methods for describing sets of data

7/26/2019 Methods for Describing Sets of Data

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Chapter2

MethodsforDescribingSetsofData


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DescriptiveStatistics

Descriptivestatisticsutilizesnumericalandgraphicalmethodsto

lookforpatternsinadataset,tosummarizetheinformation

revea e na atasetan topresentt at n ormat on na

convenientform.

DescribedatabyusingGraphs(Graphical)

Boxplots,Histograms,PieChart,BarChart,Scatterplots,etc.

Describedatabyusingnumericalmeasures(Numerical)

CentralTendencies,Variability,Range,Outliers,etc.


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DescribingQualitative

Data

Qualitativedataarenonnumerical

Field

of

study,

Political

Party,

Gender,

Eye

color,

etc.

Summarizedintwowa s:

ClassFrequency:thenumberofobservationsinthedataset

thatfallintoaparticularclass.

ClassRelativeFre uenc :classfre uenc dividedb thetotalnumberofobservationsinthedataset.

class relative frequency = n


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DescribingQualitativeDataxamp e:

e

nven ory

Specie FrequencyRelative

Frequency Proportion

A. chi lensi s 3 0.20 20%

N. alpina 8 0.53 53%

. om ey .


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DescribingQualitativeData

BarGraph:Thecategories(classes)ofthequalitativevariableare

xamp e:

e

nven ory

represen e y ars,w ere e e g o eac ar se er ec ass

frequency,class

relative

frequency

or

class

percentage.

8

7

6

N. dombeyi

3

5

4N. alpina

2

1

A. chilensis

0

N. dombeyiN. alpinaA. chilensis876543210


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PieChart:Thecategories(classes)ofthequalitativevariableare

xamp e:

e

nven ory

representedbyslicesofapie. Thesizeofeachsliceisproportional

tothe

class

relative

frequency.

A. chilensis (20.00%)

. alpina (53.33%)

N. dombeyi (26.67%)


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DescribingQualitativeDatas ea ng

p ots

SOURCE: Los Angeles Times, SOURCE: Gra hJam.comAugust 5, 1979.


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Contin enc Table:Crosstabulation of units based on

measurementsof

two

qualitative

variables

simultaneously.

Stac e BarGrap :Barc artwit onevaria erepresente on

thehorizontalaxis,secondvariableassubcategorieswithinbars.

ClusterBarGraph:Barchartwithonevariableformingmajor

groupingsonhorizontalaxis,secondvariableusedtomakeside

.


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ContingencyTable

AMS No AMS Total

Acet 14 104 118

Ginkgo 43 81 124

Acc+Gi 18 108 126

Total 115 372 487

AMS No AMS Total

Placebo 33.61 66.39 100Acet 11.86 88.14 100

Ginkgo 34.68 65.32 100

. .


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ClusterBarGraph StackedBarGraph

0.32

0.34

GRAD

PATCH

0.26

0.28

0.30

H2

0.22

0.24

CR RCB IB 4 IB 8 IB 16 IB 32 R-C

.


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3DBarGraph


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GraphicalMethodsforDescribing

uan a ve

a a

s ng evar a e

Dotplotsdisplayadotforeachobservationalongahorizontal

numberline

Duplicatevaluesarepiledontopofeachother

Thedotsreflectthesha eofthedistribution

Goodforsmalldatasets


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AStemandLeafDisplayshowsthenumberofobservationsthat

uan a ve

a a

s ng evar a e

shareacommonvalue(thestem)andtheprecisevalueofeach

observation(the

leaf)

Stem- and- l eaf di s l a f or A e

Number of observat i ons: 15. Mi ni mum: 11. 0. Maxi mum: 45. 0.St em uni t s: 10, l eaf di gi t s: 1 ( t he val ue 11. 00 i s r epr esent ed by 1| 1)

8 1| 12225678

2 3| 12

2 4| 25


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Thebox lotis a ra h re resentin information about certain

uan a ve

a a

s ng evar a e

percentiles

for

a

data

set

and

can

be

used

to

identify

outliers.

More later!


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Histogramsaregraphsofthefrequencyorrelative

uan a ve

a a

s ng evar a e

frequencyofavariable.

Classintervalsmakeupthehorizontalaxis

axis.

8

6

7

5

2

4

3

Frequency

0

504540353025201510

Age (years)


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InterpretingHistograms

Probability:Heightsofbarsovertheclassintervalsare

proportionaltothechancesanindividualchosenatrandom

wouldfallintheinterval.

Unimodal:Ahistogramwithasinglemajorpeak.

Bimodal:Histogramwithtwodistinctpeaks(oftenevidenceof

two

distinct

groups

of

units). Uniform:Intervalheightsareapproximatelyequal.

Symmetric:RightandLeftportionsaresameshape.

RightSkewed:Right

hand

side

extends

further.

LeftSkewed:Lefthandsideextendsfurther.


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InterpretingHistograms

Unimodal

Bimodal

Uniform

Symmetric RightSkewed

e t ewe


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GraphicalMethodsforDescribingQuantitativeData

Scatterplotshowstherelationshipbetweentwoquantitative

var a es.

Responsevariable

(y)

placed

on

the

vertical

(up/down)

axis

and

the

explanatoryvariable(x)placedonthehorizontal(left/right)axis.

50

40

45

30

35

Weight

275

20

250

15

300 400

25

375350325

Length


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Timeseriesplots:(anotherformofscatterplot)manydatasets

points.When

measurements

are

made

at

equally

spaced

time

points,

goalisoftentodescribetemporalvariation.


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Complexplots

(matrixscatterplot)


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GraphicalMethodsforDescribingQuantitative

Recommendations

Complete,clear,brief.


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Notation

Individualobservationsinadatasetaredenoted

x , x , x , x , x .

Summationnotation:we

use

asummation

symbol

often:

n

n

i

i xxxxx ++++=

=

...3211

1

tothelast(xn).

xamp e:

x1=

,

x2=

,

3=

an

x4=

,

1043214

=+++=x1=i


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NumericalDescriptiveMeasures

Numericsummariesofasetofmeasurements.

setof

measurements.

Value

or

values

around

which

the

data

tend

to

cluster.

MeasuresofVariabilitydescribethespreadordispersionofaset

ofmeasurements.Howstronglythedatacluster.

Parameters:NumericdescriptivemeasuresbasedonPopulationsof

measurements.

Statistics:Numericdescriptive

measures

based

on

Samplesof

measurements.


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NumericalMeasuresofCentralTendency

Themeanofasetofquantitativedataisthesumoftheobserved

valuesdividedbythenumberofvalues

n

ix x

N

iix

n

==N

i== 1

Example:x1 = 1,x2 = 2,x3 = 3 andx4 = 4 (i.e. n = 4)

, ,

= 1 + 2 + 3 + 4 /4 = 10/4 = 2.51

n

i

i

x

x ==

n


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Themedian(M)ofasetofquantitativedataisthevaluewhichis

locatedinthemiddleofthedata,arrangedfromlowesttohighest

values(orviceversa),with50%oftheobservationsaboveand50%

below.

50% 50%

Lowest Value Highest ValueMedian

Findingthe

Median,

M:

Arrangethenmeasurementsfromsmallesttolargest:

n so , s em enum er.

Ifniseven,Mistheaverageofthemiddletwonumbers.


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Themodeisthemostfrequentlyobservedvalue(continuous)

Themodalclassisthemidpointoftheclasswiththehighest

relativefrequency(discrete).

Mode (Age): 12

o a c ass pec e : . a p na

8 1| 122256783 2| 1882 3| 122 4| 25

Median: 18


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TrimmedMean(TM):Meanthatisbasedoncentermeasurements

.

Skewness:Sha eofthedistribution:

MoundShapedDistributions: Mode MedianMean

Right

Skewed

Distributions:

Mode

3isconsideredanoutlier.


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Example

The length of a total of 20 fruits were

measured from two different bags.

data Example ;

input ID Length;

datalines;1 122 15

ID Length Bag ID Length Bag

1 12 A 10 27 A2 15 A 17 26 B

3 22 A 9 25 A

3 22

4 18

5 226 14

7 13

5 22 A 15 23 B

6 14 A 3 22 A

7 13 A 5 22 A

8 18 A 11 21 B

9 2510 27

11 2112 17

13 25

10 27 A 20 20 B

11 21 B 16 19 B

12 17 B 4 18 A

13 25 B 8 18 A

14 18 B 14 18 B

15 2316 1917 26

18 619 20

15 23 B 12 17 B

16 19 B 2 15 A

17 26 B 6 14 A

18 6 B 7 13 A

19 20 B 1 12 A

;

proc print;run;

proc univariate data=Example

20 20 B 18 6 B plots;

var Length;run;


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Example

ID Length Bag

10 27 A

17 26 B

13 25 B

15 23 B

3 22 A

5 22 A

19 20 B

20 20 B

16 19 B

4 18 A

8 18 A

14 18 B

12 17 B

2 15 A

6 14 A

7 13 A

1 12 A

18 6 B


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Example

ID Length Bag

10 27 A

17 26 B

9 25 A

13 25 B

15 23 B3 22 A

5 22 A

26

28

11 21 B

19 20 B

20 20 B

16 19 B

4 18 A

18

22

24

20

8 18 A

14 18 B

12 17 B

2 15 A

6 14 A

10

14

16

12

1 12 A18 6 B

186

Length

8

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