Correlations

Determining Relationships:

Chapter 5:123-127

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 = heightLet y-axis = shoe size

See the point?

Draw a “best fit” line (trend line) through the dots

Make a conclusion

Scatter plots “visualize” correlationsWhen dots fall near the line, the

correlation is strongWhen dots are widely scattered,

the correlation is weakGraphs are quick, but math is

more precise (of course!)

Strong vs. Weak Relationships:

Strong Weak

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.. . .

..

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And there’s more… If the graph slopes up to the right, the

correlation is PositiveAs X gets bigger, Y gets bigger“Direct relationship”

If the graph slopes down to the right, the correlation is NegativeAs 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 StrengthSwimming Stroke # vs. TimesExpert Judge vs. Skill

AchievementTeaching Method vs. Student

Outcomes

Wellness ApplicationsSmoking vs. CAD RiskSkinfold thickness vs. Body

DensityHeart Rate vs. VO2

Flexibility vs. Lower Back Pain

Uses of correlational researchExploration: “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 rxy = N(S)xy - ([S]x)([S]y) [N(S)x2

- ([S]x)2] [N(S)y2 - ([S]y2] Thank goodness for Excel! r = -1.0 ->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 = 0 -1.0Direction: Positive or Negative r =

+ or – 0-1.0Probability: What’s the chance

this happened by chance? P< 0.05 or better

Correlation

(Almost) (Almost)

Correlation StrengthVery Strong: 0.90 – 1.0

Strong: 0.80 - .89

Moderate: 0.50 - .79

Weak: < .50

Methods of data collection:One sample populationTwo variables are “paired”

from each individualI.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 = +0.89Does 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 -> +1.0

When r = |1.0|, the relationship is perfectly strong

Summary, cont.

r = 0: There is no correlationNegative: 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) (Almost)

Lab 3: CorrelationsRead the Lab thoroughly Import Data from my Web siteDetermine your hypothesis

regarding the measurement that has the best correlation with body fatness:

Complete the lab

N.C. Wyeth, “The Spruce Ledge”