aronchpt3correlation

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Chapter 3C l i d P di iCorrelation and Prediction

Copyright © 2011 by Pearson Education, Inc. All rights reserved

CorrelationsCorrelationsCorrelationsCorrelationsCan be thought of as a descriptive statistic for h l i hi b i blthe relationship between two variables

Describes the relationship between two equal-interval numeric variablesinterval numeric variables◦ e.g., the correlation between amount of time

studying and amount learned y g◦ e.g., the correlation between number of years

of education and salary


Scatter DiagramScatter DiagramScatter DiagramScatter Diagram

Graphing a Scatter DiagramGraphing a Scatter DiagramGraphing a Scatter DiagramGraphing a Scatter DiagramTo make a scatter diagram:

Draw the axes and decide which variable goes on which axisDraw 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 axesthe 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 haveEach 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-Move up to the height for the score for the first pair of scores on the verticalaxis variable and mark a clear dot.Keep going until you have marked a dot for each person.


Linear CorrelationLinear CorrelationLinear CorrelationLinear Correlation

A linear correlation◦ relationship between two variables that shows

up on a scatter diagram as dots roughly a ro imatin a strai ht lineapproximating a straight line

Curvilinear CorrelationCurvilinear CorrelationCurvilinear CorrelationCurvilinear CorrelationCurvilinear correlation◦ any association between two variables other ◦ any association between two variables other

than a linear correlation◦ relationship between two variables that shows 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 CorrelationNo CorrelationNo correlation◦ no systematic relationship between two

variables


Positive and Negative Linear Positive and Negative Linear CorrelationCorrelation

Positive CorrelationPositive CorrelationHigh scores go with high scores.Low scores go with low scores.Medium scores go with medium scoresMedium scores go with medium scores.When graphed, the line goes up and to the right.

e.g., level of education achieved and income

Negative CorrelationgHigh 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


Importance of Identifying the Importance of Identifying the Pattern of CorrelationPattern of Correlation

Use a scatter diagram to examine the pattern, direction, and strength of a correlationand 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

l icorrelation.◦ 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.


The Correlation CoefficientThe Correlation CoefficientThe Correlation CoefficientThe Correlation Coefficient

A number that gives the exact correlation 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)

Z t diff t i bl◦ uses Z scores to compare scores on different variables


The Correlation Coefficient The Correlation Coefficient ( r )( r )

The sign of r (Pearson correlation The sign of r (Pearson correlation coefficient) tells the general trend of a relationship between two variables.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 CoefficientsStrength of Correlation CoefficientsStrength of Correlation CoefficientsStrength of Correlation Coefficients

Correlation Coefficient Value Strength of Relationship

+/- .70-1.00 Strongg

+/- .30-.69 Moderate

+/- .00-.29 None (.00) to Weak

The value of a correlation defines the strength of the correlation regardless of the signcorrelation regardless of the sign.

e.g., -.99 is a stronger correlation than .75

Formula for a Correlation Formula for a Correlation CoefficientCoefficient

r = ∑ZxZy

NZx = Z score for each person on the X variableZ Z f h h Y blZy = Z score for each person on the Y variableZxZy = cross-product of Zx and Zy

∑ZxZy = sum of the cross-products of the Z scores over all participants in the study


Steps for Figuring the Correlation Steps for Figuring the Correlation C ffi iC ffi iCoefficientCoefficient

Change all scores to Z scores.◦ Figure the mean and the standard deviation of each variable.◦ Change each raw score to a Z score◦ Change each raw score to a Z score.

Calculate the cross-product of the Z scores for each person.p◦ 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 scoresAdd up the cross products of the Z scores.Divide by the number of people in the study.y


Calculating a Correlation CoefficientCalculating a Correlation Coefficientgg

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 0 646 ‐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 ΣZXZY

SD 1 63 SD 1 91 5 14/6 ΣZXZYSD=1.63 SD=1.91 r=5.14/6 r=ΣZXZY

r=.85

Issues in Interpreting the Issues in Interpreting the Correlation CoefficientCorrelation Coefficient

Direction of causalityy◦ path of causal effect (e.g., X causes Y)

You cannot determine the direction You cannot determine the direction of causality just because two variables are correlated.variables are correlated.


Reasons Why We cannot Assume Reasons Why We cannot Assume CausalityCausality

Variable X causes variable Y.◦ e.g., less sleep causes more stress Variable Y causes variable X.◦ e.g., more stress causes people to sleep lessThere 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


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 timemore points in timee.g., evaluating number of hours of sleep at one time point and then evaluating their levels of stress at a later time point

True ExperimentTrue Experiment◦ a study in which participants are randomly

assigned to a particular level of a variable and h d h i blthen measured on another variablee.g., exposing individuals to varying amounts of sleep in a laboratory environment and then evaluating their stress levels


The Statistical Significance of a Correlation The Statistical Significance of a Correlation CoefficientCoefficientCoefficientCoefficientA correlation is statistically significant if it is unlikely that you could have gotten a unlikely that you could have gotten a correlation as big as you did if in fact there was no relationship between variables.p◦ If the probability (p) is less than some small degree

of probability (e.g., 5% or 1%), the correlation is considered statistically significant.

PredictionPredictionPredictionPrediction

Predictor Variable (X)Predictor Variable (X)variable being predicted from

e.g., level of education achieved

Criterion Variable (Y)variable being predicted to

e g incomee.g., income

If we expect level of education to predict income, the predictor variable would be level of education and h it i i bl ld b ithe criterion variable would be income.


Prediction Using Z ScoresPrediction Using Z ScoresPrediction Using Z ScoresPrediction Using Z Scores

Prediction ModelPrediction ModelA person’s predicted Z score on the criterion variable is found by multiplying the standardized regression coefficient (β) by that person’s Z score regression coefficient (β) by that person s Z score on the predictor variable.

Formula for the prediction model using Z scores:P di t d Z (β)(Z ) Predicted Zy = (β)(Zx) Predicted Zy = predicted value of the particular person’s Z score on the criterion variable Y

’ Zx = particular person’s Z score in the predictor variable X


Steps for Prediction Using Z ScoresSteps for Prediction Using Z ScoresSteps for Prediction Using Z ScoresSteps for Prediction Using Z Scores

Determine the standardized regression gcoefficient (β).Multiply the standardized regression u t p y t e sta a e eg ess o coefficient (β) by the person’s Z score on the predictor variable.p


How Are You Doing?How Are You Doing?How Are You Doing?How Are You Doing?So, let’s say that we want to try to predict a

’ l t ti b d person’s oral presentation score based on a known relationship between self-confidence and presentation ability. p y

Which is the predictor variable (Zx)? The criterion variable (Zy)?If 90 d Z 2 25 th Z ?If r = .90 and Zx = 2.25 then Zy = ?

So what? What does this predicted value tell us?


Prediction Using Raw ScoresPrediction Using Raw ScoresPrediction Using Raw ScoresPrediction Using Raw Scores

Change the person’s raw score on the predictor Change the person s raw score on the predictor variable to a Z score.Multiply the standardized regression coefficient (β) Multiply the standardized regression coefficient (β) by the person’s Z score on the predictor variable.

Multiply β by Zx.This gives the predicted Z score on the criterion variable.

Predicted Zy = (β)(Zx)

Change the person’s predicted Z score on the g p pcriterion variable back to a raw score.

Predicted Y = (SDy)(Predicted Zy) + My


Example of Prediction Using Raw Example of Prediction Using Raw S Ch R S t Z S Ch R S t Z Scores: Change Raw Scores to Z Scores: Change Raw Scores to Z ScoresScores

From 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.pChange the person’s raw score on the predictor variable to a Z score.◦ Z = (X M ) / SD◦ Zx = (X - Mx) / SDx

◦ (4-7) / 1.63 = -3 / 1.63 = -1.84


Example of Prediction Using Raw Example of Prediction Using Raw Scores: Find the Predicted Z Score Scores: Find the Predicted Z Score on the Criterion Variableon the Criterion Variableo t C t o a abo t C t o a ab

Multiply the standardized regression coefficient (β) by the person’s Z score on the predictor (β) by the person s Z score on the predictor variable.◦ Multiply β by Zx.p y β y

This gives the predicted Z score on the criterion variable.Predicted Zy = (β)(Zx) = (.85)(-1.84) = -1.56


Example of Prediction Using Raw Example of Prediction Using Raw p gp gScores: Change Raw Scores to Z Scores: Change Raw Scores to Z ScoresScoresScoresScores

Change the person’s predicted Z score on the criterion variable to a raw scorecriterion variable to a raw score.◦ Predicted Y = (SDy)(Predicted Zy) + My

◦ Predicted Y = (1 92)(-1 56) + 4 = -3 00 + 4 = ◦ Predicted Y = (1.92)(-1.56) + 4 = -3.00 + 4 = 1.00


The Correlation Coefficient and the The Correlation Coefficient and the P f V A d fP f V A d fProportion of Variance Accounted forProportion of Variance Accounted for

Proportion of variance accounted for (r2)Proportion of variance accounted for (r2)◦ To compare correlations with each other, you

have to square each correlationhave 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= .04If you have an r .2, your r .04◦ Where, a r= .4, you have an r2= .16 ◦ So, relationship with r = .4 is 4x stronger than , p g

r=.2

aronchpt3correlation

Business

correlation correlation

correlation coefficientr

thepattern of correlation

stronger correlation

low scores

pair of scores

axis axis

z scorescross products