Correlations Determining Relationships: Chapter 5:

Let’s Experiment... Take a Random Sample of 10 pairs of data from our heights and shoe sizes. Draw a “Scatter plot” Let x-axis = height Let y-axis = shoe size

See the point? Draw a “best fit” line (trend line) through the dots Make a conclusion

Scatter plots “visualize” correlations When dots fall near the line, the correlation is strong When dots are widely scattered, the correlation is weak Graphs are quick, but math is more precise (of course!)

Strong vs. Weak Relationships: Strong Weak

And there’s more… If the graph slopes up to the right, the correlation is Positive As X gets bigger, Y gets bigger “Direct relationship” If the graph slopes down to the right, the correlation is Negative As X gets bigger, Y gets smaller “Indirect or Inverse relationship”

Positive and Negative Relationships Positive Negative X Y Y X

QUESTION: Think of 5 “pairs” of data you could gather to compare (heights and shoe sizes) to observe a correlation.

Teaching Applications Free Throw % vs. Arm Strength Swimming Stroke # vs. Times Expert Judge vs. Skill Achievement Teaching Method vs. Student Outcomes

Wellness Applications Smoking vs. CAD Risk Skinfold thickness vs. Body Density Heart Rate vs. VO 2 Flexibility vs. Lower Back Pain

Uses of correlational research Exploration: “I wonder if these variables are related?” Prediction: “I wonder if I can predict one from the other?”

Quantifying the Relationship for precision Does that mean “math”? Pearson Product Moment: r=

Pearson Product Moment r xy = N(S)xy - ([S]x)([S]y) [N(S)x 2 - ([S]x) 2 ] [N(S)y 2 - ([S]y 2 ] Thank goodness for Excel! r = >1.0: When r = |1.0| is perfect correlation, and 0 is no correlation

Interpreting Correlation Coefficients (r) Strength: How close the dots are to the line r = Direction: Positive or Negative r = + or – Probability: What’s the chance this happened by chance? P< 0.05 or better

Correlation (Almost)

Correlation Strength Very Strong:0.90 – 1.0 Strong: Moderate: Weak:<.50

Methods of data collection: One sample population Two variables are “paired” from each individual I.E: Swimming speed and number of stroke cycles

Correlation is not Causation Just because two variables are found to “co-vary” with each other doesn’t mean they “Cause” the other.

Suppose: The correlation between Crime Rate and Churches in town is r = Does that mean having lots of churches causes more crime?

Summary Correlations look for relationships in variance between two variables Scatter plots are used to visualize (graph) correlations

Summary, cont... Pearson Product Moment is an example of statistical quantification of co-variance: r = > +1.0 When r = |1.0|, the relationship is perfectly strong

Summary, cont. r = 0: There is no correlation Negative: inversely related - when one gets higher, the other gets lower Positive: Both get higher or lower together

Strong vs. Weak Relationships: Strong Weak

Positive and Negative Relationships Positive Negative X Y Y X

Correlation (Almost)

Lab 3: Correlations Read the Lab thoroughly Import Data from my Web site Determine your hypothesis regarding the measurement that has the best correlation with body fatness: Complete the lab

N.C. Wyeth, “The Spruce Ledge”