Stats Chap02 bluman

8/16/2019 Stats Chap02 bluman

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Chapter 2

Frequency Distributionsand Graphs

1© McGraw-Hill, Bluman, th

ed, Chapter2


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Chapter 2 !"er"iew

#ntroduction

2-$ !r%ani&in% Data

2-2 Histo%rams, Frequency'oly%ons, and !%i"es

2-( !ther )ypes o* Graphs

2-+ 'aired Data and catter 'lots

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Chapter 2 !b ecti"es$. !r%ani&e data usin% *requency distributions.2. /epresent data in *requency distributions

%raphically usin% histo%rams, *requency

poly%ons, and o%i"es.(. /epresent data usin% 'areto charts, time

series %raphs, and pie %raphs.+. Draw and interpret a stem and lea* plot.

. Draw and interpret a scatter plot *or a set o*paired data.

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2-$ !r%ani&in% DataData collected in ori%inal *orm is calledraw dataraw data .

0 frequency distributionfrequency distribution is theor%ani&ation o* raw data in table *orm,usin% classes and *requencies.

1ominal- or ordinal-le"el data that can beplaced in cate%ories is or%ani&ed incategorical frequency distributionscategorical frequency distributions .

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Chapter 2

Frequency Distributions andGraphs

ection 2-$3ample 2-$

'a%e 4(5

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Cate%orical Frequency Distribution

)wenty-*i"e army indicates were %i"en a bloodtest to determine their blood type.

/aw Data6 0,B,B,0B,! !,!,B,0B,B

B,B,!,0,! 0,!,!,!,0B 0B,0,!,B,0

Construct a *requency distribution *or the data.

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Cate%orical Frequency Distribution

)wenty-*i"e army indicates were %i"en a bloodtest to determine their blood type.

/aw Data6 0,B,B,0B,! !,!,B,0B,B

B,B,!,0,! 0,!,!,!,0B 0B,0,!,B,0

Class )ally Frequency 'ercent

0B!

0B

######## ###### ########

78+

29(+$:

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Grouped Frequency DistributionGrouped frequency distributionsGrouped frequency distributions areused when the ran%e o* the data is lar%e.

)he smallest and lar%est possible data"alues in a class are the lower lower andupper class limitsupper class limits . Class boundariesClass boundaries separate the classes.

)o *ind a class boundary, a"era%e theupper class limit o* one class and thelower class limit o* the ne3t class.

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Grouped Frequency Distribution)he class widthclass width can be calculated by subtractin%

successi"e lower class limits ;or boundaries<successi"e upper class limits ;or boundaries<upper and lower class boundaries

)he class midpoint X class midpoint X mm can be calculated by a"era%in%upper and lower class limits ;or boundaries<

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/ules *or Classes in Grouped

Frequency Distributions$. )here should be -29 classes.2. )he class width should be an odd

number.(. )he classes must be mutually e3clusi"e.+. )he classes must be continuous.

. )he classes must be e3hausti"e.:. )he classes must be equal in width

;e3cept in open-ended distributions


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Chapter 2


ection 2-$3ample 2-2

'a%e 4+$

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Constructin% a Grouped Frequency

Distribution)he *ollowin% data represent the recordhi%h temperatures *or each o* the 9 states.

Construct a %rouped *requencydistribution *or the data usin% 7 classes.

$$2 $99 $27 $29 $(+ $$5 $9 $$9 $98 $$2$$9 $$5 $$7 $$: $$5 $22 $$+ $$+ $9 $98$97 $$2 $$+ $$ $$5 $$7 $$5 $22 $9: $$9$$: $95 $$9 $2$ $$( $29 $$8 $$$ $9+ $$$$29 $$( $29 $$7 $9 $$9 $$5 $$2 $$+ $$+

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Distribution) ' $ Determine the classes.

Find the class width by di"idin% the ran%e by

the number o* classes 7.Range = High > Low

= $(+ > $99 = (+

Width = Range ?7 = (+?7 =

/oundin% /ule6 0lways round up i* a remainder.

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DistributionFor con"enience sa@e, we will choose the lowest data"alue, $99, *or the *irst lower class limit.)he subsequent lower class limits are *ound by addin%the width to the pre"ious lower class limits.

Class Aimits $99 - $9 -

$$9 - $$ - $29 - $2 - $(9 -

$9+$98

$$+$$8$2+$28$(+

)he *irst upper class limit is oneless than the ne3t lower class limit.

)he subsequent upper class limitsare *ound by addin% the width to thepre"ious upper class limits.

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Distribution)he class boundary is midway between an upperclass limit and a subsequent lower class limit.$9+,$9+. ,$9

ClassAimits

ClassBoundaries Frequency

Cumulati"eFrequency

$99 - $9+$9 - $98

$$9 - $$+$$ - $$8$29 - $2+$2 - $28$(9 - $(+

88. - $9+.$9+. - $98.$98. - $$+.$$+. - $$8.$$8. - $2+.$2+. - $28.

$28. - $(+. 15Bluman, Chapter 2


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Distribution) ' 2 )ally the data.) ' ( Find the *requencies.

25

$5$(7$

$

ClassAimits


Cumulati"eFrequency

$99 - $9+$9 - $98

$$9 - $$+$$ - $$8$29 - $2+$2 - $28$(9 - $(+

88. - $9+.$9+. - $98.$98. - $$+.$$+. - $$8.$$8. - $2+.$2+. - $28.

$28. - $(+. 16Bluman, Chapter 2


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ClassAimits


Cumulati"eFrequency

$99 - $9+$9 - $98

$$9 - $$+$$ - $$8$29 - $2+$2 - $28$(9 - $(+


Distribution) ' + Find the cumulati"e *requencies by@eepin% a runnin% total o* the *requencies.

2$925+$+5+8

9

88. - $9+.$9+. - $98.$98. - $$+.$$+. - $$8.$$8. - $2+.$2+. - $28.

$28. - $(+.

25

$5$(7$

$ 17Bluman, Chapter 2


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2-2 Histo%rams, Frequency

'oly%ons, and !%i"es3 Most Common Graphs in Research3 Most Common Gra phs in Research

1.1. HistogramHistogram

2.2. requency !olygonrequency !olygon

3.3. Cumulati"e requency !olygon #$gi"e%Cumulati"e requency !olygon #$gi"e%

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2-2 Histo%rams, Frequency

'oly%ons, and !%i"es)he histogramhistogram is a %raph thatdisplays the data by usin% "erticalbars o* "arious hei%hts to representthe *requencies o* the classes.

)he class boundaries arerepresented on the hori&ontal a3is.

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Chapter 2


ection 2-23ample 2-+

'a%e 4 $

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Histo%rams

Construct a histo%ram to represent thedata *or the record hi%h temperatures *oreach o* the 9 states ;see 3ample 2>2

*or the data


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Histo%rams

Class

Aimits

Class

BoundariesFrequency

$99 - $9+$9 - $98$$9 - $$+$$ - $$8$29 - $2+$2 - $28$(9 - $(+

88. - $9+.$9+. - $98.$98. - $$+.

$$+. - $$8.$$8. - $2+.$2+. - $28.$28. - $(+.

25

$5

$(7$$

Histo%rams use class boundaries and*requencies o* the classes.

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Histo%rams

Histo%rams use class boundaries and*requencies o* the classes.

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2.2 Histo%rams, Frequency

'oly%ons, and !%i"es)he frequency polygonfrequency polygon is a %raph thatdisplays the data by usin% lines that

connect points plotted *or the*requencies at the class midpoints. )he*requencies are represented by the

hei%hts o* the points.)he class midpoints are represented onthe hori&ontal a3is.

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Chapter 2


ection 2-23ample 2-

'a%e 4 (

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Frequency 'oly%ons

Construct a *requency poly%on torepresent the data *or the record hi%htemperatures *or each o* the 9 states

;see 3ample 2>2 *or the data


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Frequency 'oly%ons

Class

Aimits

Class

MidpointsFrequency

$99 - $9+$9 - $98$$9 - $$+$$ - $$8$29 - $2+$2 - $28$(9 - $(+

$92$97$$2

$$7$22$27$(2

25

$5

$(7$$

Frequency poly%ons use class midpointsand *requencies o* the classes.

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Frequency 'oly%ons

Frequency poly%ons use class midpointsand *requencies o* the classes.

0 *requency poly%onis anchored on the3-a3is be*ore the *irstclass and a*ter thelast class.

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'oly%ons, and !%i"es)he ogiveogive is a %raph that representsthe cumulati"e *requencies *or the

classes in a *requency distribution.

)he upper class boundaries arerepresented on the hori&ontal a3is.

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Chapter 2


ection 2-23ample 2-:

'a%e 4 +

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!%i"es

Construct an o%i"e to represent the data*or the record hi%h temperatures *or eacho* the 9 states ;see 3ample 2>2 *or

the data


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!%i"es

!%i"es use upper class boundaries andcumulati"e *requencies o* the classes.

Class

Aimits

Class

BoundariesFrequency Cumulati"e

Frequency$99 - $9+$9 - $98$$9 - $$+$$ - $$8$29 - $2+$2 - $28$(9 - $(+

88. - $9+.$9+. - $98.$98. - $$+.

$$+. - $$8.$$8. - $2+.$2+. - $28.$28. - $(+.

25

$5

$(7$$

2$925

+$+5+8

9

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!%i"es


Class Boundaries Cumulati"e

Frequency Aess than $9+.Aess than $98.Aess than $$+.

Aess than $$8.Aess than $2+.Aess than $28.Aess than $(+.

2$925

+$+5+8

9

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!%i"es


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'rocedure )able

Constructin% tatistical Graphs$6 Draw and label the$6 Draw and label the x x andand y y a3es.a3es.

26 Choose a suitable scale *or the *requencies or26 Choose a suitable scale *or the *requencies orcumulati"e *requencies, and label it on thecumulati"e *requencies, and label it on the y y a3is.a3is.

(6 /epresent the class boundaries *or the histo%ram(6 /epresent the class boundaries *or the histo%ramor o%i"e, or the midpoint *or the *requencyor o%i"e, or the midpoint *or the *requency

poly%on, on thepoly%on, on the x x a3is.a3is.+6 'lot the points and then draw the bars or lines.+6 'lot the points and then draw the bars or lines.

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'oly%ons, and !%i"es#* proportions are used instead o**requencies, the %raphs are called

relative frequency graphsrelative frequency graphs ./elati"e *requency %raphs are usedwhen the proportion o* data "alues that

*all into a %i"en class is more importantthan the actual number o* data "aluesthat *all into that class.

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Chapter 2


ection 2-23ample 2-7

'a%e 4 7

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ClassBoundaries Frequency . - $9.$9. - $ .$ . - 29.

29. - 2 .2 . - (9.(9. - ( .( . - +9.

$2(

+(2

Construct a histo%ram, *requency poly%on,and o%i"e usin% relati"e *requencies *or thedistribution ;shown here< o* the miles that29 randomly selected runners ran durin% a%i"en wee@.

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Histo%rams


/elati"eFrequency

. - $9.$9. - $ .$ . - 29.29. - 2 .

2 . - (9.(9. - ( .( . - +9.

$2(

+(2

$?29 =2?29 =(?29 =

?29 =+?29 =(?29 =2?29 =

)he *ollowin% is a *requency distribution o*miles run per wee@ by 29 selected runners.

Σ f = 29 Σ rf = $.99

9.99.$99.$9.29.299.$9.$9

Di"ide each*requencyby the total*requency to%et therelati"e*requency.

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Histo%rams

se the class boundaries and therelati"e *requencies o* the classes.

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Frequency 'oly%ons

ClassBoundaries

ClassMidpoints

/elati"eFrequency

. - $9.$9. - $ .$ . - 29.29. - 2 .

2 . - (9.(9. - ( .( . - +9.

5$($52(25(((5


9.99.$99.$9.29.299.$9.$9

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Frequency 'oly%ons

se the class midpoints and therelati"e *requencies o* the classes.

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!%i"es


Cumulati"eFrequency

Cum. /el.Frequency

. - $9.$9. - $ .$ . - 29.29. - 2 .

2 . - (9.(9. - ( .( . - +9.

$2(

+(2

$?29 =(?29 =:?29 =

$$?29 =$ ?29 =$5?29 =29?29 =


Σ f = 29

9.99.$9.(99.9.79.89$.99

$(:

$$$$529

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!%i"es


Class Boundaries Cum. /el.

FrequencyAess than $9.Aess than $ .Aess than 29.

Aess than 2 .Aess than (9.Aess than ( .Aess than +9.

9.99.$9.(9

9.9.79.89$.99

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!%i"es

se the upper class boundaries and thecumulati"e relati"e *requencies.

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hapes o* Distributions

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2.( !ther )ypes o* Graphs

Bar Graphs

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!ther )ypes o* Graphs

'areto Charts

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)ime eries Graphs

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'ie Graphs

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tem and Aea* 'lots 0 stem and leaf plotsstem and leaf plots is a data plot thatuses part o* a data "alue as the stem

and part o* the data "alue as the lea* to*orm %roups or classes.

#t has the ad"anta%e o"er %rouped

*requency distribution o* retainin% theactual data while showin% them in%raphic *orm.

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Chapter 2


ection 2-(3ample 2-$(

'a%e 459

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0t an outpatient testin% center, thenumber o* cardio%rams per*ormed eachday *or 29 days is shown. Construct astem and lea* plot *or the data.

54Bluman, Chapter 2

2 ($ 29 (2 $($+ +( 2 7 2((: (2 (( (2 ++(2 2 ++ $ +


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55Bluman, Chapter 2

2 ($ 29 (2 $($+ +( 2 7 2(

(: (2 (( (2 ++(2 2 ++ $ +

9 2$ ( +2 9 (( $ 2 2 2 2 ( :+ ( + +

$ 2 7

nordered tem 'lot !rdered tem 'lot

9 2$ ( +2 9 (

( $ 2 : 2 ( 2 2+ ( + +

7 2 $


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2.+ catter 'lots and Correlation

0 scatter plotscatter plot is a %raph o* the orderedpairs ; x , y < o* numbers consistin% o* theindependent "ariable x and the

dependent "ariable y . 0 scatter plot is used to determine i* arelationship e3ists between the two

"ariables.

Bluman, Chapter 2 56


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3ample 2-$:6 et Bi@e 0ccidents 0 researcher is interested in determinin% i* there is arelationship between the number o* wet bi@e accidentsand the number o* wet bi@e *atalities. )he data are *or a$9-year period. Draw a scatter plot *or the data.

&tep 1' Draw and label the x and y a3es.&tep 2' 'lot each point on the %raph.


(o. of accidents) x (7: : 9 55+ $$:2 $ $( $: 9 22(: (992 +925 +9$9

(o. of fatalities) y 29 29 25 2: (+ ( : :5


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3ample 2-$:6 et Bi@e 0ccidents


(o. of accidents) x (7: : 9 55+ $$:2 $ $( $: 9 22(: (992 +925 +9$9

(o. of fatalities) y 29 29 25 2: (+ ( : :5


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0naly&in% the catter 'lot

$. 0 positi"e linear relationshippositi"e linear relationship e3ists when thepoints *all appro3imately in an ascendin%strai%ht line *rom le*t to ri%ht and both the x andy "alues increase at the same time.

2. 0 negati"e linear relationshipnegati"e linear relationship e3ists when thepoints *all appro3imately in a descendin%strai%ht line *rom le*t to ri%ht.

(. 0 nonlinear relationshipnonlinear relationship e3ists when thepoints *all in a cur"ed line.

+. #t is said that no relationshipno relationship e3ists when thereis no discernable pattern o* the points.



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0naly&in% the catter 'lot

;a< 'ositi"e linear relationship ;b< 1e%ati"e linear relationship

;c< 1onlinear relationship ;d< 1o relationship

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