Data Presentation and Analysis

7/23/2019 Data Presentation and Analysis

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Presentation

and Analysis ofStatistical Data



Statistics

• A science of conducting studies to collect,organize, summarize, analyze, and drawconclusion from data; interpreting and

presenting numerical data.• Can refer to the mere tabulation of

numeric information as in reports of stock,market, transactions, or to the body of

techniues used in processing or analyzingdata.



Two Branches of Statistics

Statistics can be organized into twobranches!

• Descriptive statistics Concerned with collecting, organizing,presenting, and analyzing numerical

data.• Inferential statistics

"ts main concern is to analyze theorganized data leading to prediction or

inferences.



Some Important Concepts

• Data

Data are the raw material which the statistician works. Datacan be found through sur#eys, e$periments, numericalrecords, and other modes of research.

• Population

%efers to the groups or aggregates of people, ob&ects,materials, e#ents or thing of any form.

• Sample

Consist of few or more members of the population.• Outliers

An outlier is a data point that lies outside the o#erall patternof a distribution.



Variables - types

Statistical data or information can be gatheredthrough di'erent ways such as inter#iewingpeople, obser#ing or inspecting items, usinguestionnaires and checklists. (he characteristic

that is being studied is called a variable. "t#aries from one person or thing to another.

• )umerical #alues and are examples ofquantitative variables.

• )on*numerical #alues or attributes are examples of qualitative variables.



Dependent and Independent

Variables

A #ariable can be dependent or independent depending on its use.

(o predict the #alue of #ariable on the other,

Independent variable is the predictor while thedependent variable is the #ariable whose #alue

is being predicted.



Scales of Measurement

of Data• ominal Data

+se numbers for the purpose of identifying name or membership in a group orcategory.

• Ordinal Data

Connote ranking or ineualities in this type of data, numbers represents

greater than- or less than- measurement, such as preferences or rankings.

• Inter!al Data

"ndicate an actual amount and there is eual unit of measurementseparating each score, specically eual inter#als. (he true zero ispresent.

• "atio Data Similar to inter#al data but has an absolute zero and multiples

are meaningful. "t include all the usual measurement of length,height, weight, area, #olume, density, #elocity, money andduration.



Types of Data

Primary Data * data collected directlyby the researcher himself. (heseare rst*hand or original sources.

(hey can be collected through!/. Direct obser#ation or measurement 0primary

source of info1.2. 3y inter#iew 0uestionnaires or rating

scales1.4. 3y mail of recording or of recording forms.5. 6$perimentation.



• Secondary Data * "nformation taken frompublished or unpublished materials pre#iouslygathered by other researchers or agencies suchas book, newspapers, magazines; &ournals,published and unpublished thesis anddissertations.



Methods in Presentin#

Data

• Textual Form * data in paragraphform.

• Tabular Form * systematicarrangement of data in rows andcolumns.

• Graphical Form * a graph or chart isa de#ice for showing numerical#alues in pictorial form.

• Semi Tabular/Semi Tabular Form *

the combination of (e$tual and



Tabular Presentation

• +sed for summarization and tabulation of data.

• 8elps in analysis of relationships, trends,

relati#e size of data.

• Parts of a (able (itle and numberCaption 0 column heading1

Stub 0row heading1

Sources and footnotes 0optional1



$raphical Presentation

(ypes of graphs or Charts!

• 9ine Plots

• 9ine Chart• 3ar Chart

• Stem and leaf plots

• Pie Charts

• 3o$ Plots

• Pictograms

• Scatter Diagrams



%ine Plots

Consists of

• a horizontal number line of the possible data #alues;• one : for each element in the data set placed o#er the

corresponding #alue on the number line.

orks well when

• the data is uantitati#e 0numerical1;• there is one group of data 0uni*#ariate1;• the data set has fewer than <= #alues;• the range of possible #alues is not too great.



%ine Plot &'ample

Suppose thirty people li#e inan apartment building. (heages of the residents arebelow.

<>, 4=, 4?, 4@, 45, 5, 4<,

5=, 5?, 5?, 4, <5, 5?, 5>,<5, <=, 4<, 5=, 4>, 5?, 5>,45, 5=, 5@, 5, 5?, 4<, 5>,5?, 5@

(he graph is easier to createwhen the ages are placed inorder from largest tosmallest as the #alues willappear on the number line.

4=, 45, 45, 4<, 4<, 4<, 4@,4?, 4>, 4, 5=, 5=, 5=, 5@,5@, 5?, 5?, 5?, 5?, 5?, 5?,5>, 5>, 5>, 5, 5, <=, <5,<5, <>



(d!anta#es of %ine Plots

• (he plot shows all the data.

• 9ine plots allow se#eral features of the datato become more ob#ious, including any

outliers, data clusters, or gaps.

• (he mode is easily #isible.

• (he range can be calculated uite easilyfrom this data display.



Disad!anta#es of %ine

Plots

• A line plot may only be used foruantitati#e 0numerical1 data.

• A line plot is not eBcient when the data islarge andor the the range is large.



%ine $raph

Consists of

• paired #alues graphed as points on a plane

dened by an x * and y *a$is;• line segments connecting the graphed points

0much like a dot*to*dot1.

orks well when

• the data is paired 0bi*#ariate1;• the data is continuous.



%ine $raph &'ample

ohn weighed @> kg in //, ?= kg in /2, ?5 kg in /4,?5 kg in /5, and ?4 kg in /<.



(d!anta#es of %ine

$raphs

• A line graph is a way to summarize howtwo pieces of information are related andhow they #ary depending on one another.

Disad!anta#es of %ine

$raphs

• Changing the scale of either a$es candramatically change the #isual impressionof the graph



Bar $raph

Consists of

• bars of the same width drawn either horizontally or#ertically;

• bars whose length 0or height1 represents thefreuencies of each #alue in a data set.

orks well when

• the data is numerical or categorical;• the data is discrete;• the data is collected using a freuency table.



Bar $raph &'ample



(d!anta#es of Bar

$raphs

• (he mode is easily #isible.

• A bar graph can be used with numerical

or categorical data.

Disad!anta#es of Bar

$raphs• A bar graph shows only the freuencies of

the elements of a data set.



Stem and %eaf Plot

Consists of

• )umbers on the left, called the stem, which are the rst halfof the place #alue of the numbers 0such as tens #alues1;

• )umbers on the right, called the leaf, which are the secondhalf of the place #alue of the numbers 0such as ones #alues1so that each leaf represents one of the data elements.

orks well when

•the data contains more than 2< elements;• the data is collected in a freuency table;

• the data #alues span many tens- of #alues.



Stem and %eaf Plot

&'ample (he number of points scored by the Eikings basketball team thisseason! ?>, @, >>, ?5, @4, >@, 2, @@, ?2, >>, >4, =, @?, >/, ><,5. riting the data in numerical

order may help to organize thedata, but is )F( a reuiredstep.

@4, @@, @?, ?2, ?5, ?>, >/, >4,><, >@, >>, >>, =, 2, 5, @

Separate each number into a

stem and a leaf. Since theseare two digit numbers, the tensdigit is the stem and the unitsdigit is the leaf.

(he number @4 would be

represented as

Hroup the numbers with thesame stems. 9ist the stems innumerical order. (itle the

graph.

Stem Leaf

6 3

Points scored by the Vikings

Stem Leaf

6 3 6 77 2 4 8

8 1 3 5 6 8 8

! 2 4 6



(d!anta#es of

Stem and %eaf Plots

• "t can be used to uickly organize a large list of data #alues.• "t is con#enient to use in determining median or mode of a

data set uickly.

• Futliers, data clusters, or gaps are easily #isible.

Disad!anta#es of

Stem and %eaf Plots• A stem and leaf plot is not #ery informati#e for a small set

of data.



Pie Chart

Consists of

• a circle di#ided into sectors 0or wedges1 that show the

percent of the data elements that are categorized similarly.

orks well when

• there is only one set of data 0uni*#ariate1;• comparing the composition of each part to the whole set of

data.



Pie Chart &'ample

"o#or $%mber

&hite 1

'#ack 25

(ray 11

)ed 18

' #%e 7*ther 1!

+ota# !

A proportion can be used to calculate the anglemeasure for each sector. +sing white as the e$ample,/ white cars compare to the total of = in the sameway that ?@ degrees compares to the total degrees04@=1 in a circle.



(d!anta#es of Pie

Charts• can be used for either numerical or categorical data.

• shows a part to whole relationship.

Disad!anta#es of Pie

Charts

• ithout technology, a pie chart may be diBcult to make.

6ach percent must be con#erted to an angle bycalculating the fraction of 4@= degrees. (hen the correctangle must be drawn.

• A circle graph does not pro#ide information aboutmeasures of central tendency or spread.



Bo' and )his*er PlotConsists of

• the #e*point summary- 0the least #alue, the greatest#alue, the median, the rst uartile, and the thirduartile1;

• a bo$ drawn to show the inter#al from the rst 02< th percentile1 to the third uartile 0?<th percentile1 with a linedrawn through the bo$ at the median;

• line segments, called the whiskers, connecting the bo$ tothe least and greatest #alues in the data distribution.

orks well when

• there is only one set of data 0uni*#ariate1;

• there are many data #alues.

B d )hi * Pl t



Bo' and )his*er Plot

&'ampleIath test scores >=, ?<, =, <, @<, @<, >=, ><, ?=,

/==.rite the data in numericalorder and nd the #e pointsummary,,

median - 8!first .%arti#e - 7!third .%arti#e - !

sma##est /a#%e - 65

#argest /a#%e - 1!!

650 650 7!0 750 8!0 8!0 850 !0 50 1!!

Place a point beneatheach of these #alues on anumber line,

Draw the bo$ andwhiskers and median line.

edian

edian of Loer Part0

irst %arti#e

edian of er Part0

+hird %arti#e

85757!65 8! ! 1!!5

85757!65 8! ! 1!!5



Bo' and )his*er Plot

&'ample (he following set ofnumbers are the amount0arranged from least togreatest1 of #ideo gamesowned by each boy in theclub.

/> 2? 45 <2 <5 < @/ @>?> >2 >< >? / 4 /==68 is the meian

(he median is the

#alue e$actly in themiddle of an orderedset of numbers.

!" is the lo#er $uartile (he lower quartile is the median ofthe lower half of the #alues 0/>2? 45 <2 <5 < @/1.

8% is the upper $uartile

(he upper quartile is the medianof the upper half of the #alues 0?>,>2, ><, >?, /, 4, /==1.



(d!anta#es of Bo' and

)his*er Plots• "mmediate #isuals of a bo$*and*whisker plot are the center, the

spread, and the o#erall range of distribution.

• 3o$ plots are useful for comparing data sets, especially when the datasets are large or when they ha#e di'erent numbers of data elements.

Disad!anta#es of Bo' and

)his*er Plots

• "t shows only certain statistics rather than all the data.

• Since the data elements are not displayed, it is impossible todetermine if there are gaps or clusters in the data.



Scatter Plot

Consists of

• paired data 0bi*#ariate1 displayed on a two*

dimensional grid.

orks well when

• multiple measurements are made for eachelement of a sample.



(dditional otes about

Scatter plots• "f the relationship is thought to be a causal one,

then the independent #ariable is representedalong the $*a$is and the dependent #ariable onthe y*a$is

• A scatter plot can show that there is a positi#e,negati#e, constant, or no relationship0correlation1 between the #ariables. Positi#e! As the #alue of one #ariable increases, so

does the other.

)egati#e! As the #alue of one #ariable increases, theother decreases.

Constant! As the #alue of one #ariable increases 0ordecreases1, the other remains constant.

)o relationship! (here is no pattern to the points.



Scatter plot &'ample



(d!anta#es of Scatter plots

• A scatter plot is one of the best ways to determine if twocharacteristics are related.

• A scatter plot may be used when there are multiple trials for thesame input #ariable in an e$periment.

Disad!anta#es of Scatter plots

• hen a scatter plot shows an association between two #ariables,there is not necessarily a cause and e'ect relationship. 3oth#ariables could be related to some third #ariable that e$plains their#ariation or there could be some other cause. Alternati#ely, anapparent association could simply be a result of chance



+isto#ram

Consists of

• eual inter#als marked on the horizontal a$is;

• bars of eual width drawn for each inter#al, with the heightof each bar representing either the number of elements or

the percent of elements in that inter#al.0(here is no space between the bars.1

orks well when

• data elements could assume any #alue in a range;• there is one set of data 0uni*#ariate1;

• the data is collected using a freuency table.



+isto#ram &'ample



(d!anta#es of

+isto#rams

• A histogram pro#ides a way to display thefreuency of occurrences of data along an inter#al.

Disad!anta#es of +isto#rams

• (he use of inter#als pre#ents the calculation of ane$act measure of central tendency.



Frequency Polygon

Consists of

"t is a line chart that is constructed by plotting

the freuencies and class mark and connectingthe plotted pointed by means of a straight line;the polygon us closed by considering anadditional class at each end and each end of thelines are brought down to the horizontal a$is atthe mid point of the additional classes.



Frequency Polygon

(Data)

C.I. f <cf X <cf f/n sector

80-89 1 40 84.5 1 0.0025 9percen

t

70-79 4 39 74.5 5 0.1000 3660-69 13 35 64.5 18 0.3250 117

50-59 13 22 54.5 31 0.3250 117

40-49 4 9 44.5 35 0.1000 36

30-39 3 5 34.5 38 0.0750 27

20-29 1 2 24.5 39 0.0250 9

10-19 1 1 14.5 40 0.0250 9

n=40 ∑rf=1





(d!anta#es of ,reuency

Poly#on

• A 7reuency Polygon pro#ides a way to display thefreuency of occurrences of data along an inter#al.

• Denotes Class Iark

Disad!anta#es of ,reuency

Poly#on

• (he use of inter#als pre#ents the calculation of ane$act measure of central tendency.



Ogive

Consists of

• Hraph of a cumulati#e freuency distribution andsometimes called a cumulati#e freuencydistribution graph

orks well when

• 9ess (han And Iore (han inter#als are reuired.



Ogive



(d!anta#es of O#i!e

• A Fgi#e pro#ides a way to display the cumulati#efreuency of occurrences of data along an inter#al.

Disad!anta#es of O#i!e

• (he use of Cumulati#e inter#als pre#ents the

calculation of an e$act freuency and measure ofcentral tendency.



M&(S."&S O, C&T"(% T&D&C/

*single number represent the gi#en data.

&'(ean J a#erage #alue of the gi#en data.

* not appropriate measures of centraltendency if there is outlier.

2. (eian J di#ide the distribution into two eualparts 0upper <=K and the lower <=K1

4. (oe J the most freuent occuring data.

* nominal #aluepart.



Iean



.$"O.P&D D(T(

2< 42 5/ <> ?> < /=< //= //2 //2//<

L >>4

//L >=.2?2?

Iean L M :

n



$"O.P&D D(T(

L 22<=

5=

Iean L <@.2<

Iean L Mf$ n



Short Cut Method

L @5.<N 0*445=1/=

L @5.<*>.2<

Iean L <@.2<

Iean L AI 0Mfdn1i



Iedian

di 0



L 5.< N 02=*/41/=

L 5.< N 0///41 /=

L 5.< N >.5@/<

Iedian L<?.@/<

Iedian L 99N0n2*Ocf1 i

f



Iode



where Q/ L di'erence between the modal class and the ne$t lowerscore.

Q2 L di'erence between the modal class and the ne$tupper score.

L <= N 0/4/4 N/41/= Iode L <<

IodeL 99 N 0Q/ Q/N Q21 h



0.("TI%&102



Three 0uartiles

• R/* 2<K /5 7irst Ruartile

• R2* <=K /2 Second Ruartile Iedian

• R4* ?<K 45 (hird Ruartile



R/ L 99 N n5 * Ocf i f

L 5.< N /=* /=

/4

L 5.< N 0//41 /=

L 5.< N =.?@2

R/ L <=.2@2



R4 L 99 N 4n5 * Ocf "

f

L <.< N 4=*22 /= /4

L <.< N 0>=/41 /=

L <.< N @./<4>

R4 L @<.@<4>



P&"C&TI%& 1P2

• P/* //==

• P2* 2/==

• P4* 4/==

• P5* 5/==• P<* </==

..

• P* /==



P24 L 99 N =.24n * Ocf i f L 5.< N .2 * /=

/4

L 5.< N 0=.2/41 /=

L 5.< N 0=.=/<41/=

L 5.< N =./<4>

P24 L 5.@<4>



Measures of Variability

•Ieasure of the scatteredness of a particulardata in a gi#en data set.

•A#erage of distance

•+sed to determine the reliability of a#erage

#alues and control #ariation

,ollowin# are the



,ollowin# are the

measures of !ariation !• %ange * Di'erence between #alues of the two e$treme

items of a series

• "nteruartile %ange J Di'erence between third and

7irst Ruartile

• Iean A#erage De#iation J the arithmetic mean of theabsolute di'erences of the #alues from their a#erage.

• Standard De#iation J positi#e Suare root of thea#erage of Suares of De#iations of #alues from theirmeans.T



"an#e



% L :9 J :s

6$ample !Age of a sample of /= sub&ects from apopulation are as follows!52, 2>, 2>, @/, 4/, 24, <=, 45, 42, 4?.

%ange L @/ J 24 L 4>.

Coe' of %ange L 0:9 J :s1 0:9 N :s1 L 0@/ J 241 0@/ N241

L =.5<2



Inter0uartile

"an#e1I0"2



"R% L 0R4

J R/

1

Semi Ruartile %ange L 0R4 J R/12

Coe' of Ruartile De#iation L 0R4 J R/1 0R4 NR/1



Mean De!iation

• 7or +ngrouped Data !



• 7or +ngrouped Data !

IAD L MU$*$U n

• 7or Hrouped Data !

IAD L Mf iU$i*$U

Mf i

Coe' of IAD !

L 0IAD Iean1



Standard De!iation

7or +ngrouped Data



7or +ngrouped Data!

V L WM0:i J :12n

7or Hrouped Data!

V L WM f i0:i J :12Mf i

Coe' of Eariation L 0V :1 : /==

Eariance L V2

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Data Presentation and Analysis