Descriptive Statistics III REVIEW Variability Range, variance, standard deviation Coefficient of variation (S/M): 2 data sets Value of standard scores?

Correlation and Prediction HPER 3150 Dr. Ayers

Correlation (Pearson Product Moment or r) Are two variables related? Car speed & likelihood of getting a ticket Skinfolds & percent body fat What happens to one variable when the other one changes? Linear relationship between two variables

Attributes of r

Scatterplot of correlation between pull-ups and chin-ups (direct relationship/+) Pull-ups (#completed) Chin-ups (#completed)

Scatterplot of correlation between body weight and pull-ups (indirect relationship/-) Weight (lb) Pull-ups (#completed)

Scatterplot of zero correlation (r = 0) Figure 4.4 X Y

Correlation Formula (page 54)

Correlation issues Causation < r < Coefficient of Determination (r 2 ) (shared variance) Linear or Curvilinear (≠ no relationship) Range Restriction Prediction (relationship allows prediction) Error of Prediction (for r ≠ 1.0) Standard Error of Estimate (prediction error)

Limitations of r Figure 4.5 Curvilinear relationship Example of variable? Figure 4.6 Range restriction

Limitations of r

Uses of Correlation Quantify RELIABILITY of a test/measure Quantify VALIDITY of a test/measure Understand nature/magnitude of bivariate relationship Provide evidence to suggest possible causality

Misuses of Correlation Implying cause/effect relationship Over-emphasize strength of relationship due to “significant” r

Correlation/Prediction REVIEW Bivariate nature Strength (-1 to 1) Linear relationships (curvilinear?) (In)Direct relationships Coefficient of determination: what is it and what does it tell you? Uses/Misuses of correlation?

Sample Correlations Excel document

Correlation and prediction Skinfolds % Fat

Variables Independent Presumed cause Antecedent Manipulated by researcher Predicted from Predictor X Dependent Presumed effect Consequence Measured by researcher Predicted Criterion Y

Equation for a line Y’ = bX + c b=slope C=Y intercept

We have data from a previous study on weight loss. Predict the expected weight loss (Y; dependent) as a function of #days dieting (X; independent)for a new program we are starting

To get regression equation, calculate b & c b=r(s y /s x )b=.90(1.5/15)b=.09 On average, we expect a daily wt loss of.09# while dieting c=Ybar–bXbarc= (65)c=2.15 Y’ = bX + c Y’ =.09x Predicted wt loss =.09(days dieting) Y=weight lossYbar=8.0#s y =1.5# X=days dietingXbar=65 dayss x =15 days r xy =.90

Correlation and prediction Skinfolds % Fat

Correlation and prediction Skinfolds % Fat

Correlation and prediction Skinfolds % Fat

Standard Error of Estimate (SEE) As r ↑, error ↓ As r ↓, error ↑ Is ↑r good? Why/Not? Is ↑ error good? Why/Not?

Correlation and prediction Skinfolds % Fat SEE = 3%