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Aronchpt3correlation

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Aronchpt3correlation

2. 2. Correlations Can be thought of as a descriptive statistic for the relationship b h l i hi between two variables i bl Describes the relationship between two equal- interval 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 Copyright © 2011 by Pearson Education, Inc. All rights reserved
3. 3. Scatter Diagram
4. 4. Graphing a Scatter DiagramTo make a scatter diagram: Draw the axes and decide which variable goes on which axis 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 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 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- vertical axis variable and mark a clear dot. Keep going until you have marked a dot for each person. Copyright © 2011 by Pearson Education, Inc. All rights reserved
5. 5. Linear CorrelationA linear correlation◦ relationship between two variables that shows up on a scatter diagram as dots roughly approximating strai ht a ro imatin a straight line
6. 6. Curvilinear Correlation Curvilinear 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
8. 8. Positive and Negative LinearCorrelationPositive Correlation High scores go with high scores. Low scores go with low scores. Medium scores go with medium scores scores. When graphed, the line goes up and to the right. e.g., level of education achieved and incomeNegative Correlation g High scores go with low scores. e.g., the relationship between fewer hours of sleep and higher levels of stressStrength of the Correlation how close the dots on a scatter diagram fall to a simple straight line Copyright © 2011 by Pearson Education, Inc. All rights reserved
9. 9. Importance of Identifying thePattern of Correlation Use a scatter diagram to examine the pattern, direction, and strength of a correlation correlation. ◦ First, determine whether it is a linear or curvilinear relationship. ◦ If linear, look to see if it is a positive or negative correlation. l i ◦ 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. Copyright © 2011 by Pearson Education, Inc. All rights reserved
10. 10. The Correlation CoefficientA number that gives the exact correlationbetween 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 t diff t i bl Copyright © 2011 by Pearson Education, Inc. All rights reserved
11. 11. The Correlation Coefficient(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
12. 12. Strength of Correlation CoefficientsCorrelation Coefficient Value Strength of Relationship+/- .70-1.00 Strong g+/- .30-.69 Moderate+/- .00-.29 None (.00) to Weak The value of a correlation defines the strength of the correlation regardless of the sign sign. e.g., -.99 is a stronger correlation than .75
13. 13. Formula for a CorrelationCoefficient r = ∑ZxZy N Zx = Z score for each person on the X variable Zy = Z score for each person on the Y variable f h h bl ZxZy = cross-product of Zx and Zy ∑ZxZy = sum of the cross-products of the Z scores over all participants in the study Copyright © 2011 by Pearson Education, Inc. All rights reserved
14. 14. Steps for Figuring the CorrelationCoefficientC ffi iChange all scores to Z scores.◦ Figure the mean and the standard deviation of each variable.◦ Change each raw score to a Z score score.Calculate the cross-product of the Z scoresfor 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 scores cross products scores.Divide by the number of people in thestudy. y Copyright © 2011 by Pearson Education, Inc. All rights reserved
15. 15. Calculating a Correlation Coefficient gNumber 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 0 61 2 ‐1.05 1 05 0.64 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 63 SD= 1.91 SD 1 91 r=5.14/6 5 14/6 ΣZXZY r=ΣZXZY r=.85
16. 16. Issues in Interpreting theCorrelation Coefficient Direction of causality y ◦ path of causal effect (e.g., X causes Y) You cannot determine the direction of causality just because two variables are correlated. Copyright © 2011 by Pearson Education, Inc. All rights reserved
17. 17. Reasons Why We cannot AssumeCausality 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 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 Copyright © 2011 by Pearson Education, Inc. All rights reserved
18. 18. Ruling Out Some PossibleDirections of CausalityLongitudinal 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 pointTrue Experiment◦ a study in which participants are randomly assigned to a particular level of a variable and then measured on another variable h d h i bl e.g., exposing individuals to varying amounts of sleep in a laboratory environment and then evaluating their stress levels Copyright © 2011 by Pearson Education, Inc. All rights reserved
19. 19. The Statistical Significance of a CorrelationCoefficientA correlation is statistically significant if it isunlikely that you could have gotten acorrelation as big as you did if in fact therewas 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.
20. 20. PredictionPredictor Variable (X) variable being predicted from e.g., level of education achievedCriterion Variable (Y) variable being predicted to e.g., e g incomeIf we expect level of education to predict income, thepredictor variable would be level of education andthe criterion variable would b i h it i i bl ld be income. Copyright © 2011 by Pearson Education, Inc. All rights reserved
21. 21. Prediction Using Z ScoresPrediction Model A person’s predicted Z score on the criterion variable is found by multiplying the standardized regression coefficient (β) by that person s Z score person’s on the predictor variable.Formula for the prediction model using Z scores: Predicted P di t d 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 Copyright © 2011 by Pearson Education, Inc. All rights reserved
22. 22. Steps for Prediction Using Z Scores Determine the standardized regression g coefficient (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 Copyright © 2011 by Pearson Education, Inc. All rights reserved
23. 23. How Are You Doing? So, let’s say that we want to try to predict a person’s oral presentation score b d on a ’ l t ti based known relationship between self-confidence and presentation ability. p yWhich is the predictor variable (Zx)? The criterion variable (Zy)? If r = .90 and Zx = 2 25 th Zy = ? 90 d 2.25 then So what? What does this predicted value tell us? Copyright © 2011 by Pearson Education, Inc. All rights reserved
24. 24. Prediction Using Raw ScoresChange the person’s raw score on the predictor person svariable to a Z score.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 Copyright © 2011 by Pearson Education, Inc. All rights reserved
25. 25. Example of Prediction Using RawScores: Change Raw Scores to ZS Ch R S tScores 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. p 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 Copyright © 2011 by Pearson Education, Inc. All rights reserved
26. 26. Example of Prediction Using RawScores: Find the Predicted Z Scoreo t C t o a abon the Criterion Variable Multiply the standardized regression coefficient (β) by the person’s Z score on the predictor person s variable. ◦ Multiply β by Zx. py y This gives the predicted Z score on the criterion variable. Predicted Zy = (β)(Zx) = (.85)(-1.84) = -1.56 Copyright © 2011 by Pearson Education, Inc. All rights reserved
27. 27. Example of Prediction Using Raw p gScores: Change Raw Scores to ZScoresChange the person’s predicted Z score on thecriterion variable to a raw score score.◦ Predicted Y = (SDy)(Predicted Zy) + My◦ Predicted Y = (1.92)(-1.56) + 4 = -3 00 + 4 = (1 92)(-1 56) -3.00 1.00 Copyright © 2011 by Pearson Education, Inc. All rights reserved
28. 28. The Correlation Coefficient and theProportion of Variance Accounted forP fV A dfProportion of variance accounted for (r2)◦ To compare correlations with each other, you have to square each correlation 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 r◦ Where, a r= .4, you have an r2= .16◦ So, relationship with r = .4 is 4x stronger than , p g r=.2