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Lecture 5

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


Department
Psychology
Course Code
PSYC 3430
Professor
Peter K Papadogiannis
Lecture
5

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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 “trick” 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 (ρ = 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 student’s GPA is perfectly predictable (100%)
from the student’s 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 = 0.70, 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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