# PSYC 3430 Lecture Notes - Lecture 5: Level Of Measurement, Statistical Hypothesis Testing, Descriptive Statistics

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**preview**shows pages 1-3. to view the full**16 pages of the document.**Instructor Notes - Chapter 15 - page 217

Chapter 15: Correlation

Chapter Outline

15.1 Introduction

The Characteristics of a Relationship (Direction, Form, and Degree)

15.2 The Pearson Correlation

The Sum of Products of Deviations

Calculation of the Pearson Correlation

The Pearson Correlation and z-Scores

15.3 Using and Interpreting the Pearson Correlation

Where and Why Correlations are Used

Interpreting Correlations

Correlation and Causation

Correlation and Restricted Range

Outliers

Correlation and the Strength of the Relationship (r2)

15.4 Hypothesis Tests with the Pearson Correlation

The Hypotheses

Degrees of Freedom for the Correlation Test

The Hypothesis Test

In the Literature - Reporting Correlations

Partial Correlations

15.5 Alternatives to the Pearson Correlation

The Spearman Correlation

Ranking Tied Scores

Special Formula for the Spearman Correlation

Testing the Significance of the Spearman Correlation

The Point-Biserial Correlation and Measuring Effect Size with r2

Point-Biserial Correlation, Partial Correlation and Effect Size for the Repeated-Measures

t Test

The Phi-Coefficient

Learning Objectives and Chapter Summary

1. Understand the Pearson correlation as a descriptive statistic that measures and describes the

relationship between two variables.

The Pearson correlation measures the direction and the degree of linear relationship. The

sign of the correlation describes the direction of relationship; a positive correlation

indicates that X and Y tend to change in the same direction, and a negative correlation

indicates that X and Y tend to change in opposite directions. The numerical value of the

correlation measures the strength or consistency of the relationship, and describes how

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Instructor Notes - Chapter 15 - page 218

closely the data points fit on a straight line. A value of 1.00 indicates a perfect fit and a

value of zero indicates no fit at all.

2. Be able to compute the Pearson correlation using either the definitional or the computational

formula for SP (the sum of products of deviations).

The main new element in the calculation of the Pearson correlation is the sum of products

of deviations, SP. However, the concept and the calculation of SP are both very similar

to the concept and calculation of SS (the sum of squared deviations). To compute SS you

must square X values (X times X). To compute SP you must find products of X times Y.

The other ātrickā to the correlation formula is to remind students to multiply the SS

values in the denominator. (In many other formulas such as pooled variance, the two SS

values are added.)

3. Students should understand the concept of a partial correlation and be able to follow the

formula to calculate the partial correlation between two variables, holding a third variable

constant.

Partial correlations are valuable when researchers suspect that the true relationship

between two variables may be distorted by the influence of a third variable. The partial

correlation reveals the relationship while holding the third variable constant. Partial

correlations also help researcher interpret the individual contributions of separate

predictor variables in a multiple-regression equation.

4. Students should be able to use a sample correlation to test a hypothesis about the

corresponding population correlation.

The null hypothesis states that there is no correlation in the population (Ļ = 0). A sample

correlation (r) that is near zero provides support for the null hypothesis. On the other

hand, a sample correlation that is much different than zero will provide evidence to reject

H0. The hypothesis test consists of comparing the sample correlation with the critical

values in the table. If the sample correlation is greater than or equal to the table value,

then it is big enough to reject the null hypothesis and conclude that there is a significant,

non-zero correlation in the population.

5. Students should understand the Spearman correlation and how it differs from the Pearson

correlation in terms of the data that it uses and the type of relationship that it measures.

The Spearman correlation measures the relationship between two variables that are

both measured on an ordinal scale (both the X values and the Y values are ranks). The

Spearman correlation measures the degree of consistency of direction for the relationship

but does not require that the points cluster around a straight line. To compute the

Spearman correlation, the Pearson formula is applied to ordinal data.

6. Students should understand how the Pearson correlation formula can be used to compute a

point-biserial correlation to measure the relationship between two variable when one variable is

dichotomous.

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Instructor Notes - Chapter 15 - page 219

A dichotomous variable has exactly two categories (for example, male and female or

succeed and fail). A point-biserial correlation measures the relationship between one

dichotomous variable and one regular, numerical variable; for example the relationship

between gender and reaction time scores. The two categories for the dichotomous

variable are usually assigned values of 0 and 1, and then the Pearson formula is used to

compute the correlation. The point-biserial correlation is strongly related to the

independent-measures t hypothesis test. The value of the correlation (r) can be squared to

obtain a measure of effect size (r2) for the independent-measures t.

6. Students should understand how the Pearson correlation formula can be used to compute a

phi-coefficient to measure the relationship between two dichotomous variables.

The phi-coefficient is used to measure the relationship between two dichotomous

variables; for example, the relationship between gender (male/female) and color-blindness

(yes/no). The two categories are each variable are assigned values of 0 and 1, and the Pearson

formula is then used to compute the coefficient. The phi-coefficient is strongly related to the

chi-square test for independence that is discussed in Chapter 18.

Other Lecture Suggestions

1. As a general introduction to correlation, you can combine the concepts of relationship and

prediction in one example. Begin by supposing that grade point average is determined

exclusively by IQ. The student with the highest IQ has the highest GPA, second highest IQ goes

with the second highest GPA, and so on. Sketch a graph showing a perfect, straight-line

relationship. Also, note that in this situation a studentās GPA is perfectly predictable (100%)

from the studentās IQ score. In the real world however, it is extremely rare for one variable to

entirely control another. Grade point average, for example, is partially determined by IQ, but it

is also partially determined by other factors such as motivation and health. Thus, the student

with the highest IQ may not have much motivation and therefore does not have the highest GPA.

Start modifying the graph by moving points off the line. The result is that we now have a degree

of relationship, and we find that GPA is only partially predictable from IQ. Correlation provides

a method for measuring the degree of relationship and determining how much of one variable

can be predicted from another.

To demonstrate different degrees of relationship, you can introduce new variables. For

example, GPA is also related to family income; students from wealthy families tend to have

higher grade point averages than students from poorer families, but this relationship is not nearly

as strong as the relationship between IQ and GPA.

2. If students are given the value of a correlation, for example r = ā0.70, they should be able to

sketch a scatterplot showing how the data would appear. In this case the points are scattered

around a line that slopes down to the right and the cluster of points is shaped roughly like a

football. (Fatter than a football indicates a correlation closer to zero, and thinner than a football

indicates a correlation closer to 1.00.) Also, if students are shown a scatterplot, they should be

able to estimate the value of the correlation.

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