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Chapter 2
Frequency Distributionsand Graphs
1© McGraw-Hill, Bluman, th
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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
Frequency Distributions andGraphs
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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Constructin% a Grouped Frequency
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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Constructin% a Grouped Frequency
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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Constructin% a Grouped Frequency
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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Constructin% a Grouped Frequency
Distribution) ' 2 )ally the data.) ' ( Find the *requencies.
25
$5$(7$
$
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. - $(+. 16Bluman, Chapter 2
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ClassAimits
ClassBoundaries Frequency
Cumulati"eFrequency
$99 - $9+$9 - $98
$$9 - $$+$$ - $$8$29 - $2+$2 - $28$(9 - $(+
Constructin% a Grouped Frequency
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$
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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
Frequency Distributions andGraphs
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$$
Histo%rams use class boundaries and*requencies o* the classes.
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Histo%rams
Histo%rams use class boundaries and*requencies o* the classes.
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2.2 Histo%rams, Frequency
'oly%ons, and !%i"es)he frequency polygonfrequency polygon is a %raph thatdisplays the data by usin% lines that
connect points plotted *or the*requencies at the class midpoints. )he*requencies are represented by the
hei%hts o* the points.)he class midpoints are represented onthe hori&ontal a3is.
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Chapter 2
Frequency Distributions andGraphs
ection 2-23ample 2-
'a%e 4 (
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Frequency 'oly%ons
Construct a *requency poly%on torepresent the data *or the record hi%htemperatures *or each o* the 9 states
;see 3ample 2>2 *or the data
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Frequency 'oly%ons
Class
Aimits
Class
MidpointsFrequency
$99 - $9+$9 - $98$$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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2.2 Histo%rams, Frequency
'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
Frequency Distributions andGraphs
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$$
2$925
+$+5+8
9
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!%i"es
!%i"es use upper class boundaries andcumulati"e *requencies o* the classes.
Class Boundaries Cumulati"e
Frequency Aess than $9+.Aess than $98.Aess than $$+.
Aess than $$8.Aess than $2+.Aess than $28.Aess than $(+.
2$925
+$+5+8
9
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!%i"es
!%i"es use upper class boundaries andcumulati"e *requencies o* the classes.
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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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2.2 Histo%rams, Frequency
'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
Frequency Distributions andGraphs
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
ClassBoundaries Frequency
/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
)he *ollowin% is a *requency distribution o*miles run per wee@ by 29 selected runners.
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
ClassBoundaries Frequency
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 =
)he *ollowin% is a *requency distribution o*miles run per wee@ by 29 selected runners.
Σ f = 29
9.99.$9.(99.9.79.89$.99
$(:
$$$$529
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!%i"es
!%i"es use upper class boundaries andcumulati"e *requencies o* the classes.
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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!ther )ypes o* Graphs
)ime eries Graphs
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!ther )ypes o* Graphs
'ie Graphs
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!ther )ypes o* Graphs
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
Frequency Distributions andGraphs
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.
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2 ($ 29 (2 $($+ +( 2 7 2((: (2 (( (2 ++(2 2 ++ $ +
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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.
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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.
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(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
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(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