Guide to Pearson Correlation: Understanding Variable Relationships - Prof. Morris Okun, Study notes of Statistics

This study guide provides an in-depth exploration of the pearson correlation coefficient, a statistical measure used to determine the linear relationship between two continuous variables. The difference between experimental and correlational approaches, the question addressed with correlational research, the bivariate distribution and scatter plots, the definition and interpretation of pearson r, and the computation of the correlation coefficient. It also includes examples and exercises to help students understand the concepts.

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Pre 2010

Uploaded on 09/02/2009

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Okun
PSY 230
STUDY GUIDE #8
Pearson Correlation Coefficient
I. Pearson Correlation Coefficient
1. What is the difference between the experimental and correlational approaches to
research?
2. What question is addressed with correlational research with continuous variables?
“Is there a relationship between the relative standing of individuals on one variable (low,
medium, or high) and their relative standing (low, medium, or high) on a second
variable?”
Person # of hours slept Mood
(X) (Y)
______________________________________________________________
Aaron 9 10
Brooke 8 7
Carole 5 4
Donald 3 2
Elvis 1 1
_______________________________________________________________
3. When is the correlational approach useful?
4. What is a bivariate distribution? How can a bivariate distribution be plotted on a graph?
A bivariate distribution involves pairs of scores from each individual in the sample or
population.
A scatter plot is a graph that represents the pair of observations for each person. Each pair of
observations for an individual is represented by a dot placed at the intersection of the X score
and the Y score. Thus, a scatter plot is a graph of a bivariate distribution.
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Okun PSY 230 STUDY GUIDE # Pearson Correlation Coefficient I. Pearson Correlation Coefficient

  1. What is the difference between the experimental and correlational approaches to research?
  2. What question is addressed with correlational research with continuous variables? “Is there a relationship between the relative standing of individuals on one variable (low, medium, or high) and their relative standing (low, medium, or high) on a second variable?” Person # of hours slept Mood (X) (Y) ______________________________________________________________ Aaron 9 10 Brooke 8 7 Carole 5 4 Donald 3 2 Elvis 1 1 _______________________________________________________________
  3. When is the correlational approach useful?
  4. What is a bivariate distribution? How can a bivariate distribution be plotted on a graph? A bi variate distribution involves pairs of scores from each individual in the sample or population. A scatter plot is a graph that represents the pair of observations for each person. Each pair of observations for an individual is represented by a dot placed at the intersection of the X score and the Y score. Thus, a scatter plot is a graph of a bi variate distribution.

Person Manual Number of Typing Errors (Y) Dexterity (X)


Fred 50 15 Gene 50 12 Heidi 100 12 Irene 150 6 Janet 150 0


Mean 100 9 Standard Deviation 50 6


  1. How can the Pearson r be defined? The Pearson Product Moment Correlation Coefficient is an index with a sign that indicates the direction and a numerical value that indicates the strength of the linear relationship between two variables. The population symbol for the Pearson Product Moment Correlation Coefficient is rho []. The sample symbol for this correlation coefficient is r. The Pearson r can take on values ranging from -1.0 to +1.0. The direction of the relationship is indicated by the sign, positive or negative, associated with the Pearson r. The strength of the relationship is indicated by the absolute value of the numerical value associated with the Pearson r. A perfect positive relationship is associated with a Pearson r of +1.0, whereas a perfect negative relationship is associated with a Pearson r of -1.0. A Pearson r of .00 indicates the absence of a linear relationship. A correlation coefficient indicates whether each person’s relative standing in the X distribution is related to his or her relative standing in the Y distribution.

A researcher is interested in determining whether there is a relationship between a manual dexterity test and number of errors made on a typing test. She draws a sample of 5 high school students learning to type. Below are the scores on the manual dexterity test and number of errors made on the typing test. In addition means and standard deviations are provided for both variables. Raw Data for Computing the Correlation between 2 Interval Variables Person Manual # of Typing Errors (Y) Dexterity (X)


Fred 50 15 Gene 50 12 Heidi 100 12 Irene 150 6 Janet 150 0


Mean 100 9 Standard Deviation 50 6


for the X variable, z = [Xi- M x] /Sx for the Y variable, z = [Yi- M y] /Sy

  1. What is the null hypothesis? =.
  2. What needs to be computed to test this null hypothesis?
  3. How many degrees of freedom are associated with this test? Df = n-
  4. How can the critical value be determined given an alpha level? [See Page 8 of the Study Guide.]
  5. How can a conclusion be drawn regarding the null hypothesis?
  6. What cautions need to be considered in interpreting Pearson r s?

Steps in Computing Power. STEP 1: Determine gamma (), the population effect size. In the case of the correlation coefficient, gamma equals the absolute value of A. STEP 2: Determine delta (). Delta combines the estimate of the population effect size and sample size.


delta = /A/ x  N- STEP 3: Enter Table H with an a priori specified value for  and the value of delta that was just computed.

Given the desired power, , and A how can the sample size be determined for a study examining a Pearson correlation coefficient? STEP 1: Entering Table I with the desired power and a priori specified value of , determine delta. STEP 2: Solve for n using the formula, n = (DELTA/A)^2 + 1.