1. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved Chapter 3 Investigating the Relationship of Scores

2. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-2 Chapter Objectives After completing this chapter, you should be able to 1. Define correlation, linear correlation, interpret the correlation coefficient, and use the rank-difference and product-moment methods to determine the relationship between two variables. 2. Construct a scattergram and interpret it.

3. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-3 Linear Correlation Correlation - a statistical technique used to express the relationship between two sets of scores (two variables) Linear correlation – the degree to which a straight line best describes the relationship between two variables Examples: longevity and exercise, smoking and cancer, intramural participation and grades, number of miles run per week and time on 5K. Correlation coefficient - number that represents the correlation

4. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-4 Correlation Coefficient 1.The values of the coefficient will always range from +1.00 to -1.00. Rare that coefficients of +1.00, -1.00, and 0.00 are found. 2. A positive coefficient indicates direct relationship. 3. A negative relationship indicates inverse relationship. 4. A correlation coefficient near.00 indicates no relationship.

5. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-5 Correlation Coefficient 5. The number indicates the degree of relationship and the sign indicates the type of relationship. The number +.88 indicates the same degree of relationship as the number -.88. The signs indicate that the directions of the relationships are different. 6. A correlation coefficient indicates relationship. After determining a correlation coefficient, you cannot infer that one variable causes something to happen to the other variable.

6. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-6 Scattergram Graph use to illustrate the relationship between two variables. See Figure 3.1 Scattergram can indicate a positive relationship, a negative relationship, or a zero relationship.

7. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-7 Scattergram Positive relationship - points will tend to cluster along a diagonal line that runs from the lower left-hand corner of scattergram to the upper right-hand corner. Negative relationship - points will tend to cluster along a diagonal line that runs from the upper left-hand corner to the lower right-hand corner. The closer the points cluster along the diagonal line, the higher the relationship. Zero relationship - points are scatter throughout the scattergram. See Figure 3.2.

8. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-8 Spearman Rank-Difference Correlation Coefficient Also called rank-order. Used when one or both variables are rank or ordinal scales. Difference (D) between ranks of two sets of scores is used to determine correlation coefficient. Examples - golf driving distance and order of finish in golf tournament; height and IQ score; weight and order of finish in 400 meter race; number of calories consumed and weight lost

9. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-9 Spearman Rank-Difference Correlation Coefficient Symbol: Greek rho (  ) or r rho To determine  : 1. List each set of scores in a column. 2. Rank the two sets of scores. 3. Place the appropriate rank beside each score. 4. Head a column D and determine the difference in rank for each pair of scores. (Sum of the D column should always be 0)

10. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-10 Spearman Rank-Difference Correlation Coefficient 5. Square each number in the D column and sum the values (  D 2 ). 6. Calculate the correlation coefficient by subtracting the values in the formula  = 1.00 - 6 (  D 2 ) Table 3.1 N(N 2 – 1) illustrates use of rank- difference correlation coefficient for sit-up and push-up scores.

11. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-11 Pearson Product-Moment Correlation Coefficient Also called Pearson r. Used when measurement results are reported in interval or ratio scale scores. Has many variations. Symbol is r. Examples - study time and test grade; leg strength and standing long jump; running long jump and time for 100 meters

12. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-12 Pearson Product-Moment Correlation Coefficient Calculation procedure (see Table 3.2) 1. Label columns for name, X, X 2, Y, Y 2, XY. 2. Designate one set of scores as X, designate the other set as Y, and place the appropriate paired scores by the individual’s name. 3. Find the sums of the X and Y columns (  X and  Y). 4. Square each X score, place squared scores in the X 2 column, and find the sum of the column (  X 2 ).

13. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-13 Pearson Product-Moment Correlation Coefficient 5. Square each Y score, place squared scores in the Y 2 column, and find the sum of the column (  Y 2 ). 6. Multiple each X score by the Y score, place the product in the XY column, and find the sum of the column (  XY). 7. Substitute the values in the formula r = N(  XY) - (  X)(  Y) N(  X 2 ) - (  X) 2 N(  Y 2 ) - (  Y) 2

14. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-14 Interpretation of the Correlation Coefficient *The purpose for which the correlation is computed must be considered. Following ranges can be used as general guidelines for interpretation of the correlation coefficient. r = below.20 (extremely low relationship) r =.20 to.39 (low relationship) r =.40 to.59 (moderate relationship) r =.60 to.79 (high relationship) r =.80 to 1.00 (very high relationship)

15. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-15 Significance of the Correlation Coefficient *Statistical significance, or reliability, of the correlation coefficient should be considered. *In determining statistical significance, you are answering the question: If the study were repeated, what is the probability of obtaining a similar relationship? *When r is calculated, the number of paired scores is important. With small number of paired scores, it is possible that a high r can occur by chance.

16. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-16 Significance of the Correlation Coefficient *With small number of paired scores, r must be large to be significant. *With large number of paired scores, a small r may be significant. *A table of values is used to determine the statistical significance of a correlation coefficient. *Must determine degrees of freedom and level of significance.

17. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-17 Significance of the Correlation Coefficient *Degrees of freedom (df) equal N-2;.05 and.01 levels of significance. *If correlation coefficient significant at the.05 level, it will occur only 5 in 100 times by chance. *If significant at the.01 level, it will occur only 1 in 100 times by chance.

18. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-18 Significance of the Correlation Coefficient *Use appendix A to compare obtained r. *If obtained r is larger than table values found at the.05 and.01 level, r is significant at the.01 level. *If obtained r falls between these two table values, r is significant at the.05 level.

19. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-19 Significance of the Correlation Coefficient *Significant correlation coefficients lower than.50 can be useful for indicating nonchance relationships among variables, but they probably are not large enough to be useful in predicting individual scores. *Table 3.2 : r =.90; df = 15-2 = 13 *Note difference in table values with the increased number of paired scores and at the.05 and.01 levels.

20. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-20 Coefficient of Determination *Square of the correlation coefficient (r 2 ). *Represents the common variance between two variables (the proportion of variance in one variable that can be accounted for by the other variable. *Example: Correlation of.85 between long jump test and leg strength test; r 2 =.72; 72% of the variability in the standing long jump scores is associated with leg strength; 72% of both variables come from common factors. *Use of coefficient of determination shows that a high correlation coefficient is needed to indicate a substantial to high correlation between two variables.

21. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-21 Negative Correlation Coefficients *Occasions when negative correlation coefficient is expected. *Negative correlation: Small score that is considered to be a better score is correlated with a large score that also is considered to be a better score. Examples: Relationship between time to run 5K and maximum O 2 consumption; weight and pull-ups

22. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-22 Correlation, Regression, and Prediction Linear correlation – how close the relationship between two variables is to a straight line If relationship found, a score for one variable can be used to predict the score for other variable – linear regression analysis Standard error of estimate - numerical value that indicates the amount of error to be expected in predicted score; confidence limits; use in the same way as standard deviation is used with a group of scores

23. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 3-23 Correlation, Regression and Prediction Through multiple correlation-regression analysis, we can predict a score using several other scores. May predict college freshman year grade point average with SAT of ACT score, high school grade point average, and class rank. Predict health problems through lifestyle.