aron chpt 3 correlation compatability version f2011
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Aron, Coups, & Aron Aron, Coups, & Aron
Chapter 3Correlation and
Prediction
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CorrelationsCorrelationsCan be thought of as a descriptive
statistic for the relationship between two variables
Describes the relationship between two equal-interval numeric variables◦e.g., the correlation between amount
of time studying and amount learned ◦e.g., the correlation between number
of years of education and salary
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Scatter DiagramScatter Diagram
Graphing a Scatter Graphing a Scatter DiagramDiagram
To make a scatter diagram: Draw the axes and decide which variable goes on which axis.
The values of one variable go along the horizontal axis and the values of the other variable go along the vertical axis.
Determine the range of values to use for each variable and mark them on the axes.Numbers should go from low to high on each axis starting from
where the axes meet .Usually your low value on each axis is 0.Each axis should continue to the highest value your measure can
possibly have. Make a dot for each pair of scores.
Find the place on the horizontal axis for the first pair of scores on the horizontal-axis variable.
Move up to the height for the score for the first pair of scores on the vertical-axis variable and mark a clear dot.
Keep going until you have marked a dot for each person.
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Linear CorrelationLinear CorrelationA linear correlation
◦relationship between two variables that shows up on a scatter diagram as dots roughly approximating a straight line
Curvilinear CorrelationCurvilinear CorrelationCurvilinear correlation
◦any association between two variables other than a linear correlation
◦relationship between two variables that shows up on a scatter diagram as dots following a systematic pattern that is not a straight line
No CorrelationNo CorrelationNo correlation
◦no systematic relationship between two variables
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Positive and Negative Linear Positive and Negative Linear CorrelationCorrelation
Positive CorrelationHigh scores go with high scores.Low scores go with low scores.Medium scores go with medium scores.When graphed, the line goes up and to the right.
e.g., level of education achieved and income Negative Correlation
High scores go with low scores. e.g., the relationship between fewer hours of sleep and higher levels of stress
Strength of the Correlationhow close the dots on a scatter diagram fall to a simple
straight line
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Importance of Identifying the Importance of Identifying the Pattern of CorrelationPattern of CorrelationUse a scatter diagram to examine the
pattern, direction, and strength of a correlation.◦ First, determine whether it is a linear or curvilinear
relationship.◦ If linear, look to see if it is a positive or
negative correlation.◦ Then look to see if the correlation is large,
small, or moderate.Approximating the direction and strength of
a correlation allows you to double check your calculations later.
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The Correlation CoefficientThe Correlation Coefficient
A number that gives the exact correlation between two variables
◦ can tell you both direction and strength of relationship between two variables (X and Y)
◦ uses Z scores to compare scores on different variables
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The Correlation Coefficient The Correlation Coefficient ( r )( r )The sign of r (Pearson correlation
coefficient) tells the general trend of a relationship between two variables. + sign means the correlation is positive. - sign means the correlation is negative.
The value of r ranges from -1 to 1. A correlation of 1 or -1 means that the variables are
perfectly correlated. 0 = no correlation
Strength of Correlation Strength of Correlation CoefficientsCoefficients
Correlation Coefficient Value Strength of Relationship
+/- .70-1.00 Strong
+/- .30-.69 Moderate
+/- .00-.29 None (.00) to Weak
The value of a correlation defines the strength of the correlation regardless of the sign.
e.g., -.99 is a stronger correlation than .75
Formula for a Correlation Formula for a Correlation CoefficientCoefficientr = ∑ZxZy
N Zx = Z score for each person on the X variable Zy = Z score for each person on the Y variable ZxZy = cross-product of Zx and Zy
∑ZxZy = sum of the cross-products of the Z scores over all participants in the study
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Steps for Figuring the Correlation Steps for Figuring the Correlation CoefficientCoefficient
Change all scores to Z scores.◦ Figure the mean and the standard deviation of each
variable.◦ Change each raw score to a Z score.
Calculate the cross-product of the Z scores for each person.◦ Multiply each person’s Z score on one variable by his
or her Z score on the other variable.Add up the cross-products of the Z
scores.Divide by the number of people in the
study.Copyright © 2011 by Pearson Education, Inc. All rights reserved
Calculating a Correlation CoefficientCalculating a Correlation Coefficient
Number of Hours Slept (X) Level of Mood (Y) Calculate r
X Zscore Sleep Y Zscore Mood Cross Product ZXZY
5 -1.23 2 -1.05 1.28
7 0.00 4 0.00 0.00
8 0.61 7 1.57 0.96
6 -0.61 2 -1.05 0.64
6 -0.61 3 -0.52 0.32
10 1.84 6 1.05 1.93
MEAN=7 MEAN=4 5.14 ZXZY
SD=1.63 SD=1.91 r=5.14/6 r=ZXZY
r=.85
Issues in Interpreting the Issues in Interpreting the Correlation CoefficientCorrelation CoefficientDirection of causality
◦path of causal effect (e.g., X causes Y)
You cannot determine the direction of causality just because two variables are correlated.
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Reasons Why We cannot Reasons Why We cannot Assume CausalityAssume CausalityVariable X causes variable Y.
◦e.g., less sleep causes more stress Variable Y causes variable X.
◦e.g., more stress causes people to sleep less
There is a third variable that causes both variable X and variable Y.◦e.g., working longer hours causes
both stress and fewer hours of sleep
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Ruling Out Some Possible Ruling Out Some Possible Directions of CausalityDirections of Causality
Longitudinal Study◦a study where people are measured at
two or more points in time e.g., evaluating number of hours of sleep at one time
point and then evaluating their levels of stress at a later time point
True Experiment◦a study in which participants are randomly
assigned to a particular level of a variable and then measured on another variable e.g., exposing individuals to varying amounts of sleep in
a laboratory environment and then evaluating their stress levels
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The Statistical Significance of a The Statistical Significance of a Correlation CoefficientCorrelation Coefficient
A correlation is statistically significant if it is unlikely that you could have gotten a correlation as big as you did if in fact there was no relationship between variables.◦ If the probability (p) is less than some
small degree of probability (e.g., 5% or 1%), the correlation is considered statistically significant.
PredictionPrediction
Predictor Variable (X)variable being predicted from
e.g., level of education achievedCriterion Variable (Y)
variable being predicted toe.g., income
If we expect level of education to predict income, the predictor variable would be level of education and the criterion variable would be income.
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Prediction Using Z ScoresPrediction Using Z Scores
Prediction ModelA person’s predicted Z score on the
criterion variable is found by multiplying the standardized regression coefficient () by that person’s Z score on the predictor variable.
Formula for the prediction model using Z scores:Predicted Zy = ()(Zx) Predicted Zy = predicted value of the particular
person’s Z score on the criterion variable YZx = particular person’s Z score in the
predictor variable X
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Steps for Prediction Using Z Steps for Prediction Using Z ScoresScoresDetermine the standardized
regression coefficient ().Multiply the standardized
regression coefficient () by the person’s Z score on the predictor variable.
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How Are You Doing?How Are You Doing?So, let’s say that we want to try to
predict a person’s oral presentation score based on a known relationship between self-confidence and presentation ability.
Which is the predictor variable (Zx)? The criterion variable (Zy)?
If r = .90 and Zx = 2.25 then Zy = ?
So what? What does this predicted value tell us?
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Prediction Using Raw Prediction Using Raw ScoresScores
Change the person’s raw score on the predictor variable to a Z score.
Multiply the standardized regression coefficient () by the person’s Z score on the predictor variable.Multiply by Zx.
This gives the predicted Z score on the criterion variable. Predicted Zy = ()(Zx)
Change the person’s predicted Z score on the criterion variable back to a raw score.Predicted Y = (SDy)(Predicted Zy) + My
Copyright © 2011 by Pearson Education, Inc. All rights reserved
Copyright © 2011 by Pearson Education, Inc. All rights reserved
Example of Prediction Using Example of Prediction Using Raw Scores: Change Raw Raw Scores: Change Raw Scores to Z ScoresScores to Z ScoresFrom the sleep and mood study example, we
known the mean for sleep is 7 and the standard deviation is 1.63, and that the mean for happy mood is 4 and the standard deviation is 1.92.
The correlation between sleep and mood is .85.
Change the person’s raw score on the predictor variable to a Z score.◦ Zx = (X - Mx) / SDx
◦ (4-7) / 1.63 = -3 / 1.63 = -1.84
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Example of Prediction Using Example of Prediction Using Raw Scores: Find the Raw Scores: Find the Predicted Z Score on the Predicted Z Score on the Criterion VariableCriterion VariableMultiply the standardized regression
coefficient () by the person’s Z score on the predictor variable.◦Multiply by Zx.
This gives the predicted Z score on the criterion variable. Predicted Zy = ()(Zx) = (.85)(-1.84) = -1.56
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Example of Prediction Using Example of Prediction Using Raw Scores: Change Raw Raw Scores: Change Raw Scores to Z ScoresScores to Z ScoresChange the person’s predicted Z score
on the criterion variable to a raw score.◦Predicted Y = (SDy)(Predicted Zy) + My
◦Predicted Y = (1.92)(-1.56) + 4 = -3.00 + 4 = 1.00
The Correlation Coefficient and The Correlation Coefficient and the Proportion of Variance the Proportion of Variance Accounted forAccounted for
Proportion of variance accounted for (r2)◦To compare correlations with each other,
you have to square each correlation.◦This number represents the proportion of
the total variance in one variable that can be explained by the other variable.
◦If you have an r= .2, your r2= .04◦Where, a r= .4, you have an r2= .16 ◦So, relationship with r = .4 is 4x stronger
than r=.2