- ouse#olds

1 3 1 1 1

1 2 2 1

1 1

1 $ 3 %

3 3 1 1

1 1 % 1

3 % % 1 2

2 3 1 1

" 1 1 2 2

3 2 "

.

S)luti)n" 0in$in' the Stan$ar$ Deviati)n

%)r r)upe$ Data


138/156


x f xf

1


139/156


%)r r)upe$ Data

• Deter,ine the su, of squares.

x f

1 1.5 H 1.5


140/156


141/156


+,- #+ -O.,SE/0 8

a'e 11*

.

9uartiles


142/156


• Fractiles are nu,bers that !artition


143/156


The nu,ber of nuclear !ower !lants in the to! 1$ nuclear !ower9!roducin+ countries in the world are listed. Find

the first) second) and third quartiles of the data set.

' 15 11 % $@ 1' 15 $" 1" 2 31 5 1 1$ 1@

Solution:

• +2 divides the data set into two halves.

% ' 5 1 11 1$ 1' 15 15 1@ 2 31 $" $@ 1"

Q&

Gower half !!er half

.

S)luti)n" 0in$in' 9uartiles


144/156


• The first and third quartiles are the ,edians of the lower and

u!!er halves of the data set.

% ' 5 1 11 1$ 1' 15 15 1@ 2 31 $" $@ 1"

Q&

Gower half !!er half

Q Q9

About one fourth of the countries have 1 or less)

about one half have 15 or lessI and about three fourths

have 31 or less.

.


146/156

Ea!ple" 0in$in' the #nteruartile ,an'e


147/156


Find the interquartile ran+e of the data set.;ecall +1 H 1) +2 H 15) and +3 H 31

Solution:

• 7; H +3 +1 H 31 1 H 21

The nu,ber of !ower !lants in the ,iddle !ortion of

the data set vary by at ,ost 21.

.


148/156


+,- #+ -O.,SE/0 3

a'e 124

.

):an$:Whis>er l)t


149/156


o;%and%


150/156


1. Find the five9nu,ber su,,ary of the data set.2. Construct a hori6ontal scale that s!ans the ran+e of

the data.

3. (lot the five nu,bers above the hori6ontal scale.

". Draw a bo above the hori6ontal scale fro, +1 to +3

and draw a vertical line in the bo at +2.

$. Draw whisers fro, the bo to the ,ini,u, and

,ai,u, entries.hiser hiser

#ai,u,

entry

#ini,u,

entry

Eo

#edian) +2 +3+1

.

Ea!ple" Drain' a ):an$:Whis>er

l)t


151/156


Draw a bo9and9whiser !lot that re!resents the 1$ dataset.

#in H %) +1 H 1) +2 H 15) +3 H 31) #a H 1")

Solution:

About half the scores are between 1 and 31. Ey looin+at the len+th of the ri+ht whiser) you can conclude 1"

is a !ossible outlier.

.


152/156


+,- #+ -O.,SE/0 4

a'e 125

.

ercentiles an$ Other 0ractiles


153/156


Fractiles Summary Symbolsuartiles Divides data into " equal

!arts+1) +2) +3

Deciles Divides data into 1 equal

!arts

,1) ,2) ,3)/) ,@

(ercentiles Divides data into 1 equal !arts

P 1 - P 2) P 3)/) P @@

.

ercentile that c)rresp)n$s t) a speci%ic

$ata entr&=


154/156


&=

And the round to the nearest whole nu,ber

.

Ea!ple" 0in$in' a percentile


155/156


For the data set below) find the !ercentile that corres!onds to >3)

35 33 " "2 3" 2' "" 35 32 3" "$ 32 23 "% 2' 23 3 2' "1

22 2% "$ 31 2% 1@

Solution:

The tuition cost of >3) corres!onds to the 3%th !ercentile

7nter!retation * The tuition cost of >3) is +reater than 3%0 of the other tuition

costs.

.


156/156

+,- #+ -O.,SE/0 6

a'e 127

chapter 1 - power point slide staistic

Documents