Descriptive Statistics III REVIEW Variability Range variance standard

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

Variables Independent • • • (categorical: name) Presumed cause Antecedent Manipulated by researcher Predicted from Predictor X Dependent • • • (ordinal/continuous: #) Presumed effect Consequence Measured by researcher Predicted Criterion Y

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: • 1 measure of 2 separate variables or 2 measures of 1 variable

Attributes of r

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

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

Scatterplot of zero correlation (r = 0) Figure 4. 4 Y X

Correlation Formula (page 54)

Correlation issues • Causation • -1. 00 < r < +1. 00 • Coefficient of Determination (r 2) (shared variance) • r=. 70 • • • r 2=. 49 49% variance in Y accounted for by X Linear or Curvilinear (≠ no relationship; fig 4. 5) Range Restriction (fig 4. 6; ↓ r) 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 and prediction % Fat Skinfolds

Sample Correlations Excel document

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

We have data from a previous study on weight loss. Predict the expected weight loss as a function of #days dieting for a new program we are starting

Y=weight loss X=days dieting rxy=. 90 Ybar=8. 0# Xbar=65 days sy=1. 5# sx=15 days To get regression equation, calculate b & c b=r(sy/sx) b=. 90(1. 5/15) b=. 09 On average, we expect a daily wt loss of. 09# while dieting c=Ybar–b(Xbar) c=8. 0 -. 09(65) c=2. 15 Y’ = b. X + c Y’ =. 09 x + 2. 15 Predicted wt loss =. 09(days dieting) + 2. 15 7 days on program yields 2. 78# weight loss

Correlation and prediction % Fat Skinfolds

Standard Error of Estimate (SEE)

Correlation and prediction % Fat 20 23 17 SEE = 3% Skinfolds 40