Chapter 3

Investigating the Relationship

between Scores

Chapter ObjectivesAfter completing this chapter, you should be able to1 . 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.

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Linear Correlation

• Correaltion - 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

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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.

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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.

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Scattergram

Graph use to illustrate the relationship between twovariables.

See Figure 3.1

Scattergram can indicate a positive relationship, a negativerelationship, or a zero relationship.

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ScattergramPositive 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.

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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

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Spearman Rank-Difference Correlation Coefficient

Symbol: Greek rho () or rrho

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)

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Spearman Rank-Difference Correlation Coefficient

5. Square each number in the D column and sum the values (D2).6. Calculate the correlation coefficient by subtracting the values in the formula

= 1.00 - 6 ( D2) Table 3.1 N(N2 – 1) illustrates use of rank- difference correlation coefficient for sit-up and push-up scores.

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Pearson Product-MomentCorrelation 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

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Pearson Product-Moment Correlation Coefficient

Calculation procedure (see Table 3.2)1. Label columns for name, X, X2, Y, Y2, 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 X2 column, and find the sum of the column (X 2).

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Pearson Product-Moment Correlation Coefficient

5. Square each Y score, place squared scores in the Y2 column, and find the sum of the column (Y2).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(Y2) - (Y)2

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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)

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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.

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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.

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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.

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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.

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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.

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Coefficient of Determination*Square of the correlation coefficient (r2).

*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; r2 = .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.

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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 O2 consumption; weight and pull-ups

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Correlation, Regression, and PredicitonLinear 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

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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.

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