Aron chpt 3 correlation compatability version f2011

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Aron, Coups, & Aron Aron, Coups, & Aron Chapter 3 Correlation and Prediction Copyright © 2011 by Pearson Education, Inc. All rights reserved

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Transcript of Aron chpt 3 correlation compatability version f2011

Page 1: Aron chpt 3 correlation compatability version f2011

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

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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

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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

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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

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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

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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

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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

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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.

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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

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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 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

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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