You can calculate: Central tendency Variability You could graph the data

You can calculate: Central tendency Variability You could graph the data

Bivariate Distribution

Positive Correlation

Regression Line

Correlation r = 1.00

Regression Line..... r =.64

Regression Line.... r =.64.

Practice

Regression Line

.....

.....

Negative Correlation

r =

Negative Correlation..... r = -.85

Zero Correlation

..... r =.00

Correlation Coefficient The sign of a correlation (+ or -) only tells you the direction of the relationship The value of the correlation only tells you about the size of the relationship (i.e., how close the scores are to the regression line)

Excel Example

Which is a bigger effect? r =.40 or r = -.40 How are they different?

Interpreting an r value What is a “big r” Rule of thumb: Smallr =.10 Mediumr =.30 Larger =.50

Practice Do you think the following variables are positively, negatively or uncorrelated to each other? Alcohol consumption & Driving skills Miles of running a day & speed in a foot race Height & GPA Forearm length & foot length Test #1 score and Test#2 score

Statistics Needed Need to find the best place to draw the regression line on a scatter plot Need to quantify the cluster of scores around this regression line (i.e., the correlation coefficient)

Covariance Correlations are based on the statistic called covariance Reflects the degree to which two variables vary together –Expressed in deviations measured in the original units in which X and Y are measured

Note how it is similar to a variance –If Ys were changed to Xs it would be s 2 How it works (positive vs. negative vs. zero)

Computational formula

Ingredients: ∑XY ∑X ∑Y N

N = 5

∑XY = 84 ∑Y = 23 ∑X = 15 N = 5

∑XY = 84 ∑Y = 23 ∑X = 15 N = 5

∑XY = 84 ∑Y = 23 ∑X = 15 N = 5

∑XY = 84 ∑Y = 23 ∑X = 15 N = 5

∑XY = 84 ∑Y = 23 ∑X = 15 N = 5

Problem! The size of the covariance depends on the standard deviation of the variables COV XY = 3.75 might occur because –There is a strong correlation between X and Y, but small standard deviations –There is a weak correlation between X and Y, but large standard deviations

Solution Need to “standardize” the covariance Remember how we standardized single scores

Correlation

Practice You are interested in if candy intake is related to childhood depression. You collect data from 5 children.

Practice CandyDepression Charlie555 Augustus743 Veruca459 Mike3108 Violet465 S candy = 1.52S depression = 24.82

Practice Candy (X) Depression (Y) XY Charlie Augustus Veruca Mike Violet ∑

Practice Candy (X) Depression (Y) XY Charlie Augustus Veruca Mike Violet ∑

∑XY = 1396 ∑Y = 330 ∑X = 23 N = 5

∑XY = 1396 ∑Y = 330 ∑X = 23 N = 5

Correlation COV = Sx = 1.52 Sy = 24.82

Correlation COV = Sx = 1.52 Sy = 24.82

Hypothesis testing of r Is there a significant relationship between X and Y (or are they independent) –Like the X 2

Steps for testing r value 1) State the hypothesis 2) Find t-critical 3) Calculate r value 4) Calculate t-observed 5) Decision 6) Put answer into words

Practice Determine if candy consumption is significantly related to depression. –Test at alpha =.05

Practice CandyDepression Charlie555 Augustus743 Veruca459 Mike3108 Violet465 S candy = 1.52S depression = 24.82

Step 1 H 1 : r is not equal to 0 –The two variables are related to each other H 0 : r is equal to zero –The two variables are not related to each other

Step 2 Calculate df = N - 2 Page 747 –First Column are df –Look at an alpha of.05 with two-tails

t distribution df = 3 0

t distribution t crit = t crit =

t distribution t crit = t crit =

Step 3 COV = Sx = 1.52 Sy = 24.82

Step 4 Calculate t-observed

Step 4 Calculate t-observed

Step 4 Calculate t-observed

Step 5 If t obs falls in the critical region: –Reject H 0, and accept H 1 If t obs does not fall in the critical region: –Fail to reject H 0

t distribution t crit = t crit =

t distribution t crit = t crit =

Step 5 If t obs falls in the critical region: –Reject H 0, and accept H 1 If t obs does not fall in the critical region:If t obs does not fall in the critical region: –Fail to reject H 0

Step 6 Determine if candy consumption is significantly related to depression. –Test at alpha =.05 Candy consumption is not significantly related to depression –Note: this finding is due to the small sample size

Practice Is there a significant (.05) relationship between aggression and happiness?

Mean aggression = 14.50; S 2 aggression = Mean happiness = 6.00; S 2 happiness = 4.67

Answer Cov = r = -.76 t crit = Thus, fail to reject Ho Aggression was not significantly related to happiness

Practice Situation 1 Based on a sample of 100 subjects you find the correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero. Situation 2 Based on a sample of 600 subjects you find the correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero.

Step 1 Situation 1 H 1 : r is not equal to 0 –The two variables are related to each other H 0 : r is equal to zero –The two variables are not related to each other Situation 2 H 1 : r is not equal to 0 –The two variables are related to each other H 0 : r is equal to zero –The two variables are not related to each other

Step 2 Situation 1 df = 98 t crit = and Situation 2 df = 598 t crit = and -1.96

Step 3 Situation 1 r =.15 Situation 2 r =.15

Step 4 Situation 1 Situation 2

Step 5 Situation 1 If t obs falls in the critical region: –Reject H 0, and accept H 1 If t obs does not fall in the critical region: –Fail to reject H 0 Situation 2 If t obs falls in the critical region: –Reject H 0, and accept H 1 If t obs does not fall in the critical region: –Fail to reject H 0

Step 6 Situation 1 Based on a sample of 100 subjects you find the correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero. There is not a significant relationship between extraversion and happiness Situation 2 Based on a sample of 600 subjects you find the correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero. There is a significant relationship between extraversion and happiness.