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# Aron chpt 3 correlation

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### Aron chpt 3 correlation

2. 2. Can be thought of as a descriptive statistic for the relationship between two variables<br />Describes the relationship between two equal-interval numeric variables<br />e.g., the correlation between amount of time studying and amount learned <br />e.g., the correlation between number of years of education and salary<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Correlations<br />
3. 3. Scatter Diagram<br />
4. 4. To make a scatter diagram:<br />Draw the axes and decide which variable goes on which axis.<br />The values of one variable go along the horizontal axis and the values of the other variable go along the vertical axis.<br />Determine the range of values to use for each variable and mark them on the axes.<br />Numbers should go from low to high on each axis starting from where the axes meet .<br />Usually your low value on each axis is 0.<br />Each axis should continue to the highest value your measure can possibly have.<br />Make a dot for each pair of scores.<br />Find the place on the horizontal axis for the first pair of scores on the horizontal-axis variable.<br />Move up to the height for the score for the first pair of scores on the vertical-axis variable and mark a clear dot.<br />Keep going until you have marked a dot for each person.<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Graphing a Scatter Diagram<br />
5. 5. A linear correlation<br />relationship between two variables that shows up on a scatter diagram as dots roughly approximating a straight line<br />Linear Correlation<br />
6. 6. Curvilinear Correlation<br />Curvilinear correlation<br />any association between two variables other than a linear correlation<br />relationship between two variables that shows up on a scatter diagram as dots following a systematic pattern that is not a straight line<br />
8. 8. Positive Correlation<br />High scores go with high scores.<br />Low scores go with low scores.<br />Medium scores go with medium scores.<br />When graphed, the line goes up and to the right.<br />e.g., level of education achieved and income <br />Negative Correlation<br />High scores go with low scores. <br />e.g., the relationship between fewer hours of <br />sleep and higher levels of stress<br />Strength of the Correlation<br />how close the dots on a scatter diagram fall to a simple straight line<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Positive and Negative Linear Correlation<br />
9. 9. Use a scatter diagram to examine the pattern, direction, and strength of a correlation.<br />First, determine whether it is a linear or curvilinear relationship.<br />If linear, look to see if it is a positive or negative correlation.<br />Then look to see if the correlation is large, small, or moderate.<br />Approximating the direction and strength of a correlation allows you to double check your calculations later.<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Importance of Identifying the Pattern of Correlation<br />
10. 10. A number that gives the exact correlation between two variables<br />can tell you both direction and strength of relationship between two variables (X and Y)<br />uses Z scores to compare scores on different variables<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />The Correlation Coefficient<br />
11. 11. The Correlation Coefficient ( r )<br />The sign of r (Pearson correlation coefficient) tells the general trend of a relationship between two variables.<br />+ sign means the correlation is positive.<br />- sign means the correlation is negative.<br />The value of r ranges from -1 to 1.<br />A correlation of 1 or -1 means that the variables are perfectly correlated.<br />0 = no correlation<br />
12. 12. Strength of Correlation Coefficients<br />The value of a correlation defines the strength of the correlation regardless of the sign.<br />e.g., -.99 is a stronger correlation than .75<br />
13. 13. r = ∑ZxZy<br />N<br />Zx = Z score for each person on the X variable<br />Zy = Z score for each person on the Y variable<br />ZxZy = cross-product of Zx and Zy <br />∑ZxZy = sum of the cross-products of the Z scores over all participants in the study<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Formula for a Correlation Coefficient<br />
14. 14. Change all scores to Z scores.<br />Figure the mean and the standard deviation of each variable.<br />Change each raw score to a Z score.<br />Calculate the cross-product of the Z scores for each person.<br />Multiply each person’s Z score on one variable by his or her Z score on the other variable.<br />Add up the cross-products of the Z scores.<br />Divide by the number of people in the study.<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Steps for Figuring the Correlation Coefficient<br />
15. 15. Calculating a Correlation Coefficient<br />
16. 16. Direction of causality<br />path of causal effect (e.g., X causes Y)<br />You cannot determine the direction of causality just because two variables are correlated.<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Issues in Interpreting the Correlation Coefficient<br />
17. 17. Variable X causes variable Y.<br />e.g., less sleep causes more stress <br />Variable Y causes variable X.<br />e.g., more stress causes people to sleep less<br />There is a third variable that causes both variable X and variable Y.<br />e.g., working longer hours causes both stress and fewer hours of sleep<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Reasons Why We cannot Assume Causality<br />
18. 18. Longitudinal Study<br />a study where people are measured at two or more points in time<br />e.g., evaluating number of hours of sleep at one time point and then evaluating their levels of stress at a later time point<br />True Experiment<br />a study in which participants are randomly assigned to a particular level of a variable and then measured on another variable<br />e.g., exposing individuals to varying amounts of sleep in a laboratory environment and then evaluating their stress levels <br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Ruling Out Some Possible Directions of Causality<br />
19. 19. The Statistical Significance of a Correlation Coefficient<br />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.<br /> If the probability (p) is less than some small degree of probability (e.g., 5% or 1%), the correlation is considered statistically significant.<br />
20. 20. Predictor Variable (X)<br />variable being predicted from<br />e.g., level of education achieved<br />Criterion Variable (Y)<br />variable being predicted to <br />e.g., income<br />If we expect level of education to predict income, the predictor variable would be level of education and the criterion variable would be income.<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Prediction<br />
21. 21. Prediction Model<br />A 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.<br />Formula for the prediction model using Z scores:<br />Predicted Zy = ()(Zx) <br />Predicted Zy = predicted value of the particular person’s Z score on the criterion variable Y<br />Zx = particular person’s Z score in the predictor variable X<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Prediction Using Z Scores<br />
22. 22. Determine the standardized regression coefficient ().<br />Multiply the standardized regression coefficient () by the person’s Z score on the predictor variable.<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Steps for Prediction Using Z Scores<br />
23. 23.
24. 24. 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. <br />Which is the predictor variable (Zx)? The criterion variable (Zy)?<br />If r = .90 and Zx = 2.25 then Zy = ?<br />So what? What does this predicted value tell us?<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />How Are You Doing?<br />
25. 25. Change the person’s raw score on the predictor variable to a Z score.<br />Multiply the standardized regression coefficient () by the person’s Z score on the predictor variable.<br />Multiply  by Zx.<br />This gives the predicted Z score on the criterion variable.<br />Predicted Zy = ()(Zx)<br />Change the person’s predicted Z score on the criterion variable back to a raw score.<br />Predicted Y = (SDy)(Predicted Zy) + My<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />Prediction Using Raw Scores<br />
26. 26. Example of Prediction Using Raw Scores: Change Raw Scores to Z Scores<br />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. <br />The correlation between sleep and mood is .85.<br />Change the person’s raw score on the predictor variable to a Z score.<br />Zx = (X - Mx) / SDx<br />(4-7) / 1.63 = -3 / 1.63 = -1.84<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />
27. 27. Example of Prediction Using Raw Scores: Find the Predicted Z Score on the Criterion Variable<br />Multiply the standardized regression coefficient () by the person’s Z score on the predictor variable.<br />Multiply  by Zx.<br />This gives the predicted Z score on the criterion variable.<br />Predicted Zy = ()(Zx) = (.85)(-1.84) = -1.56<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />
28. 28. Example of Prediction Using Raw Scores: Change Raw Scores to Z Scores<br />Change the person’s predicted Z score on the criterion variable to a raw score.<br />Predicted Y = (SDy)(Predicted Zy) + My<br />Predicted Y = (1.92)(-1.56) + 4 = -3.00 + 4 = 1.00<br />Copyright © 2011 by Pearson Education, Inc. All rights reserved<br />
29. 29. Proportion of variance accounted for (r2)<br />To compare correlations with each other, you have to square each correlation.<br />This number represents the proportion of the total variance in one variable that can be explained by the other variable.<br />If you have an r= .2, your r2= .04<br />Where, a r= .4, you have an r2= .16 <br />So, relationship with r = .4 is 4x stronger than r=.2<br />The Correlation Coefficient and the Proportion of Variance Accounted for<br />