Basic concepts of statistics

Basic Concepts ofBasic Concepts of
StatisticsStatistics
and Hypothesisand Hypothesis Inference & HypothesisInference & Hypothesis
 Basic Concepts –Basic Concepts –
 Null & Alternative HypothesisNull & Alternative Hypothesis
 Type-I & Type-II ErrorsType-I & Type-II Errors
 One-tailed & Two-tailed TestsOne-tailed & Two-tailed Tests
 Power of A TestPower of A Test
 Effect SizeEffect Size
 Level of SignificanceLevel of Significance
 Degrees of FreedomDegrees of Freedom
 Critical RegionCritical Region
 Steps in Testing HypothesisSteps in Testing Hypothesis
Tuesday, February 6, 2018Tuesday, February 6, 2018All Disciplines. Book Zone Publication, ISBN: 978-984-33-9565-8, Chittagong-4203, Bangladesh. smskabir218@gmail.com; smskabir@psy.jnu.ac.bdAll Disciplines. Book Zone Publication, ISBN: 978-984-33-9565-8, Chittagong-4203, Bangladesh. smskabir218@gmail.com; smskabir@psy.jnu.ac.bd

DEFINITION OFDEFINITION OF
STATISTICSSTATISTICS
Simply statistics is the scientific study of data - howSimply statistics is the scientific study of data - how
to collect, summarize and present it.to collect, summarize and present it.
Different scientists defined statistics differently such as -Different scientists defined statistics differently such as -
 Statistics can be defined as the collection presentation andStatistics can be defined as the collection presentation and
interpretation of numerical data [Croxton and Crowed].interpretation of numerical data [Croxton and Crowed].
 Statistics are numerical statement of facts in anyStatistics are numerical statement of facts in any
department of enquiry placed interrelation to each otherdepartment of enquiry placed interrelation to each other
[Bouly].[Bouly].
 Statistics are measurement, enumerations or estimates ofStatistics are measurement, enumerations or estimates of
natural or social phenomena systematically arrangement tonatural or social phenomena systematically arrangement to
exhibit their inner relation [Conner].exhibit their inner relation [Conner].
 The science of statistics is essentially a branch of appliedThe science of statistics is essentially a branch of applied
mathematics and can be regarded as a mathematicsmathematics and can be regarded as a mathematics
applied to observation data [R.A. Fisher].applied to observation data [R.A. Fisher].
All Disciplines. Book Zone Publication, ISBN: 978-984-33-9565-8, Chittagong-4203, Bangladesh. smskabir218@gmail.com; smskabir@psy.jnu.ac.bdAll Disciplines. Book Zone Publication, ISBN: 978-984-33-9565-8, Chittagong-4203, Bangladesh. smskabir218@gmail.com; smskabir@psy.jnu.ac.bdTuesday, February 6, 2018Tuesday, February 6, 2018

Statistics can be defined in two senses i.e.Statistics can be defined in two senses i.e.
SingularSingular andand PluralPlural..
 InIn Singular SenseSingular Sense it can be defined as theit can be defined as the
various methods and techniques forvarious methods and techniques for
attaining and analyzing the numericalattaining and analyzing the numerical
information.information.
The techniques and method meansThe techniques and method means

Collection of DataCollection of Data

OrganizationOrganization

PresentationPresentation

Analysis andAnalysis and

Interpretation of numerical data.Interpretation of numerical data.
DefinitionofStatisticsDefinitionofStatistics

AA Plural SensePlural Sense, statistics means the, statistics means the
aggregates of numerical facts collectedaggregates of numerical facts collected
systematically.systematically.
According to this definition statistics in plural sense shouldAccording to this definition statistics in plural sense should
have following features –have following features –
StatisticsStatistics are aggregate of factsare aggregate of facts
Affected by multiplicity of causesAffected by multiplicity of causes
Numerically expressNumerically express
Collected in a systematic MannerCollected in a systematic Manner
They are collected for the pre-determined purposeThey are collected for the pre-determined purpose
Enumerated or Estimated according to reasonableEnumerated or Estimated according to reasonable
standard of accuracystandard of accuracy

Statistics is the branch of applied advancedStatistics is the branch of applied advanced
mathematics which provides logical andmathematics which provides logical and
scientific basis for doing research and providesscientific basis for doing research and provides
scientific methods in - proper formulation of thescientific methods in - proper formulation of the
problem under study; outlining the fieldproblem under study; outlining the field
procedures for data collection; processing ofprocedures for data collection; processing of
data; getting estimates from the data; obtainingdata; getting estimates from the data; obtaining
the hypothesis under consideration; makingthe hypothesis under consideration; making
interpretations of obtained results; providinginterpretations of obtained results; providing
recommendations and conclusions on individualrecommendations and conclusions on individual
and collective issues connected to the problemand collective issues connected to the problem
under investigation.under investigation.

INFERENTIAL STATISTICSINFERENTIAL STATISTICS
Statistical procedures can be divided intoStatistical procedures can be divided into
two major categories –two major categories –
1. Descriptive Statistics1. Descriptive Statistics
2. Inferential Statistics.2. Inferential Statistics.
Basically -Basically -
 Descriptive statistics describes theDescriptive statistics describes the
features of the sample data quantitatively.features of the sample data quantitatively.
 Inferential statistics does inferences forInferential statistics does inferences for
the population data from which the giventhe population data from which the given
samples were taken.samples were taken.

 Inferential statistics is concerned with makingInferential statistics is concerned with making
predictions or inferences about a population frompredictions or inferences about a population from
observations and analyses of a sample. This providesobservations and analyses of a sample. This provides
information about the further surveys andinformation about the further surveys and
experiments.experiments.
 Inferential statistics enables the researcher to drawInferential statistics enables the researcher to draw
conclusions before the implementation of someconclusions before the implementation of some
particular organizational policy.particular organizational policy.
 For example - suppose we want to have an idea aboutFor example - suppose we want to have an idea about
the percentage of illiterates in our country. We take athe percentage of illiterates in our country. We take a
sample from the population and find the proportion ofsample from the population and find the proportion of
illiterates in the sample. This sample proportion withilliterates in the sample. This sample proportion with
the help of probability enables us to make somethe help of probability enables us to make some
inferences about the population proportion.inferences about the population proportion.

It is used the powerful tool of statisticalIt is used the powerful tool of statistical
methods based on the theory of probability.methods based on the theory of probability.
Examples of inferential statistics include –Examples of inferential statistics include –
Linear regression analysesLinear regression analyses
Logistic regression analysesLogistic regression analyses
ANOVAANOVA
Correlation analysesCorrelation analyses
Structural equation modelingStructural equation modeling
Survival analysis, to name a few.Survival analysis, to name a few.

HYPOTHESISHYPOTHESIS
 Hypothesis consists of two words: Hypo + thesis =Hypothesis consists of two words: Hypo + thesis =
Hypothesis.Hypothesis.
 ‘‘Hypo’ means tentative or subject to the verification andHypo’ means tentative or subject to the verification and
‘Thesis’ means statement about solution of a problem.‘Thesis’ means statement about solution of a problem.
 The term hypothesis is a tentative statement about theThe term hypothesis is a tentative statement about the
solution of the problem. Hypothesis offers a solution ofsolution of the problem. Hypothesis offers a solution of
the problem that is to be verified empirically and basedthe problem that is to be verified empirically and based
on some rationale.on some rationale.
 Another meaning of the word hypothesis which isAnother meaning of the word hypothesis which is
composed of two words – ‘Hypo’ means composition ofcomposed of two words – ‘Hypo’ means composition of
two or more variables which is to be verified. ‘Thesis’two or more variables which is to be verified. ‘Thesis’
means position of these variables in the specific framemeans position of these variables in the specific frame
of reference. This is the operational meaning of the termof reference. This is the operational meaning of the term
hypothesis.hypothesis.

 A hypothesis is a tentative statement about theA hypothesis is a tentative statement about the
relationship between two or more variables.relationship between two or more variables.
 A hypothesis is a specific, testable prediction about whatA hypothesis is a specific, testable prediction about what
you expect to happen in your study.you expect to happen in your study.
 To be complete the hypothesis must include threeTo be complete the hypothesis must include three
components - ●The variables; ●The population; andcomponents - ●The variables; ●The population; and
●●The relationship between the variables.The relationship between the variables.
 Remember, a hypothesis does not have to be correct.Remember, a hypothesis does not have to be correct.
When conducting an experiment, researchers mightWhen conducting an experiment, researchers might
explore a number of different factors to determine whichexplore a number of different factors to determine which
ones might contribute to the ultimate outcome. In manyones might contribute to the ultimate outcome. In many
cases, researchers may find the results of an experimentcases, researchers may find the results of an experiment
do not support the original hypothesis.do not support the original hypothesis.
HypothesisHypothesis

A research hypothesis is a prediction of the outcome of aA research hypothesis is a prediction of the outcome of a
study. The prediction may be based on an educated guessstudy. The prediction may be based on an educated guess
or a formal theory. Example 1 is a hypothesis for a non-or a formal theory. Example 1 is a hypothesis for a non-
experimental study.experimental study.
 Example 1:Example 1: It is hypothesized that first grade girls willIt is hypothesized that first grade girls will
show better reading comprehension than first grade boys.show better reading comprehension than first grade boys.
The author is predicting that s/he will find higherThe author is predicting that s/he will find higher
comprehension among girls than boys. To test it, a non-comprehension among girls than boys. To test it, a non-
experimental study would be appropriate because nothingexperimental study would be appropriate because nothing
in the hypothesis suggests that treatments will be given.in the hypothesis suggests that treatments will be given.
A simple research hypothesis predicts a relationshipA simple research hypothesis predicts a relationship
between two variables. Two variables in Example 1 are (1)between two variables. Two variables in Example 1 are (1)
gender and (2) reading comprehension. The hypothesisgender and (2) reading comprehension. The hypothesis
states that reading comprehension is related to gender.states that reading comprehension is related to gender.
ExamplesofHypothesisExamplesofHypothesis

Example 2 is a hypothesis for an experimental study.Example 2 is a hypothesis for an experimental study.
 Example 2:Example 2: It is hypothesized that children who arcIt is hypothesized that children who arc
shown a video with mild violence will be moreshown a video with mild violence will be more
aggressive on the playground than those who areaggressive on the playground than those who are
shown a similar video without the violence.shown a similar video without the violence.
In Example 2, theIn Example 2, the independent variableindependent variable is violenceis violence
(mild vs. none), and the(mild vs. none), and the dependent variabledependent variable isis
aggressiveness on the playground.aggressiveness on the playground.
The hypotheses in Examples 1 and 2 are examples ofThe hypotheses in Examples 1 and 2 are examples of
directional hypotheses. In a direc-tional hypothesis,directional hypotheses. In a direc-tional hypothesis,
we predict which group will be higher or have morewe predict which group will be higher or have more
of something.of something.

Sometimes we have a non-directional hypothesis.Sometimes we have a non-directional hypothesis.
 Example 3:Example 3: It is hypothesized that the child-rearingIt is hypothesized that the child-rearing
practices of Tribe A are different from those of Tribe B.practices of Tribe A are different from those of Tribe B.
The author of Example 3 is saying that there will be aThe author of Example 3 is saying that there will be a
difference but does not predict the direction of thedifference but does not predict the direction of the
difference. This is perfectly acceptable when there is nodifference. This is perfectly acceptable when there is no
basis for making an educated guess.basis for making an educated guess.
The choice of hypothesis is purely a matter ofThe choice of hypothesis is purely a matter of
personal taste - all are acceptable in the scientificpersonal taste - all are acceptable in the scientific
community.community.
Of course, when we are willing to predict the outcomeOf course, when we are willing to predict the outcome
of a study, we should state a directional hypothesis.of a study, we should state a directional hypothesis.

BASIC CONCEPTSBASIC CONCEPTS
Null HypothesisNull Hypothesis
 Hypothesis that proposes no relationship or differenceHypothesis that proposes no relationship or difference
between two variables.between two variables.
 There is no relationship between two groups that theThere is no relationship between two groups that the
researcher compares on a certain variable.researcher compares on a certain variable.
 There is no significant difference when different groups areThere is no significant difference when different groups are
compared with respect to a particular variable.compared with respect to a particular variable.
 Example, ‘There is no difference in the academic performance ofExample, ‘There is no difference in the academic performance of
high school students who participate in extracurricular activitieshigh school students who participate in extracurricular activities
and those who do not participate in such activities’ is a nulland those who do not participate in such activities’ is a null
hypothesis. It asserts that there is no true difference in the samplehypothesis. It asserts that there is no true difference in the sample
statistic and population parameter under consideration and thatstatistic and population parameter under consideration and that
the difference found is accidental arising out of fluctuations ofthe difference found is accidental arising out of fluctuations of
sampling.sampling.
 The word ‘null’ which means invalid, void, or a mounting toThe word ‘null’ which means invalid, void, or a mounting to
nothing. It is denoted asnothing. It is denoted as HH00..

Decision on
Null Hypothesis
States of Nature
Null Hypothesis
True
Null Hypothesis
False
Accept Correct Decision
Probability = 1-
α
Type II error
Probability = β
Reject Type I error
Probability = α
(α is called
significance level)
Correct Decision
Probability = 1-
β
(1- β is called
power of a test)
The rejection of the null hypothesis indicates that the differences have statisticalThe rejection of the null hypothesis indicates that the differences have statistical
significance and the acceptance of the null hypothesis indicates that thesignificance and the acceptance of the null hypothesis indicates that the
differences are due to chance.differences are due to chance.
States of Nature and Decisions on Null HypothesisStates of Nature and Decisions on Null Hypothesis
Null HypothesisNull Hypothesis BASIC CONCEPTSBASIC CONCEPTS
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Alternate or Research HypothesisAlternate or Research Hypothesis
 This hypothesis proposes a relationship between two orThis hypothesis proposes a relationship between two or
more variables, symbolized as Hmore variables, symbolized as H11..
 For example, if a researcher was interested inFor example, if a researcher was interested in
examining the relationship between music and emotion,examining the relationship between music and emotion,
s/he may believe that there is a relationship betweens/he may believe that there is a relationship between
music and emotion.music and emotion.
HH11 (the research/alternate hypothesis): Music at a fast tempo is(the research/alternate hypothesis): Music at a fast tempo is
rated by participants as being happier than music at a slow tempo.rated by participants as being happier than music at a slow tempo.
HH00 (the null hypothesis): Music at a fast tempo and at a slow tempo(the null hypothesis): Music at a fast tempo and at a slow tempo
is rated the same in happiness by participants.is rated the same in happiness by participants.
 The two hypotheses we propose to test must beThe two hypotheses we propose to test must be
mutually exclusive;mutually exclusive; i.e., when one is true the other musti.e., when one is true the other must
be false. And we see that they must bebe false. And we see that they must be exhaustiveexhaustive; they; they
must include all possible occurrences.must include all possible occurrences.

Statistical HypothesisStatistical Hypothesis
Statistical hypothesis is anStatistical hypothesis is an
assumption about statisticalassumption about statistical
populations that one seeks topopulations that one seeks to
support or refute.support or refute.
The null hypothesis andThe null hypothesis and
alternative hypothesis togetheralternative hypothesis together
are called statistical hypothesis.are called statistical hypothesis.

Type I ErrorType I Error
 Also known as an error of the first kind, occurs whenAlso known as an error of the first kind, occurs when
the null hypothesis (the null hypothesis (HH00) is true, but is rejected.) is true, but is rejected.
 It is asserting something that is absent, a false hit.It is asserting something that is absent, a false hit.
 A false positive error, or in short false positive,A false positive error, or in short false positive,
commonly called a ‘false alarm’, is a result thatcommonly called a ‘false alarm’, is a result that
indicates a given condition has been fulfilled, when itindicates a given condition has been fulfilled, when it
actually has not been fulfilled.actually has not been fulfilled.
 The rate of the type I error is called the size of the testThe rate of the type I error is called the size of the test
and denoted by the Greek letterand denoted by the Greek letter αα (alpha).(alpha).
 It usually equals the significance level of a test. In theIt usually equals the significance level of a test. In the
case of a simple null hypothesiscase of a simple null hypothesis αα is the probability of ais the probability of a
type I error.type I error.
Type -I & Type - II ErrorsType -I & Type - II Errors

Type II ErrorType II Error
 Also known as an error of the second kind, occursAlso known as an error of the second kind, occurs
when the null hypothesis is false, but erroneously failswhen the null hypothesis is false, but erroneously fails
to be rejected.to be rejected.
 It is failing to assert what is present, a miss.It is failing to assert what is present, a miss.
 A false negative error, or in short false negative, isA false negative error, or in short false negative, is
where a test result indicates that a condition failed,where a test result indicates that a condition failed,
while it actually was successful.while it actually was successful.
 A false negative error is a type II error occurring in testA false negative error is a type II error occurring in test
steps where a single condition is checked for and thesteps where a single condition is checked for and the
result can either be positive or negative.result can either be positive or negative.
 The rate of the type II error is denoted by the GreekThe rate of the type II error is denoted by the Greek
letterletter ββ (beta) and related to the power of a test (which(beta) and related to the power of a test (which
equals 1−equals 1−ββ).).

Null hypothesis (HNull hypothesis (H00))
is trueis true
Alternative hypothesisAlternative hypothesis
(H(H11) is true) is true
AcceptAccept
HH11
Type I error (FalseType I error (False
Alarm)Alarm)
Correct DecisionCorrect Decision
AcceptAccept
HH00
Correct DecisionCorrect Decision Type II error (Miss)Type II error (Miss)
When a statistical hypothesis is tested, there are four possible resultsWhen a statistical hypothesis is tested, there are four possible results ––
(a)(a)When we acceptWhen we accept HH11 andand HH11 is true, this is a correct decisionis true, this is a correct decision
about nature.about nature.
(b) When we accept(b) When we accept HH00 andand HH00 is true, this is also a correctis true, this is also a correct
decision about nature.decision about nature.
(c) The acceptance of(c) The acceptance of HH11 whenwhen HH00 is true, is called ais true, is called a ‘‘Type IType I’’
error. Iterror. It’’s incorrect decision about nature.s incorrect decision about nature.
(d) The acceptance of(d) The acceptance of HH00 whenwhen HH11 is true, is called ais true, is called a ‘‘Type IIType II’’
error. Iterror. It’’s also incorrect decision about nature.s also incorrect decision about nature.
Tabularized relations between truth/falseness of the nullTabularized relations between truth/falseness of the null
hypothesis and outcomes of the test-hypothesis and outcomes of the test-
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In 1948, Frederick Mosteller (1916–2006) argued that a ‘third kind of error’ wasIn 1948, Frederick Mosteller (1916–2006) argued that a ‘third kind of error’ was
required to describe circumstances he had observed, namely-required to describe circumstances he had observed, namely-
 Type I error: Rejecting the null hypothesis when it is true.Type I error: Rejecting the null hypothesis when it is true.
 Type II error: Accepting the null hypothesis when it is false.Type II error: Accepting the null hypothesis when it is false.
 Type III error: Correctly rejecting the null hypothesis for theType III error: Correctly rejecting the null hypothesis for the
wrong reason.wrong reason.
Type I and type II errors are part of the process of hypothesisType I and type II errors are part of the process of hypothesis
testing. Although the errors cannot be completely eliminated,testing. Although the errors cannot be completely eliminated,
we can minimize one type of error. Typically when we try towe can minimize one type of error. Typically when we try to
decrease the probability one type of error, the probability fordecrease the probability one type of error, the probability for
the other type increases. We could decrease the value ofthe other type increases. We could decrease the value of
alpha from 0.05 to 0.01, corresponding to a 99% level ofalpha from 0.05 to 0.01, corresponding to a 99% level of
confidence. However, if everything else remains the same,confidence. However, if everything else remains the same,
then the probability of a type II error will nearly alwaysthen the probability of a type II error will nearly always
increase.increase.

QuestionsQuestions
AnswersAnswers
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