Analyzing Chi-Squares and Correlations Dr. K. A. Korb University of Jos.

Analyzing Chi-Squares and Analyzing Chi-Squares and Correlations Correlations

Dr. K. A. KorbDr. K. A. Korb

University of JosUniversity of Jos

OutlineOutline

Chi-SquareChi-Square– PurposePurpose– Null HypothesesNull Hypotheses– InterpretingInterpreting– ReportingReporting

CorrelationCorrelation– PurposePurpose– InterpretingInterpreting– Null HypothesesNull Hypotheses– ReportingReporting

Dr. K. A. KorbUniversity of Jos

ANCOVA: Analysis of Covariance

ANOVA: Analysis of Variance

t-test CorrelationRegression

Yes

No

No

Yes

Yes

No

NoYes

Do you have more than 2 variables?

Do you have more than 2

groups or independent variables?

Do you have pre- and post-

tests?

Do you have categorical data only?

Chi-Square(χ2)

Yes

No

Do you have independent

and dependent variables?


Chi-SquareChi-Square


Purpose of Chi-SquarePurpose of Chi-Square

A descriptive research design is to determine the A descriptive research design is to determine the type of instruction that Nigerian children prefer.type of instruction that Nigerian children prefer.– The three possible types of instruction include online The three possible types of instruction include online

presentation, lecture, and discussion.presentation, lecture, and discussion.– This type of variable is called This type of variable is called nominal, nominal, or or categorical.categorical.

Nominal Nominal scale is one in which categories have no scale is one in which categories have no quantitative value. In other words, categories are just quantitative value. In other words, categories are just names.names.

– Since we cannot meaningfully assign numbers to the Since we cannot meaningfully assign numbers to the category of variables, we must use a Chi-Square test.category of variables, we must use a Chi-Square test.


Purpose of Chi-SquarePurpose of Chi-Square

The purpose of conducting a Chi-Square The purpose of conducting a Chi-Square analysis is to determine whether frequency analysis is to determine whether frequency counts are equivalently distributed.counts are equivalently distributed.– Two conditions must be met to conduct a Chi-SquareTwo conditions must be met to conduct a Chi-Square

Each and every data point falls into only one categoryEach and every data point falls into only one category

The sample size must be largeThe sample size must be large

The Chi-Square statistic compares the actual The Chi-Square statistic compares the actual frequency distribution to an expected frequency frequency distribution to an expected frequency distribution.distribution.


Chi-Square Null HypothesesChi-Square Null Hypotheses

The first step in creating null hypotheses for a The first step in creating null hypotheses for a Chi-Square is to determine the expected Chi-Square is to determine the expected frequency of your datafrequency of your data– The expected frequency distribution must be set The expected frequency distribution must be set

before before collecting your data.collecting your data.

Once you have set your expected frequency, the Once you have set your expected frequency, the null hypothesis is that there is no significant null hypothesis is that there is no significant difference between the expected and observed difference between the expected and observed frequency distributions.frequency distributions.


Chi-Square Null HypothesesChi-Square Null HypothesesExpected Frequency Expected Frequency

Distribution for Students’ Distribution for Students’ Instructional PreferenceInstructional Preference

Type of InstructionType of Instruction FrequencyFrequency

Online PresentationOnline Presentation 5050

LectureLecture 5050

DiscussionDiscussion 5050

There will be 150 students in the sample. There will be 150 students in the sample. The expected frequency distribution is that equal The expected frequency distribution is that equal numbers of students will prefer each type of instruction.numbers of students will prefer each type of instruction.Null hypothesis: Null hypothesis: There are no significant differences There are no significant differences between the frequency of students who prefer each of between the frequency of students who prefer each of the three types of instruction.the three types of instruction.Dr. K. A. Korb

University of Jos

Chi-Square Null HypothesesChi-Square Null Hypotheses

Expected Expected Frequency Frequency Distribution for Students’ Distribution for Students’ Instructional PreferenceInstructional Preference

Type of Type of InstructionInstruction

FrequencyFrequency

Online Online PresentationPresentation 5050

LectureLecture 5050


Observed Observed Frequency Frequency Distribution for Students’ Distribution for Students’ Instructional PreferenceInstructional Preference

Type of Type of InstructionInstruction

FrequencyFrequency

Online Online PresentationPresentation 7575

LectureLecture 1010


The Chi-Square statistic quantifies the discrepancy The Chi-Square statistic quantifies the discrepancy between the between the expectedexpected and and observedobserved frequency charts. frequency charts.


Interpreting Chi-SquareInterpreting Chi-Square

When calculating a Chi-Square, there are three When calculating a Chi-Square, there are three important statisticsimportant statistics– The The χχ22 (symbol for Chi-Square) (symbol for Chi-Square)

The The χχ22 is a function of the amount of discrepancy between the is a function of the amount of discrepancy between the expected and observed dataexpected and observed data

– The degrees of freedomThe degrees of freedomDegrees of freedom is the number of categories compared minus 1Degrees of freedom is the number of categories compared minus 1

– The example has 3 categories: online presentation, lecture, and The example has 3 categories: online presentation, lecture, and discussiondiscussion

– The degrees of freedom is (3 – 1 =) 2The degrees of freedom is (3 – 1 =) 2

– Probability Probability (p): (p): This tells whether the This tells whether the χχ22 is large enough for the is large enough for the results to generalize beyond the sampleresults to generalize beyond the sample

For Chi-Square statistics, you also must report the size For Chi-Square statistics, you also must report the size of your sample of your sample (N)(N)


Interpreting Chi-SquareInterpreting Chi-Square

The The p p value for a Chi-Square is interpreted exactly the value for a Chi-Square is interpreted exactly the same as for inferential statisticssame as for inferential statisticsIf the If the p p value is less than .05, the results are considered value is less than .05, the results are considered significant.significant.– The significance level that is set prior to conducting statistics is The significance level that is set prior to conducting statistics is

the alpha (the alpha (αα))– If the calculated If the calculated p p value is less than the alpha, then the null value is less than the alpha, then the null

hypothesis is rejected.hypothesis is rejected.Significant Significant p-values indicate that the results are NOT due to p-values indicate that the results are NOT due to chance and thus represent a meaningful difference between chance and thus represent a meaningful difference between the expected and observed frequencies.the expected and observed frequencies.It can then be concluded that a difference It can then be concluded that a difference does does exist exist between the expected and observed frequencies.between the expected and observed frequencies.


Reporting Chi-SquareReporting Chi-Square

There was a significant difference between the There was a significant difference between the expected and observed frequencies of students’ expected and observed frequencies of students’ preference of instruction, indicating that students preference of instruction, indicating that students do have one preferred method of instruction do have one preferred method of instruction ((χχ22(2, (2, NN = 150) = 10.51, = 150) = 10.51, pp = .03). = .03).– 2 is the degrees of freedom2 is the degrees of freedom– N N = 150 is the sample size= 150 is the sample size– 10.51 is the 10.51 is the χχ22 value value– p p = .03 is the significance= .03 is the significance


CorrelationCorrelation


Purpose of CorrelationsPurpose of Correlations

The purpose of a correlation is to The purpose of a correlation is to determine the relationship between two determine the relationship between two naturally occurring variablesnaturally occurring variables

The correlation quantifies two aspects of The correlation quantifies two aspects of the relationship between variables: the the relationship between variables: the nature and the strengthnature and the strength



Correlational research must meet two Correlational research must meet two conditions:conditions:– The variables must be naturally occurring (i.e. no The variables must be naturally occurring (i.e. no

treatment or control groups)treatment or control groups)– Both variables must be continuous in nature (i.e. vary Both variables must be continuous in nature (i.e. vary

on a continuum from low to high levels with many on a continuum from low to high levels with many possible points between)possible points between)

For example, a typical correlation between motivation and For example, a typical correlation between motivation and grade for junior secondary students would grade for junior secondary students would not not be possible be possible because only three grades exist – JS1, JS2, and JS3because only three grades exist – JS1, JS2, and JS3However, a correlation between motivation and age would be However, a correlation between motivation and age would be possible because age represents a continuous variablepossible because age represents a continuous variable



A correlational research study examines A correlational research study examines the relationship between the relationship between interestinterest in maths in maths and maths and maths performanceperformance..– Data is collected by administering a Data is collected by administering a

questionnaire to assess questionnaire to assess interestinterest in maths in maths– The researcher would also collect maths The researcher would also collect maths

performance data from school recordsperformance data from school records


Interpreting CorrelationsInterpreting Correlations

Correlational research one important statisticCorrelational research one important statistic– Correlation: Indicates the strength and direction of a Correlation: Indicates the strength and direction of a

relationshiprelationship

A A p p value can also be calculated for a correlationvalue can also be calculated for a correlation– The The p p value depends on both the magnitude of the value depends on both the magnitude of the

correlation and the size of the samplecorrelation and the size of the sample– The The p p value indicates whether the value indicates whether the correlation is correlation is

significant enough for the results to generalize to significant enough for the results to generalize to beyond the samplebeyond the sample

However, the correlation is the most meaningful However, the correlation is the most meaningful statistic for correlational research designs.statistic for correlational research designs.



Correlations range between -1.0 to +1.0Correlations range between -1.0 to +1.0NatureNature– Positive: Two variables increase or decrease Positive: Two variables increase or decrease

togethertogether– Negative: As one variable increases, the other Negative: As one variable increases, the other

decreasesdecreases

StrengthStrength– Closer to -1 or +1 is stronger relationshipCloser to -1 or +1 is stronger relationship– 0 is no relationship0 is no relationship



NatureNature– Positive: Two variables increase or decrease togetherPositive: Two variables increase or decrease together– Negative: As one variable increases, the other Negative: As one variable increases, the other

decreasesdecreases

StrengthStrength– Closer to -1 or +1 is stronger relationshipCloser to -1 or +1 is stronger relationship– 0 is no relationship0 is no relationship

Negative PositiveNature:

Strength:

0-1 +1



CorrelationCorrelation NatureNature StrengthStrength InterpretationInterpretation .95.95 PositivePositive StrongStrong The strong correlation indicates The strong correlation indicates

that as interest increases, that as interest increases, performance also increases.performance also increases.

-.10-.10 NegativeNegative WeakWeak There is very little relationship There is very little relationship between interest and between interest and performance.performance.

-.50-.50 NegativeNegative ModerateModerate The moderate correlation The moderate correlation indicates that as interest indicates that as interest increases, performance tends to increases, performance tends to decrease.decrease.

.05.05 PositivePositive Very WeakVery Weak There is no relationship between There is no relationship between interest and performance.interest and performance.

-.75-.75 NegativeNegative StrongStrong The strong correlation indicates The strong correlation indicates that as performance increases, that as performance increases, interest tends to decrease.interest tends to decrease.


Null Hypothesis for CorrelationsNull Hypothesis for Correlations

Since correlations quantify the Since correlations quantify the relationship, null hypotheses for relationship, null hypotheses for correlations are in terms of the relationship correlations are in terms of the relationship between variables.between variables.

There is no significant relationship There is no significant relationship between maths interest and maths between maths interest and maths achievement.achievement.


Reporting CorrelationsReporting Correlations

A correlation of .54 between maths A correlation of .54 between maths interest and maths performance was found interest and maths performance was found (df = 67, (df = 67, p p = .03), indicating that students = .03), indicating that students who are more interested in maths tend to who are more interested in maths tend to have higher maths performance.have higher maths performance.


Reporting CorrelationsReporting Correlations

Scatterplot of Maths Interest and Maths Performance

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100

0 2 4 6 8 10

Maths Interest

Math

s P

erf

orm

ance

This scatter plot graph created in Excel This scatter plot graph created in Excel demonstrates that as maths interest increases, so demonstrates that as maths interest increases, so does maths performance, a positive correlation.does maths performance, a positive correlation.


Analyzing Chi-Squares and Correlations Dr. K. A. Korb University of Jos.

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