# Psychological Assessment - Chapter 3 Book Notes

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Published on 13 May 2011
School
UTSC
Department
Psychology
Course
PSYC37H3
Professor
Correlation and Regression
The Scatter Diagram
Univariate distribution: involve only one variable for each individual under study
Bivariate distribution: involves two scores for each individual
Scatter diagram: is a picture of the relationship between two variables
Bivariate distribution of scores
Each point shows were a particular individual scored on both x and y
Each point represents the performance of one person who has been assessed on two measures
Useful when the relationships between x and y are not described by a straight line
Correlation
In correlational analysis, we ask whether two variables covey and is designed mainly to examine linear
relationships between variables
Correlation coefficient: describes the direction and magnitude of a relationship
Positive correlation: high scores on y are associated with high scores on x, low scores on y are
associated with low scores on x
Negative correlation: higher scores on y are associated with lower scores on x, lower scores on y
are associated with higher scores on x
No correlation: variables are not related
Calculation of the correlation coefficient involve pairs of observations
For each observation on one variable, there is an observation on one other variable for the same
person
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Regression
Regression: is used to make predictions about scores on one variable from knowledge of scores on another
variable
The Regression Line
Regression line: the best-fitting straight line through a set of points in a scatter diagram
Found by using the principle of least squares, which minimizes the squared deviation around the
regression line
The mean is the point of least squares for any single variable, so the sum of the squared deviations
around the mean will be less than it is around any value other than the mean
The line can be referred to as the running mean in two dimensions or in the space created by two
variables
The least square method in regression finds the straight line that comes as close to as many of the
y means
Thus, regression line = the line for which the squared deviations around the line are at a minimum
Regression coefficient (b): is the slope of the regression line
Can be expressed as the ratio of the sum of squares for the covariance to the sum of squares for x
Sum of squares: the sum of squared deviations around the mean
For x = the sum of the squared deviations around the x variable
Covariance: expresses how much two measures vary together
Slope: how much change is expected in y each time increases by one unit
Regression coefficient is sometimes expressed in different notations
Beta is used for a population estimate of the regression coefficient
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Intercept: is the value of y when x is 0, or the point at which the regression line crosses the y axis
The Best-Fitting Line
The actual and predicted scores on y are rarely the same
A person actually received a score of 4 on y and the regression equation predicted the person
The difference between the observed and predicted score (y-y’) is called the residual
The best-fitting line keeps the residuals to a minimum by minimizing the deviation between
observed and predicted scores
Since residuals can be (+) or (-) and will cancel to zero if averaged, each residual is squared
The best-fitting line is obtained by keeping the squared residuals as small as possible, which is known as
the principle of least squares
Correlation is a special case of regression in which the scores of both variables are standardized, or in z
units
Having the scores in z units eliminates the need to find the intercept
In correlation, the intercept is always zero
The standardized unit allows ease in interpreting the slope in correlation
When calculating the correlation coefficient, we can avoid the step of changing all the scores in z
units
Pearson product moment correlation: coefficient is a ratio used to determine the degree of variation in one
variable that can be estimated from knowledge about variation in the other variable
The correlation coefficient can take on any value from -0.1 to 1.0
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