• Variability• Range, variance, standard deviation

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

Descriptive Statistics IIIREVIEW

2SS

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MXZ

zT 1050

1

2

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n

MXS

)(50 percentilezp

Correlation and Prediction

HPHE 3150Dr. 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

Xiv

Ydv

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•Provides support for a test’s validity and reliability

Attributes of rmagnitude & direction

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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in-u

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Scatterplot of correlation betweenbody weight and pull-ups

(indirect/inverse relationship/-)

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120 130 140 150 160 170 180

Weight (lb)

Pu

ll-u

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Scatterplot of zero correlation (r = 0) Figure 4.4

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X

Y

Correlation Formula(page 60)

r n XY – X Y

n X 2 – X 2

n Y 2 – Y 2

Correlation issues

• Correlation ≠ causation

• -1.00 < r < +1.00

• Coefficient of Determination (r2) (shared variance)• r=.70 r2=.49 49% variance in Y accounted for by X

Xiv

Ydv

• Negative correlation possibly due to:• Opposite scoring scales• True negative relationship

• Linear or Curvilinear (≠ no relationship; fig 4.6)

• Range Restriction (fig 4.7; ↓ r)

• Prediction (relationship allows prediction to some degree)

• Error of Prediction (for r ≠ 1.0)

• Standard Error of Estimate (prediction error)

Limitations of r

Figure 4.6Curvilinear relationship

Example of variable?

Figure 4.7Range restriction

Limitations of r

Correlation & Prediction IREVIEW

• Bivariate nature of correlations• X (iv) & Y (dv)• +/- relationships• Range of r?

• Coefficient of Determination (r2) (shared variance)

• Coefficient of variation (S/M): 2 data sets• Low V (.1-.2=homo): M accounts for most variability in scores

• Curvilinear relationship? Fitness/PA

• Correlation/Causation?

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

Skinfolds

% Fat

Sample Correlations

Excel document

Standard Error of Estimate(SEE)

Average error in the process of predicting Y from XStandard Deviation of error

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

Is ↑r good? Why/Not?Is ↑ error good? Why/Not?