DATA ANALYSIS Descriptive_Statistics_part_1.pdf

This is a pdf that will help with data analysis and statistics

1. DESCRIPTIVE STATISTICS:
TABULAR AND GRAPHICAL
PRESENTATIONS

2. ◦SummarizingQualitativeData
◦SummarizingQuantitativeData

3. Summarizing Qualitative Data
◦Frequency Distribution
◦Relative Frequency Distribution
◦Percent Frequency Distribution
◦Bar Graphs
◦Pie Charts

4. Frequency Distribution
Frequencyisthenumberoftimesadatavalueoccurs.
Forexample,iftenstudentsscore80instatistics,thenthescoreof
80hasafrequencyof10.Frequencyisoftenrepresentedbythe
letterf.
FrequencyDistribution:Afrequencydistributionisatabularor
graphicalsummaryofdatathatdisplaysthenumberof
observationswithinagivencategoryorinterval.
◦Theobjectiveistoprovideinsightsaboutthedatathatcannotbe
quicklyobtainedbylookingonlyatrawdata.

5. Can you make any inferences from
just looking at this raw data?
Malay Chinese
Indian Others
Malay Chinese
Indian Others
Malay Chinese
Indian Others
Malay Others

6. An example of data summarized into gender and ethnic group,
then the frequency tables can get as below :
Observation Frequency
Male 28
Female 22
Total 50
Observation Frequency
Malay 33
Chinese 9
Indian 6
Others 2
Total 50

7. Relative Frequency
Therelativefrequencyofaclassisthefractionor
proportionofthetotalnumberofdataitems
belongingtotheclass.
Percent Frequency
Arelativefrequencydistributionisatabular
summaryofasetofdatashowingtherelative
frequencyforeachclass.

8. Ex: Ten people were asked about their marital status:
Single, Single, Single, Divorced, Single, Married, Widowed, Married,
Widowed, Divorced

9. Bar Chart
Barchartisusedtodisplaythefrequencydistribution
inthegraphicalform.Itconsistsoftwoorthogonal
axesandoneoftheaxesrepresenttheobservations
whiletheotheronerepresentsthefrequencyofthe
observations.Thefrequencyoftheobservationsis
representedbyabar.

10. Pie Chart
PieChartisusedtodisplaythefrequencydistribution.
Itdisplaystheratiooftheobservations.Itisacircle
consistsofafewsectors.Thesectorsrepresentthe
observationswhiletheareaofthesectorsrepresent
theproportionofthefrequenciesofthatobservations.
GradeNumber of
Students
Relative
Frequency
Central Angle
A 9
9
/30= 30% 0.30 • 360 = 108°
B 13
13
/30= 43% 0.43 • 360 = 155°
C 6
6
/30= 20% 0.20 • 360 = 72°
D 2
2
/30= 7% 0.07 • 360 = 25°
Total = 30= 100% =360
°

11. Summarizing Quantitative Data
◦Frequency Distribution
◦Relative Frequency Distribution
◦Percent Frequency Distribution
◦Histogram
◦Cumulative Distributions
◦Ogive

12. Frequency Distribution for Quantitative Data
Tosummarizequantitativedata,weusea
frequencydistributionjustlikethoseforqualitative
data.However,sincethesedatahavenonatural
categories,wedividethedataintoclasses.
Classesareintervalsofequalwidththatcoverall
valuesthatareobservedinthedataset.
Thelowerclasslimitofaclassisthesmallestvalue
thatcanappearinthatclass.
Theupperclasslimitofaclassisthelargestvalue
thatcanappearinthatclass.
Theclasswidthisthedifferencebetween
consecutivelowerclasslimits.

13. Guidelines for Choosing Classes
There are many ways to construct a frequency distribution, and they will
differ depending on the classes chosen. Following are guidelines for
choosing the classes.
•Every observation must fall into one of the classes.
•The classes must not overlap.
•The classes must be of equal width.
•There must be no gaps between classes. Even if there are no
observations in a class, it must be included in the frequency distribution.

14. Constructing a Frequency Distribution
Following are the general steps for constructing a frequency distribution.
Step 1:Choose a class width.
Step 2: Choose a lower class limit for the first class. This should be a
convenient number that is slightly less than the minimum data value.
Step 3: Compute the lower limit for the second class, by adding the class
width to the lower limit for the first class:
Lower limit for second class = Lower limit for first class + Class width
Step 4: Compute the lower limits for each of the remaining classes, by
adding the class width to the lower limit of the preceding class. Stop
when the largest data value is included in a class.
Step 5: Count the number of observations in each class, and construct the
frequency distribution.

15. Example: Frequency Distribution
The emissions for 65 vehicles, in units of grams of particles
per gallon of fuel, are given. Construct a frequency
distribution using a class width of 1.
1.50
1.48
1.40
3.12
0.25
0.87
1.06
1.37
2.37
0.53
1.12
1.11
1.81
2.12
3.36
1.25
2.15
1.14
2.68
3.47
3.46
0.86
1.63
1.17
2.74
1.11
1.81
3.67
3.34
1.88
1.12
1.47
0.55
3.79
5.94
0.88
1.24
2.67
1.28
4.24
1.29
1.63
2.63
2.10
3.52
0.94
2.14
3.03
6.55
3.59
0.64
6.64
1.23
1.18
3.10
1.31
4.04
1.04
3.06
3.33
2.49
2.48
1.63
0.48
4.58

16. Example: Frequency Distribution (Continued 1)
Since the smallest value in the data set is 0.25, we choose 0.00 as the lower
limit for the first class. The class width is 1 and the first lower class limit is 0.00,
so the lower limit for the second class is 0.00 + 1 = 1.00.
The remaining lower class limits are as follows.
1.00 + 1 = 2.00
2.00 + 1 = 3.00
3.00 + 1 = 4.00
4.00 + 1 = 5.00
5.00 + 1 = 6.00
6.00 + 1 = 7.00
Since the largest data value is 6.64, every data value is now contained in a
class.

17. Example: Frequency Distribution (Continued 2)
Lastly, we count the number of observations in each
class to obtain the frequency distribution.
Class Frequency
0.00 –0.99 9
1.00 –1.99 26
2.00 –2.99 11
3.00 –3.99 13
4.00 –4.99 3
5.00 –5.99 1
6.00 –6.99 2

18. Relative Frequency Distribution
Given a frequency distribution, a relative frequency distribution can
be constructed by computing the relative frequency for each class.
Relative Frequency =
Frequency
Sumofallfrequencies
Class Frequency Relative
Frequency
0.00 –0.99 9 0.138
1.00 –1.99 26 0.400
2.00 –2.99 11 0.169
3.00 –3.99 13 0.200
4.00 –4.99 3 0.046
5.00 –5.99 1 0.015
6.00 –6.99 2 0.031

19. Histogram
Once we have a frequency distribution or a relative
frequency distribution, we can put the information in
graphical form by constructing a histogram.
A histogram is constructed by drawing a rectangle for
each class. The heights of the rectangles are equal to
the frequencies or the relative frequencies, and the
widths are equal to the class width.

20. Example: Histogram
The frequency histogram and relative frequency
histogram are given for the particulate emissions
data.
Note that the two histograms have the same shape.
The only difference is the scale on the vertical axis.

21. Choosing the Number of Classes
There are no hard and fast rules for choosing the number of
classes. In general, it is good to have more classes rather than
fewer, but it is also good to have reasonably large frequencies
in some of the classes. There are two principles that can guide
the choice.
◦Too few classes produce a histogram lacking in detail.
◦Too many classes produce a histogram with too much detail,
so that the main features of the data are obscured.

22. Choosing the Number of Classes (Continued)
The following histograms illustrate too many
classes and too few.

23. Shape of a Data Set
A histogram gives a visual impression of the “shape” of a data set.
Statisticians have developed terminology to describe some of the
commonly observed shapes.
A histogram is symmetric if its right half is a
mirror image of its left half. There are very few
histograms that are perfectly symmetric, but
many are approximately symmetric.
A histogram with a long right-hand tail is said
to be skewed to the right, or positively
skewed.
A histogram with a long left-hand tail is said
to be skewed to the left, or negatively
skewed.

24. Unimodal and Bimodal Histograms
Apeak,orhighpoint,ofahistogramisreferredtoasamode.A
histogramisunimodalifithasonlyonemode,andbimodalifithas
twoclearlydistinctmodes.

25. You Should Know . . .
◦How to construct a frequency and relative frequency
distribution for quantitative data
◦How to construct and interpret histograms
◦The guiding principles for choosing the number of classes
in a histogram
◦Some possible shapes of a data set including:
◦Symmetric
◦Skewed to the right (positively skewed)
◦Skewed to the left (negatively skewed)
◦Unimodal
◦Bimodal