BIOSTAT Chapter2

8/17/2019 BIOSTAT Chapter2

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Chapter 2:Frequency

Distributions

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Frequency Distributions

After collecting data, the first task for aresearcher is to organize and simplify thedata so that it is possible to get a generaloverview of the results. This is the goal ofdescriptive statistical techniques.

One method for simplifying and organizing data

is to construct a frequency distribution.

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Frequency Distributions (cont.)

A frequency distribution is an organizedtabulation showing exactly how manyindividuals are located in each category on

the scale of measurement. A frequency distribution presents an

organized picture of the entire set ofscores, and it shows where eachindividual is located relative to others inthe distribution.

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A table that organizes data values into classesor intervals along with number of values that

fall in each class fre!uency, f ".

1. Ungrouped requency !istribution " fordata sets with few different values. #ach

value is in its own class.

$. %rouped requency !istribution& for datasets with many different values, which

are grouped together in the classes.

FREQUENCY DISTRIBUTIONS(CONT.)


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Grouped and Ungrouped

Frequency Distributions

CoursesTaken

Frequency, f

1 25

2 38

3 217

4 1462

5 932

6 15

Ungrouped

Age ofVoters

Frequency, f

18-3 22

31-42 58

43-54 62

55-66 413

67-78 158

78-9 32

%rouped


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Ungrouped requency !istributions

!u"#er of $eas %n a $ea $o&'a"()e '%*e+ 5

5 5 4 6 4

3 7 6 3 5

6 5 4 5 5

6 2 3 5 5

5 5 7 4 3

4 5 4 5 6

5 1 6 2 6

6 6 6 6 4

4 5 4 5 3

5 5 7 6 5

$eas (er (o& Freq, f $eas (er (o&

Freq,f

1 1

2 2

3 5

4 9

5 18

6 12

7 3


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Frequency Distribution Tables

A frequency distribution table consists of at leasttwo columns # one listing categories on the scale ofmeasurement $" and another for fre!uency f".

%n the $ column, values are listed from the highest

to lowest, without s&ipping any. 'or the fre!uency column, tallies are determined

for each value (how often each X value occursin the data set). (hese tallies are the fre!uenciesfor each $ value.

(he sum of the fre!uencies should e!ual n.

)


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Grouped Frequency

Distribution Soeties! ho"e#er! a set o$ scores co#ersa "ide range o$ #alues. %n these situations!a list o$ all the & #alues "ould be quite long

' too long to be a siple presentation o$the data.

To reedy this situation! a grouped

frequency di!ri"u!ion table is used.


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Grouped Frequency Distribution(cont.)

%n a grouped table! the & colun listsgroups o$ scores! called c#$ in!er%$#!rather than indi#idual #alues.

These inter#als all ha#e the sae "idth!usually a siple nuber such as 2! *! +,!and so on.

-ach inter#al begins "ith a #alue that is aultiple o$ the inter#al "idth. The inter#al"idth is selected so that the table "ill ha#eapproiately ten inter#als.


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/ey Concepts:

Data in its original $or and structureare called r$& d$!$' ungrouped d$!$.

E$p#e* Conider !+e fo##o&ing d$!$on ,- &oen !e!o!erone eru #e%e#

(core) e$ured in g'd#.

/0 12 1, 34 34 ,2 /4 ,1 ,3 50

02 3, 50 24 5, ,5 55 ,5 /3 /4

,- 56 56 /4 12 ,0 34 15 /, ,-

,0 35 21 55 // 32 21 ,2 32 30


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0 To con!ruc! $ frequencydi!ri"u!ion of !+e gi%en r$& d$!$7&e 8r! 8nd !+e +ig+e! eru

#e%e# $nd prep$re $ co#un of !+e!+ee #e%e# "eginning fro !+e+ig+e! %$#ue $nd ending $! !+e#o&e! one. Since !+e +ig+e!eru #e%e# i 02 $nd !+e #o&e! i247 &e +$%e*


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Seru9e%e#

f Seru9e%e#

f

1 % 32 %%%

4* % *1 %%

45 % ** %

4 %% *5 %

61 % *+ %%66 % 51 %%

63 % 54 %

65 % 53 %

62 %%% 5* %%

31 % 5 %%

3* % 5, %%

35 % 4 %%

3 %% 2 %


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7hen these data are placed into a syste "herein theyare organi8ed! then these parta9e the nature o$grouped d$!$. This procedure o$ organi8ing data into

groups is called a frequency di!ri"u!ion !$"#e(FDT).

E$p#e*

The $ollo"ing presents a $requency distribution table o$

the grouped data o$ the urine aylase (scores) o$ +*patients in aylase unitshour.

SCORES FREQUENCY

+,'+1 *

2,'21 5

,'1

5,'1 2

*,'*1 +

+*


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Coponen! of $ Frequency T$"#e

C#$ In!er%$#: these are the nubers de;ning theclass< consists o$ the end nubers called the c#$#ii! naely the upper #ii! and the #o&er #ii!.

C#$ frequency: sho"s the nuber o$ obser#ation

$alling in the class C#$ Bound$rie: these are the so called true class

liits! classi;ed as:

9o&er C#$ Bound$ry (9CB): de;ned as the

iddle #alue o$ the lo"er class liit o$ the classand the upper class liit o$ the preceding class

Upper C#$ Bound$ry: de;ned as theiddle #alue bet"een the upper class liit o$

the class and the lo"er liit o$ the net class


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C#$ i;e: the di=erence bet"een t"o consecuti#eupper liits or t"o consecuti#e lo"er liits

C#$$r


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Steps in ConstructingFDT:

Step +) Deterine the nuber o$ classes. For ;rstapproiation! it is suggested to use the STUR>ESAROI=ATION FOR=U9A.

62.244 #og n

"here $pproi$!e nu"er of c#$e

n nu"er of c#$

# +


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E$p#e* Ge no& con!ruc! !+e FDTof !+e !e!o!erone eru #e%e# of ,-&oen $ +o&n in !+e r$& d$!$*


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Step 2) Deterine the range @:

"here R $iu %$#ue:iniu %$#ue

R $:in

R 02:24

R 36


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Step ) Deterine the approiate class si8e C usingthe $orula

C R'

where R= range & K= Sturges Approximation

Formula

No!e* %t is usually con#enient to round o= C to anearest "hole nuber.

C R' C 36'3

C 6-.63/

C6-


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Step 5) Deterine the lo"est class inter#al (or the;rst class). This class should include the iniu#alue in the data set. For uni$ority! let us agreethat $or our purposes! the lo"er liit o$ the lo"est

class inter#al should start at the iniu #alue.

9e! u decide !o !$r! $! !+e iniu%$#ue. T+u !+e #o&e! c#$ i !+ec#$ 24:,6.


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Step *) Deterine all class liits by adding theclass si8e C to the liits o$ the pre#ious class.

The classes constructed by adding +, each class

liit. Thus "e ha#e:

24 ,6

,4 56

54 36

34 /6

/4 16

14 06

04 6-6

Step 3: Tally the scores or obser#ation $alling in each class


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Step 3: Tally the scores or obser#ation $alling in each class.

C#$e

24 : ,6

,4 : 56

54 : 36

34 : /6

/4 : 16

14 : 06

04 :6-6

T$##y

%%%%

%%%%'%%%%

%%%%

%%%%'%%%

%%%%'%%%

%%%%

%

Frequency

5

0

,

1

1

5

6

N,-


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T+e fo##o&ing !$"#e preen! !+e cop#e!e frequencydi!ri"u!ion !$"#e indic$!ing !+e c#$ "ound$rie7 !+ec#$ $r


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Frequency Distribution

Graphs %n a frequency di!ri"u!ion gr$p+! the score

categories (& #alues) are listed on the & ais

and the $requencies are listed on the A ais. G+en !+e core c$!egorie coni! of

nueric$# core fro $n in!er%$# or r$!ioc$#e7 !+e gr$p+ +ou#d "e ei!+er $

+i!ogr$ or $ po#ygon.


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Bistogras

%n a +i!ogr$! a bar is centered abo#e eachscore (or class inter#al) so that the height o$the bar corresponds to the $requency and the

"idth etends to the real liits! so thatadacent bars touch.


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olygons

%n a po#ygon! a dot is centered abo#e eachscore so that the height o$ the dot

corresponds to the $requency. The dots arethen connected by straight lines. Enadditional line is dra"n at each end to bringthe graph bac9 to a 8ero $requency.

2*


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?ar graphs

7hen the score categories (& #alues) areeasureents $ro a noinal or an

ordinal scale! the graph should be a bargraph.

E "$r gr$p+ is ust li9e a histograecept that gaps or spaces are le$t

bet"een adacent bars.

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@elati#e $requency

any populations are so large that it isipossible to 9no" the eact nuber o$

indi#iduals ($requency) $or any speci;ccategory.

%n these situations! population distributionscan be sho"n using re#$!i%e frequency

instead o$ the absolute nuber o$ indi#iduals$or each category.

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Sooth cur#e

%$ the scores in the population are easured onan inter#al or ratio scale! it is custoary to

present the distribution as a oo!+ cur%e rather than a agged histogra or polygon.

The sooth cur#e ephasi8es the $act that thedistribution is not sho"ing the eact $requency$or each category.

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Frequency distribution

graphs Frequency distribution graphs are use$ul

because they sho" the entire set o$ scores.

Et a glance! you can deterine the highestscore! the lo"est score! and "here the scoresare centered.

The graph also sho"s "hether the scores areclustered together or scattered o#er a "ide

range.

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Shape

E graph sho"s the +$pe o$ the distribution.

E distribution is ye!ric$# i$ the le$t sideo$ the graph is (roughly) a irror iage o$ theright side.

ne eaple o$ a syetrical distribution isthe bell'shaped noral distribution.

n the other hand! distributions are


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ositi#ely andIegati#elyS9e"ed Distributions

%n a poi!i%e#y


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ercentiles! ercentile

@an9s!and %nterpolation The relati#e location o$ indi#idual scores

"ithin a distribution can be described bypercentiles and percentile ran9s.

The percen!i#e r$n


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ercentiles! ercentile

@an9s!and %nterpolation (cont.) To ;nd percentiles and percentile ran9s! t"o ne"

coluns are placed in the $requency distribution table:ne is $or cuulati#e $requency (c$) and the other is $orcuulati#e percentage (cJ).

-ach cuulati#e percentage identi;es the percentileran9 $or the upper real liit o$ the corresponding scoreor class inter#al. 7hen scores or percentages do notcorrespond to upper real liits or cuulati#epercentages! you ust use interpolation to deterinethe corresponding ran9s and percentiles. In!erpo#$!ion is a atheatical process based on the assuption thatthe scores and the percentages change in a regular!linear $ashion as you o#e through an inter#al $ro oneend to the other.

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%nterpolation

7hen scores or percentages do notcorrespond to upper real liits orcuulati#e percentages! you ust useinterpolation to deterine thecorresponding ran9s and percentiles.

In!erpo#$!ion is a atheatical processbased on the assuption that the scores

and the percentages change in a regular!linear $ashion as you o#e through aninter#al $ro one end to the other.

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Ste'and'>ea$ Displays

E !e:$nd:#e$f dip#$y pro#ides a #eryeKcient ethod $or obtaining anddisplaying a $requency distribution.

-ach score is di#ided into a !e consisting o$ the ;rst digit or digits! and a

#e$f consisting o$ the ;nal digit. Finally! you go through the list o$ scores!

one at a tie! and "rite the lea$ $or eachscore beside its ste.

The resulting display pro#ides anorgani8ed picture o$ the entire distribution. The nuber o$ lea$s beside each stecorresponds to the $requency! and theindi#idual lea$s identi$y the indi#idualscores.


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D i ti St ti ti


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Descripti#e Statistics

Class E''%Ls o$ + Students

+,2 ++*

+24 +,1++ 41

14 +,3

+5, ++1

1 16++,

Class ?''%Ls o$ + Students

+26 +32

++ +,13 +++

4, +,1

1 46

+2, +,*+,1

'a"()e ))ustrat%on+

.%c. /rou( %s '"arter0

Each individual may be different. If you try to understand agroup by remembering the qualities of each member, you

become overwhelmed and fail to understand the group.


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Descripti#e Statistics

Which group is smarter now?

Class A--Average IQ Class B--Average IQ

110.54 110.23

They’re roughly the same!

With a summary descriptive statistic, it ismuch easier to answer our question.


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ther Graphs

Beide Hi!ogr$7 !+ere $re o!+ere!+od of gr$p+ing qu$n!i!$!i%ed$!$*

0 S!e $nd 9e$f #o!0 Do! #o!

0 Tie Serie


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Ste and >ea$ lots

Repreen! d$!$ "y ep$r$!ing e$c+ d$!$ %$#ue in!o !&op$r!* !+e !e (uc+ $ !+e #ef!o! digi!) $nd !+e #e$f (uc+$ !+e rig+!o! digi!)

Larson/Farber 4th ed. 49


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Constructing Ste and >ea$

lots0 Split each data #alue at the sae place #alue to $or the

!e and a #e$f . (7ant *'2, stes).

Errange all possible stes #ertically so there are noissing stes.

7rite each lea$ to the right o$ its ste! in order.

Create a 9ey to recreate the data.

Mariations o$ ste plots:1. Split stems

2. Back to back stem plots.

+

C t ti St d


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Constructing a Ste'and'>ea$ lot

1


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Dot lots

Do! p#o!

0 Consists o$ a graph in "hich each data #alue isplotted as a point along a scale o$ #alues

igure $'(

i i


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Tie Series(aired data)

Tie Serie

Data set is coposed o$ quantitati#e entriesta9en at regular inter#als o#er a period o$ tie.

e.g.! The aount o$ precipitation easuredeach day $or one onth.

Use a !ie erie c+$r! to graph.

tie

L u a n t i t a t i #

e

d a

t a

Time')eries %raph


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Time )eries %raph

/umber of 0creens at rive#%n ovies

(heaters

igure $'*


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Graphing Lualitati#e Data Sets

ie C+$r!

E circle is di#ided intosectors that representcategories.

+areto hart

A vertical bar graph in which theheight of each bar represents

fre!uency or relative fre!uency.

Categories

F r e q u e n

c y

C t ti i Ch t


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Constructing a ie Chart

Find !+e !o!$# $p#e i;e.

Con%er! !+e frequencie !o re#$!i%e frequencie (percen!).

-

ar%ta) 'tatus Frequency, f( %n "%))%ons

e)at%e frequency

!eer arr%e& 553

arr%e& 1277

%&oe& 139

%orce& 228

(otal4 215.)

55325 or 25

2197≈

1277

2197≈

139

2197≈

228

2197≈


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Constructing areto Charts

0 Create a bar $or each category! "here the heighto$ the bar can represent $requency or relati#e$requency.

The bars are o$ten positioned in order o$

decreasing height! "ith the tallest bar positionedat the le$t.

BIOSTAT Chapter2

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