Descriptive Statistics IIIREVIEW

• Variability• Range, variance, standard deviation

• Coefficient of variation (S/M): 2 data sets• Value of standard scores?

2SS

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zT 1050

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Correlation and Prediction

HPER 3150Dr. 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

Negative Positive

-1.0 1.000-.70 0.700-.30 0.30

Perfect PerfectHigh HighLow LowZero

Scatterplot of correlation between pull-ups and chin-ups

(direct relationship/+)

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Pull-ups (#completed)

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Scatterplot of correlation betweenbody weight and pull-ups

(indirect relationship/-)

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Weight (lb)

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Scatterplot of zero correlation (r = 0) Figure 4.4

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Correlation Formula(page 54)

r n XY – X Y

n X 2 – X 2

n Y 2 – Y 2

Correlation issues

• Causation• -1.00 < r < +1.00• Coefficient of Determination (r2) (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 4.5Curvilinear relationship

Example of variable?

Figure 4.6Range 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/PredictionREVIEW

• 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=slopeC=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(sy/sx) b=.90(1.5/15) b=.09On average, we expect a daily wt loss of .09# while dieting

c=Ybar–bXbar c=8.0-.09(65) c=2.15

Y’ = bX + cY’ = .09x + 2.15

Predicted wt loss = .09(days dieting) + 2.15

Y=weight loss Ybar=8.0# sy=1.5#X=days dieting Xbar=65 days sx=15 daysrxy=.90

Correlation and prediction

Skinfolds

% Fat

Correlation and prediction

Skinfolds

% Fat

Correlation and prediction

Skinfolds

% Fat

Standard Error of Estimate(SEE)

21* rSySe As r ↑, error ↓As r ↓, error ↑

Is ↑r good? Why/Not?

Is ↑ error good? Why/Not?

Correlation and prediction

Skinfolds

% Fat

SEE = 3%

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