MEASURES OF RELATIONSHIP Correlations

Key Concepts Pearson Correlation  interpretation  limits  computation  graphing Factors that affect the Pearson Correlation Coefficient of Determination (r 2 ) – ‘variance explained’ Correlation vs. Causation

Correlations A correlation measures a linear relationship between two variables

Correlation: Scatterplots Scatterplots are graphic representations of the relationship between two continuous variables

Correlation: Coefficients Correlation coefficients are number between and representing the relationship between two variables 0 +1

Stop and think What types of variables are correlated in education? Can you provide some examples of both positive and negative relationships?

The Ugly Formula This formula calculates the correlation between X and Y It builds on your knowledge of variance; showing how the variation in X & Y along with the covariation between X & Y make up the Pearson correlation coefficient. …the variance formula for r

Example Step 1: Layout the Problem

Step 2: Compute the Mean for both variables Sum of X = 58 Number of X = 9 Mean of X = 6.44 Sum of Y = 530 Number of Y = 9 Mean of Y = Example

Step 3: Compute the difference of each score from its Mean Mean of X = 6.44Mean of Y = Note: The sum of (X-Xbar) should equal 0 and the sum of (Y-Ybar) should equal 0. Why?

Step 4: Compute the square of each mean difference

Step 5: Sum the squares differences from the means Sum (X-Xbar) 2 = Sum (Y-Ybar) 2 =

Step 6: Compute the cross-product of the differences (for the numerator)

Step 7: Sum the cross product of the differences Sum (X-Xbar)(Y-Ybar) =

Step 8: Collect the partial values together, and substitute each into the formula. Solve the formula. Sum (X-Xbar) 2 = Sum (Y-Ybar) 2 = Sum (X-Xbar)(Y-Ybar) =

Last Step: Check the computed r for reasonableness, then interpret the value (sign and magnitude) The value of r must be between -1 and +1  Computed r =.49, which is between -1 and +1 The sign of r is positive  The relationship among the two variables is positive  “In general, younger people weigh less than older people.”  “In general, older people weigh more than younger people.” The magnitude of r is “moderate”  Although age and weight are related, the relationship is not very strong. Some of the variation in age has nothing to do with weight, and some of the variation in weight has nothing to do with age.

Variables with a curvilinear relationship will be underestimated if r is applied. Size of the group does not affect the size of the correlation coefficient. Cautions

ES = the correlation coefficient, squared (r 2 ) The proportion of the total variance of one variable that can be associated with the variance in the other variable. It is the proportion of shared or common variance between two variables. Example: calorie intake & weight r =.60 r 2 =.36 or 36% Effect Size CAL WEIGHT r 2 =.36

Correlation does not indicate causation  correlation indicates a relationship or association Correlation & Causality

Practice Compute the Pearson correlation and r squared value for the following example. Be sure to try to draw a rough sketch of a scatterplot to see if the relationship looks linear. X3, 7, 8, 2, 5 Y5, 8, 10, 3, 9 Interpret your results.

xyX-XbarY-Ybar(X-Xbar) 2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar) - numerator

xyX-XbarY-Ybar(X-Xbar) 2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar) Xbar=5Ybar=7

xyX-XbarY-Ybar(X-Xbar) 2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar) Xbar=5Ybar=7Check=0

xyX-XbarY-Ybar(X-Xbar) 2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar) Xbar=5Ybar=7Check=0 Sum=26Sum=34

xyX-XbarY-Ybar(X-Xbar) 2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar) Xbar=5Ybar=7Check=0 Sum=26Sum=34Sum=27

xyX-XbarY-Ybar(X-Xbar) 2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar) Xbar=5Ybar=7Check=0 Sum=26Sum=34Sum=27 r = 27 / sqrt((26)(34)) r = 27 / 29.7 r =.908, r 2 =.82

Key Points Correlation is a measure of relationship, and ranges from -1 to 1. Sign indicates direction, and the coefficient indicates strength of relationship. r 2 represents the shared variance Correlations do not imply causality