Lecture 2 - Descriptive Statistics Part I.pdf

Descriptive Statistics

1. DESCRIPTIVE
STATISTICS
ENS185 2
nd
Semester

2. DESCRIPTIVE STATISTICS
❑Organizing and Summarizing Data
❑Numerically Summarizing Data
❑Describing Relationship between Two
Variables

3. PROCESS OF STATISTICS
Introduction to Data Analysis 3
Identify the
research objective
Collect the data
needed
Describe the
data
Perform
inference
Aresearchermustdeterminethequestionsheorshewantanswered.The
questionsmustclearlyidentifythepopulationthatistobestudied.
Conductingthedataonthewholepopulationisimpracticalandexpensive.
However,appropriatedatacollectiontechniquesmustalsobefollowed.
Describethedatacollectedusingnumericalandvisualtools.Itgivesusan
overviewofthedataandcanhelpusdeterminewhichstatisticaltoolstouse
forinference.
Applytheappropriatetechniquestoextendtheresultsobtainedfromthe
sampletothepopulationandreportalevelofreliabilityoftheresults.

4. ORGANIZING AND
SUMMARIZING YOUR DATA
Organizing Qualitative Data
Organizing Quantitative Data
Graphical Misrepresentations of Data

5. ORGANIZING QUALITATIVE
DATA

6. 6
Recall:Qualitativedataprovidemeasures
thatcategorizeorclassifydatabasedona
characteristicorattribute.

7. ORGANIZING QUALITATIVE DATA
Descriptive Statistics 7
1. USE FREQUENCY DISTRIBUTION TABLES
A frequency distribution lists each
category of data and the number of
occurrences for each category of data.
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Day of the
Week Frequency
Relative
Frequency
Monday 52 52/400=0.1300
Tuesday 66 66/400=0.1650
Wednesday 72 72/400=0.1800
Thursday 57 57/400=0.1425
Friday 57 67/400=0.1425
Saturday 43 43/400=0.1075
Sunday 53 53/400=0.1325
Total 400 1.00
Whenrawqualitativedataiscollected,weoftenfirstdeterminethecategoriesandthe
numberofindividualsorunitsineachcategory.Wethenorganizethisintoatable.
Relative frequency is the proportion of
observations within a category.

8. ORGANIZING QUALITATIVE DATA
Descriptive Statistics 8
2. CONSTRUCT BAR GRAPHS
A bar graph is constructed by labeling
each category of data on either the
horizontal or vertical axis and the
frequency or relative frequency of the
category on the other axis. Rectangles
of equal width are drawn for each
category. The height of each rectangle
represents the category’s frequency or
relative frequency.
Oncethedataisorganizedinatable,wecancreategraphs.

9. ORGANIZING QUALITATIVE DATA
Descriptive Statistics 9
2.CONSTRUCTBARGRAPHS:APARETOCHART
A Pareto chartis a bar
graph whose bars are
drawn in decreasing order
of frequencyor relative
frequency.

10. ORGANIZING QUALITATIVE DATA
Descriptive Statistics 10
2.CONSTRUCTBARGRAPHS:SIDE-BY-SIDEBARGRAPHS
Side-by-side bar graphs can
also be used to compare
two different data sets.

11. ORGANIZING QUALITATIVE DATA
11
2.CONSTRUCTBARGRAPHS:Horizontalbargraphs
Horizontal bar graphs can be
drawn when the category names
are lengthy.
Descriptive Statistics

12. ORGANIZING QUALITATIVE DATA
12
3. CONSTRUCT PIE CHARTS
A pie chart is a circle divided
into sectors. Each sector
represents a category of
data. The area of each sector
is proportional to the
frequency of the category.
Piechartsareusedtopresenttherelativefrequencyofqualitativedata.
Descriptive Statistics

13. ORGANIZING QUANTITATIVE
DATA

14. IN SUMMARIZING QUANTITATIVE
DATA, WE FIRST DETERMINE
WHETHER THE DATA IS
DISCRETE OR CONTINUOUS.
14

15. 1. ORGANIZE DISCRETE DATA IN TABLES
15
Use the values of the discrete
variable to create the classes
when the number of distinct
data values is small.
Iftherearerelativefewdifferentvalues,thenthecategories(calledclasses)willbethe
observations(justlikequalitativedata).
Descriptive Statistics

16. 2. CONSTRUCT HISTOGRAMS OF DISCRETE DATA
16
A histogram is constructed by
drawing rectangles for each class
of data. The height of each
rectangle is the frequency or
relative frequency of the class. The
width of each rectangle is the
same and the rectangles touch
each other.
Thehistogramissimilarwithabargraph.
Relativefrequencycanalsobeusedto
createahistogram.
Descriptive Statistics

17. 3. ORGANIZE CONTINUOUS DATA IN TABLES
17
Classesarecategoriesintowhichnumericaldataaregrouped.
Descriptive Statistics
Lower class limit is
the smallest value
within the class.
Upper class limit is
the largest value
within the class.
Class width is the
difference between
consecutive lower
class limits
45-40=5
Note:Open-endedtables
aretableswherethefirst
classhasnolowerclass
limitorthelastclasshas
noupperclasslimit.

18. 3. ORGANIZE CONTINUOUS DATA IN TABLES
18
GUIDELINESINCONSTRUCTING CONTINUOUS DATATABLES
1.Decideonthenumberofclasses.Generally,thereshouldbebetween5and20
classes.Youcanalsousethe2krule.
2
??????
≥�
Wherek=numberofclasses
n=samplesize
2.Determinetheclasswidthbycomputing
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??????��??????���−�??????�??????���
���������������
Roundupthisvaluetoamoreconvenientnumber.
3.Chooseavaluebelowthesmallestvalueasthelowerclasslimitforthefirstclass
limit.
Descriptive Statistics

19. 3. ORGANIZE CONTINUOUS DATA IN TABLES
19
EXAMPLE
1.Thesamplesizeisn=50.
2.Usingthe2krule.
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�
≥�??????
3.Theclasswidthis:
??????�����??????��ℎ=
322.61−65
6
=42.94≈43
Descriptive Statistics

20. 3. ORGANIZE CONTINUOUS DATA IN TABLES
20
EXAMPLE
4.Startingwith65,
65 –102.99
108 –150.99
151 –193.99
194 –236.99
237 –279.99
280 –322.99
Descriptive Statistics
Alltheclassesshouldincludeallthemembersof
thedataset.
Extendthesignificantfiguresbasedonthegiven
data.
Onceyouhaveestablishedyourtable,countthe
numberofdatapercategory.

21. 4. CONSTRUCT HISTOGRAMS OF CONTINUOUS DATA
21Descriptive Statistics
Onceyouhavecreatedyourtable,youcancreateahistogram.Unlikethebargraph,a
histogramhasnospacesbetweenthebars.
Create a histogram using a bar chart
in Excel(start at 12:14)
CreateahistogramusingtheEXCEL
histogramchart

22. 5. DRAW A DOT PLOT
22Descriptive Statistics
Adotplotisdrawnbyplacingeachobservationhorizontallyinincreasingorderand
placingadotabovetheobservationeachtimeitisobserved.
A dot plot is generally used for
discrete data. It can also be
used for continuous data if
there are relatively few values.

23. 5. CONSTRUCT A FREQUENCY POLYGON
23Descriptive Statistics
Class Midpoint –the sum of consecutive
lower class limits divided by 2.
Frequency polygon –a graph that uses
points, connected by line segments, to
represent the frequencies for the classes. It
is constructed by plotting a point above
each class midpoint on a horizontal axis at
a height equal to the frequency of the
class. Next, line segments are drawn
connecting consecutive points. Two
additional line segments are drawn
connecting each end of the graph with the
horizontal axis.

24. 5. CONSTRUCT CUMULATIVE FREQUENCY TABLES
24Descriptive Statistics
Since quantitative data can be ordered, they can be summarized using cumulative
frequency.
Acumulativefrequency
distributiondisplaysthe
aggregatefrequencyofthe
category.
Acumulativerelative
frequency distribution
displaystheaggregate
frequencyofthecategory.

25. 5. CONSTRUCT AN OGIVE
25Descriptive Statistics
Anogive(readas“ohjive”)isa
graphthatrepresentsthe
cumulativefrequencyor
cumulativerelativefrequency
fortheclass.Itisconstructedby
plottingpointswhosex-
coordinatesaretheupperclass
limitsandwhosey-coordinates
arethecumulativefrequencies
or cumulative relative
frequenciesoftheclass.

26. IDENTIFY THE SHAPE OF THE DISTRIBUTION
26Descriptive Statistics
One wayto
describeavariable
isthroughthe
shapeofits
distribution.

27. GRAPHICAL
MISREPRESENTATIONS OF
DATA

28. WHAT CAN MAKE A GRAPH MISLEADING?
28Descriptive Statistics
One method of distorting the truth through graphics.
Amisleadinggraphicunintentionallycreatesawrongimpression.
Adeceivinggraphicpurposelycreatesanincorrectimpression.
CommonGraphicalMisrepresentations
a)Inconsistentscale
b)Misplacedorigin
*Incrementsbetweentickmarksshouldbeconstant.

29. MISREPRESENTATION OF DATA
A home security company located in Minneapolis,
Minnesota, develops a summer ad campaign with the slogan
“When you leave for vacation, burglars leave for
work.” According to the city of Minneapolis, roughly 20% of
home burglaries occur during the peak vacation months of
July and August. The advertisement contains the
graphic shown in Figure 20. Explain what is wrong with the
graphic.
Descriptive Statistics 29

30. MISREPRESENTING DATA
Descriptive Statistics 30

31. MANIPULATING THE VERTICAL
SCALE
A national news organization developed the graphic shown
below to illustrate the change in the highest marginal tax rate
effective January 1, 2018. Why might this graph be considered
misleading?
Descriptive Statistics 31

32. MANIPULATING THE VERTICAL
SCALE
Descriptive Statistics 32

33. MISLEADING GRAPHS
Soccer seems to be losing popularity as a sport in the United
States. In 2010, there were approximately 14 million
participants in the United States. By 2017, this number had
dropped to 12 million. To illustrate this decrease, we could
create a graphic like the one shown below. Describe how the
graph may be misleading.
Descriptive Statistics 33

34. MISLEADING GRAPHS
Descriptive Statistics 34
In order to avoid misleading graphs, one can add data values or choose a
consistent unit to depict the values.

35. MISREPRESENTATIONS IN 3D
SCALE
Descriptive Statistics 35

36. MISREPRESENTATION IN 3D SCALE
Descriptive Statistics 36

37. GUIDELINES FOR CONSTRUCTING GOOD GRAPHICS
37Descriptive Statistics
•Titleandlabelthegraphicaxesclearly,providingexplanationsifneeded.
•Includeunitsofmeasurementandadatasourcewhenappropriate.
•Avoiddistortion.Neverlieaboutthedata.
•Minimizetheamountofwhitespaceinthegraph.Usetheavailablespacetolet
thedatastandout.Ifyoutruncatethescales,clearlyindicatethistothereader.
•Avoidclutter,suchasexcessivegridlinesandunnecessarybackgroundsorpictures.
•Don’tdistractthereader.
•Avoidthreedimensions.Three-dimensionalchartsmaylooknice,buttheydistract
thereaderandoftenleadtomisinterpretationofthegraphic.
•Donotusemorethanonedesigninthesamegraphic.Sometimesgraphsusea
differentdesigninoneportionofthegraphtodrawattentiontothatarea.
•Don’ttrytoforcethereadertoanyspecificpartofthegraph.Letthedataspeakfor
themselves.
•Avoidrelativegraphsthataredevoidofdataorscales.