Post on 08-Apr-2018
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EDUCATIONAL STATISTICS
Dr. Joseph Mercado
Special Lecturer
PUP GS
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Research Process
Identification of the Research ProblemFormulation of HypothesesIdentification of Necessary DataData CollectionAnalysisSummarizing Results
Drawing Conclusions and ImplicationsExpanded, Revised, and New TheoryNew Knowledge
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Research Problems
Classroom Leadership, Attitude Towards TeachingScience and other Correlates of Teaching Competenceof Science High School Teachers in the NCR
Principals Attributes : Their Effects on TeachersEmpowerment and Learners Achievement in PublicSchools in the NCR
Evaluation of the Quality of Research and ExtensionPrograms of Selected Higher Education Institutions inthe CALABARZON Region
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Ethical Responsibility in the Work Behavior of SchoolManagers in Four selected Universities in the LuzonArea
An Assessment of the Expanded ROTC Program s andCivic Welfare Service and Its Implications to theNational Peace and Development Plan (NPDP)
Total Quality Management Practices in SelectedHigher Education Institutions in Metro Manila
Principal Empowerment in public Secondary Schools:Basis For The Development of a Primer
Prediction Models of School Based Management,Teaching Behavior and Professionalism
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Factors Affecting the Results of the National AchievementTest (NAT) in Selected City Schools in the NCR
Teachers Beliefs Regarding The Implementation of Constructivism in the Classroom
Analysis of Leadership Behavior and Self-Efficacy of Principals of Catholic Secondary Schools
Study Habits and Academic Achievements of IntermediatePupils in Guadalupe Nuevo Elementary School in Makati:An Assessment
Teachers Behavior, Teachers Collective Efficacy, StudentsBehavior and Their Relationship to Students AcademicAchievement in Selected Public Secondary Schools inBulacan
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Personal Needs, Job Satisfaction and Work-Related Characteristics as Correlates of TeacherCommitment Among Mathematics FacultyMembers of State Universities and Colleges inthe NCR
Teacher s Behavioral Pattern and TeachingStyles: Their Influence on Pupils AcademicAchievement in the District of Tanza
Leadership Styles of Administrators in ThreeSelected State Universities in Manila and TheirEffect on Faculty Performance
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An Analysis of the Effectiveness of Field Study Coursesof the Revised Teacher Education Curriculum AmongSelected Higher Education Institutions (HEIs) in the
National Capital Region
An Evaluation of the International Training Program inthe Hospitality Education of Selected Higher EducationInstitutions (HEIs) in Metro Manila
Personological Attributes Affecting the ManagementStyles of Educational Managers in Selected Schoolsin the Province of Laguna
Relationship Between Some Selected Variables and
Conflict Management Styles of the Administrators of the Polytechnic University of the Philippines System
Quality Assessment of Student Development andServices Program in Local Colleges and Universities inMetro Manila
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Nature of Statistics
Statistics may be defined as the science of artof collecting, presenting, analyzing andinterpreting data in a certain field, such aseducation, science, psychology, business,economics, engineering, medicine, or any otherarea.
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Statistics can also be used in makingcorrect decisions during the time of uncertainty. One may apply the differentstatistical methods so as to arrive at thecorrect result with appropriate criticaljudgment.
Meaning of Statistics
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Branches of Statistics
Descriptive Statistics the branch thatsummarizes and organizes raw data into ameaningful information.
To calculate the average score of your studentsor can summarize the percentages of the score.
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Branches of Statistics
Inferential Statistics statistical inference isthe process of obtaining information about a
larger group from the study of a smaller group.The total group of people, things, or characteristics you are interested instudying, understanding, or predicting is called population .
A sample is a group of representative items chosen from the populationand used to predict the behavior or characteristics of the total populationwith help of inferential statistics.
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The Statistical Treatment
The design of the studydeterminesdetermines what statisticaltechniques should be employed,not vice versa.
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The Statistical Treatment
The kind of statistical treatmentthat can be done in a study is
mainly constrainedconstrained by the level(or scale) of data measurementemployed in the study.
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The Statistical Treatment
Scales of Data Measurement
Nominal scale
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The Statistical Treatment
Scales of Data Measurement
1. Nominal scale2. Ordinal Scale
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The Statistical Treatment
Scales of Data Measurement
1. Nominal scale2. Ordinal Scale3. Interval and Ratio Scale
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The Statistical Treatment
Scale of Data MeasurementNominal data (the least sophisticated):
data assumes no natural ordering;
largely allied to measuring qualitativecharacteristics such as eye colo r, hair colo r, g en d e r, nati on a l it y o r even l if es t ylegr oups , i.e ., si ng les , youn g marri e d,r e tir e d.
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The Statistical TreatmentScale of Data Measurement
Nominal data: Example
Eye Color Number of Men %
Blue 60 30Brown 80 40Green 30 15Grey 20 10Hazel 10 5
TOTAL 200 100
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The Statistical TreatmentScale of Data Measurement
Ordinal data: Example
Rail travelers might be asked to give theirviews on the quality of the MRT service
according to a scale of 1 5 where:1 = very poor2 = poor3 = adequate
4 = good5 = very good
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The Statistical TreatmentScale of Data Measurement
Interval and Ratio Data
(the most sophisticated)data where values progress bothin order and according to a seriesof equal steps.
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The Statistical TreatmentScale of Data Measurement
Interval and Ratio DataExamples:
number of babies born to different
familiesnumber of people pre-purchasingtheater tickets at a particular venueover different periodsage, weight and height of a sample of human beings
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The Statistical TreatmentScale of Data Measurement
Statistical procedure is selected on thebasis of its appropriateness for
answering the question involved in thestudy.Nothing is gained by using a complicatedprocedure when a simple one will do just aswell. Statistics are to serve research, not todominate it.
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The Statistical TreatmentScale of Data Measurement
Interval and Ratio Data
Examples:number of babies born to differentfamiliesnumber of people pre-purchasingtheater tickets at a particular venue
over different periodsage, weight and height of a sample of human beings
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Sources of Data
1. S econ dar y Data : Data which arealready available. An example:statistical abstract of USA.
Advantage : less expensive.Disadvantage : may not satisfyyour needs.2. P rimar y Data : Data which must becollected.
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P reliminary Steps in Statistical Study
Define the problemDetermine the population/subject of
the studyDevise the set of questionsDetermine the sampling design
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G uidelines in the Selection of a Research P roblem
or Topic
The research problem must be chosen by theresearcher himself so that he will not make
excuses for all the obstacles he will encounter.The problem must be within the interest of the
researcher so that he will give all the time and
effort in the research work.
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G uidelines in the Selection of a Research P roblemor Topic
The problem must be within the specialization of the researcher. It will make the work easier for theresearcher because he is familiar in the area and it
will help him improve his specialization, skill andcompetence in his own area.The research problem must be within thecompetence of the researcher. The researcher must
know the procedures in making research and howto apply them. He must have a workableunderstanding of his study.
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G uidelines in the Selection of a Research P roblemor Topic
The researcher must have the ability and capacityto finance the research problem to make sure thatthe study will be completed on the target time.
The research problem must be manageable. Thedata must be available or within the capacity of theresearcher to gather data. The data must beaccurate, objective and not biased. The data should
help the researcher answer the question beinginvestigated.
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G uidelines in the Selection of a Research P roblemor Topic
The research problem must be completedwithin the period set by the researcher.
The research problem must be significant,important and relevant to the present time aswell as to the future. This means that theresearch problem must have an impact to the
situation and people it is intended for.
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G uidelines in the Selection of a Research P roblemor Topic
The results of the study must be practical andimplementable.
The study must contribute to the humanknowledge. The facts and knowledge must be aproduct of research.
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Different Ways to Draw a Sample
(2 Types of Sampling)
1 . Random Sampling or Probability Sampling
In probability sampling, the sample is aproportion of the population and such sample isselected from the population by means of systematic way in which every element of thepopulation has a chance of being included in thesample.
2 . Non-Random Sampling or Non-Probability Sampling
In a non-probability sampling, the sample is nota proportion of the population and there isno system in selecting the sample. The selectiondepends on the situation.
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b. Systematic Random Sampling - Select some startingpoint and then select every K th element in thepopulation
Random Sampling or Probability Sampling
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c. Stratified Sampling - subdivide the population intosubgroups that share the same characteristic, thendraw a sample from each stratum
Random Sampling or Probability Sampling
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d. Cluster Sampling - divide the population into sections(or clusters); randomly select some of those clusters;choose all members from selected clusters
Random Sampling or Probability Sampling
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N on-Random Sampling or N on-ProbabilitySampling
1 . Accidental SamplingIn this type of sampling, there is no system of selection butonly those whom the researcher or interviewer meets bychance are included in the sample. This type of samplinglacks representativeness where the sample may be biased. If the interviewer goes to a business section, most people whowill be interviewed are likely from the business and probablyrich people hence the respondents will be from well-to-dopeople. But if the interviewer stays in a slum area, then it ispossible that the respondents are poor people. In a research,
every section of the population must be equally representedin the sample. This method is being used when there is noalternative.
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2 . Quota SamplingIn this type of sampling, specified number of personsof certain types is included in the sample. Supposethe reactions of the people for a particular issue, suchas the effects of drug addiction in a certain locality,can be decided from a sample that constitutes 10doctors, 9 lawmakers, 15 parents and 20 drugaddicts.In quota sampling, many sectors of the populationare represented. However, the representation is
doubtful because there is no proportionalrepresentation since there are no guidelines in theselection of the respondents. Anyone who is selectedto participate will do. Quota sampling may be usedonly when any of the more desirable types of
sampling will not do.
N on-Random Sampling or N on-Probability Sampling
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N on-Random Sampling or N on-Probability Sampling
3 . Convenience SamplingConvenience sampling is a process of picking out peoplein the most convenient and fastest way to get reactionsimmediately. This method can be done by telephoneinterview to get the immediate reactions of a certaingroup of sample for a certain issue.
This kind of method is biased and not representative.This is quite different from gathering data by interviewwhereby the interview can be done through the
telephone. In the interview method, people who areinterviewed through the telephone are properlyselected to be included in the sample.
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Sample Size Determination
The extent of the population will depend on the nature of theproblem. The census survey will require all individual in the populationthat is considered while the sample survey will consider a fewrepresentative of the population.
In determining the sample size, the formula which can be applied is asfollows: (Slovins Formula)
n = sample size
N = population sizee = desired margin for error
(per cent allowance for non-precision because a sample is used)
21 NeN
n u
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Example 1A researcher wants to make use of a student
population of 3 , 000 for his study in the mathematicalachievement test. If he allows a 5% margin of error,how many students must he take for his sample?
Solution:The formula given can be used:
21 N eN n u
2)05.0(300013000
un
)0025.0(300013000
un
5.713000
un
5.83000
un
35394.352 or n u
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DESCRIPTIVE STATISTICS
Summarizes or describes the important characteristicsof a known set of data.
Measures of Central Tendency
A measure of tendency for a collection of datavalues is number that is meant to convey the idea ofcentralness for the data set.
Numerical values that are indicative of the centralpoint or the greatest frequency concerning a set ofdata. The most common measures of centraltendency are the mean, median, and mode.
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Measures of Central Tendency and Scales ofmeasurement
T he mode requires only nominal data - and you cancompute it for ordinal, interval, and ratio.
T he Median requires ordinal data - and you can compute itfor interval and ratio. You cannot compute the Median for nominal data.
T he Mean requires interval or ratio data - you cannotcompute it for either nominal or ordinal data.
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Example
1. What is the mean for the following samplevalues? 3, 8, 6, 14, 0, -4, 0, 12, -7, 0, -10.
Solution:
211
)10(0)7(120)4(014683!!
X
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2. Find the median for the ages of the following
eight college students:23 19 32 25 26 22 24 20
Solution: First order the values, The orderedarray is
19 20 22 23 24 25 26 32
median = (23 + 24)/2 = 23.5
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What is the mode for the following sample
values?3 5 1 4 2 9 6 10The data set has no mode
What is the mode for the following samplevalues?
3 5 1 4 2 9 6 10 5
3 4 3 9 3 6 1The value of the is mode is 3,Unimodal
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What is the mode for the following sample values?6 10 5 3 4 3 9 3 6 1 6
The values of the mode are 3 and 6, bimodal
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Mean: Grouped DataT he calculation of mean from a frequencydistribution is almost the same as that from anungrouped data (raw data), only in a distribution, theindividual values are not known. When the number of items is too large, it is best to compute for themeasures of central tendency using a frequencydistribution.
n
fX n
ii
X
!! 1
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Ex ample:
Calculation of Mean from Frequency Distribution of Sample Test Scores of 40 Students in E ducational Statistics
ifX
Scores f Xi fXi
70 74 2 72 144
65 69 2 67 134
60 64 3 62 186
55 59 2 57 114
50 54 8 52 41645 49 9 47 423
40 44 2 42 84
35 39 4 37 148
30 34 5 32 160
25 29 3 27 81
n = 40 = 1,890
25.4740
18901 !!!
!
n
f X n
ii
X
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Median: Grouped Data
if
F n
LMdnm
m
m
!12
Formula:
Where:Mdn = median
lower class boundary of the median class= less than cumulative frequency of the class immediately preceding the
median classi = the size of the intervaln = the total number of scores
!mL
1mF
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Scores f
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Mode: Grouped Data
id d
d LM
m oo
!
21
1
Formula:
Where:lower class boundary of the modal class
= numerical difference between the frequency of the modal classand the frequency of the adjacent lower class
= numerical difference between the frequency of the modal classand the frequency of the adjacent higher class
i = size of the class interval
!m oL
1d
2d
Ex ample:
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Scores f
70 74 2
65 69 2
60 64 3
55 59 2
50 54 8
45 49 9
40 44 235 39 4
30 34 5
25 29 3
n = 40
Ex ample:Calculation of Mode from Frequency Distribution of Sample Test Scores of 40Students in E ducational Statistics
88.48517
75.44
21
1 !
!
! i
d d
d LM
m oo
P roperties of the different central tendency measures:
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P roperties of the different central tendency measures:
1. T he mean is the standard measure of central tendency in statistics. Itis most frequently used.
2. T he mean is not necessarily equal to any score in the data set
3. T he mean is the most s table measure from s ample to s ample .
4.T
he mean is very influenced by Outlier s
--T
hat is, the mean will bestrongly influenced by the presence of e x treme scores.
5. T he median is not s en s itive to outlier s .
6. T he mean is based on all scores from the sample but the mode and
the median are not.
7. T he M ode is the lea s t s table measure from sample to sample.
8. T he median is the best measure of central tendency if the distributionis skewed.
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Measures of P ositionT he Quartiles
Quartiles are the points which divide the total number of scores into four equal parts. E ach set of scores has three quartiles. T wenty-five percentfalls below the first quartile (Q 1), fifty percent is below the second quartile(Q 2), and seventy-five percent is below the third quartile (Q 3).
T he steps in finding the quartiles of raw scores can be summarized asfollows:
1. Arrange the scores from highest to lowest or lowest to highest.
2. Determine Q k, where Q k is the k th quartile and k = 1, 2, 3
2.1 If is an integer, scores
2.2 If is not an integer, Q k = ith score where I is the closestinteger greater than
4nk
2
14
thth
k
nk k n
Q
!
4nk
4nk
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Ex ample
Calculation of Quartiles from Sample Raw Scores of E ight Students inE ducational Statistics and Nine Students in Applied Statistics
Edu catio nal Statistics Applie d Statistics
17 15
17 19
26 20
28 24
30 28
30 30
31 32
37 32
40
5.212
26172
3224
1811 !!!p!! scor esQQ
th
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5.30
23130
276
64
3833 !!!p!! scor esQQ
th
For Applied Statistics:
20325.2
4
1911
!!p!! scor eQQ r d
32775.6
439
33 !!p!! scor eQQth
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For Grouped Data:
Formula: First Quartile
if
F n
LQQ
m
Q
!1
1
1
1 4
1Q
1QL 1Q
1mF 4n
1Qf 1Q
Where:
= the First Quartile
= lower class boundary where lies
= less than cumulative frequency approaching or equal to but not e x ceeding
= the frequency where
i = the size of the interval
n = the total number of scores
lies
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For Grouped Data:Formula: T hird Quartile
if
F n
LQQ
mQ
!3
3
13
4
3
3Q
13QL
3Q
1mF
43n
3Qf 3Q
Where:
= the T hird Quartile
= lower class boundary where lies
= less than cumulative frequency approaching or equal to but not e x ceeding
= the frequency where
I = the size of the interval
n = the total number of scores
lies
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Scores f
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3754
8105.344
1
1
1
1!
!!
! i
f
F n
LQQ
m
Q
88.53582330
5.494
3
3
3
1
3 !
!
! if
F n
LQ Q
m
Q
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Measures of Variability
A measure of variability for a collection of datavalues is a number that is meant to convey theidea of spread for the data set. The mostcommonly used measures of variability forsample data are the range, the interquartilerange, the mean absolute deviation, the
variance or standard deviation, and thecoefficient of variation.
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Mean Absolute Deviation isthe average of the absolutedeviation values from themean.
Mean Absolute Deviationutilizes deviations of thedata values from the meanin its computation.
nxMAD x! //
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What is the MAD for the following sample values?
3 8 6 12 0 -4 10
Data Values, x Absolute Deviations /x-mean/
3 2
8 36 1
12 7
0 5
-4 9
10 5
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The average (absolute) distance of the samplevalues from the mean.
n
xMAD x! //
57.47
32 !!MAD
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The Variance and Standard Deviation
The Variance and Standard Deviation are the most commonand useful measures of variability. These two measuresprovide information about how the data vary about the mean.
If the data are clustered around the mean, then the varianceand standard deviation will somewhat small.
There is small variability when the data values are clusteredabout the mean
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Sample Variance
The formula says that you subtract the mean
from each data value and square thedifferences, then you add these values anddivide by the sample size minus 1.
1
2( )2 ! n
x xS
Do not let the formula frighten you. We will build a table to helpcompute the variance.
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What is the variance for the following sample
values?3 8 6 14 0 11
Solution: First of all, we need to compute the
sample mean:
7642
611014683 !!!X
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Table Used in Helping to Compute the SampleVariance
Data Deviations (x-mean) Squared Deviations(x-mean)
3 3 7 = -4 16
8 8 7 = -1 16 6 7 = -1 1
14 14 7 = 7 49
0 0 7 = -7 49
11 11 7 = -4 16
Total 0 132
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The sample variance is
1
2( )2 ! n
x xS
4.26
5
132
16
1322 !!!S
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Wh i h d d d i i f h f ll i l
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What is the standard deviation for the following samplevalues?
3 8 6 14 0 11
Solution:
1
2( )!
n
xS x
14.54.265
132 !!!S
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Coefficient of Variation
The Coefficient of Variation (CV) allows us to compare thevariation of two (or more) different variables.The sample coefficient of variation is defined as samplestandard deviation divided by the sample mean of the dataset. Usually, the result expressed as a percentage .
%100x
S
CV X !
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The sample coefficient of variation standardizes the variation bydividing it by the sample mean. The CV has no units, since thestandard deviation and the mean have the same units, and thus theunits cancel each other. Because of this property, we can use thismeasure to compare variations for different variables with differentunits.
We said that standard deviation measures the variation in a set of
data. For distributions having the same mean, the distribution withthe largest standard deviation has the greatest variation. But whenconsidering distributions with different means, decision makerscan't compare the uncertainty in distribution only by comparingstandard deviations. In this case, the coefficient of variation is used,i.e., the coefficients of variation for different distributions arecompared, and the distribution with the largest coefficient of variation value has the greatest relative variation.
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%56.5%100905
)( !! xticket sCV
%35.14
%100400,5
775)(
!!x
r evenue sCV
Since the CV is larger for the revenues, there is
more variability in the recorded revenues than in thenumber of tickets issued.
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For e x ample, Mark teaches two sections of statistics. He gives eachsection a different test covering the same material. T he mean score onthe test for the day section is 27, with a standard deviation of 3.4. T hemean score for the night section is 94 with a standard deviation of 8.0.Which section has the greatest variation or dispersion of scores?
D ay Section.................... N ight Section
M ean .......27.......................94S. D ...........3.4.....................8.0
Direct comparison of the twostandard deviations
shows that the nightsection has the greatest variation. But comparing the coefficient of variations show quite different results:
C.V.(day) = (3.4/27) x 100 = 12.6% and C.V.(night) = (8/94) x 100 = 8.5%
Example 2
E mpirical Rule:
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E mpirical Rule:
T his rule generally applies to mound-shaped data, but specifically to thedata that are normally distributed, i.e., bell shaped. T he rule is as follows:
Appro x imately 68% of the measurements (data) will fall within onestandard deviation of the mean (One-Sigma Rule), 95% fall within twostandard deviations ( T wo-Sigma Rule), and 99.7% (or almost 100% ) fallwithin three standard deviations ( T hree-Sigma Rule). See the following
figure:
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For e x ample, in the height problem, the mean height was 70 inches with astandard deviation of 3.4 inches. T hus, 68% of the heights fall between 66.6and 73.4 inches, one standard deviation, i.e., (mean + 1 standard deviation) =
(70 + 3.4) = 73.4, and (mean - 1 standard deviation) = 66.6. Ninety fivepercent (95%) of the heights fall between 63.2 and 76.8 inches, two standarddeviations. Ninety nine and seven tenths percent (99.7%) fall between 59.8and 80.2 inches, three standard deviations. See the following figure: