Correlation

What is a correlation?

• A correlation examines the relationship between two measured variables.– No manipulation by the experimenter/just observed. – E.g., Look at relationship between height and weight.

• You can correlate any two variables as long as they are numerical (no nominal variables)

• Is there a relationship between the height and weight of the students in this room?– Of course! Taller students tend to weigh more.

1) Strength of Relationships

• 2 aspects of the relationship2 aspects of the relationship: Strength and Direction.

• The relationship between any 2 variables is rarely a perfect correlation.

• Perfect correlation: +1.00 OR –1.00– strongest possible relationship– Tough to find.

• No correlation: 0.00 (no relationship).– E.g, height and social security #.

2) Direction of the Relationship• Positive relationship – Variables change in the

same direction.• As X is increasing, Y is increasing

• As X is decreasing, Y is decreasing

– E.g., As height increases, so does weight.

• Negative relationship – Variables change in opposite directions.

• As X is increasing, Y is decreasing

• As X is decreasing, Y is increasing

– E.g., As TV time increases, grades decrease

Indicated bysign; (+) or (-).

Positive Correlation–as x increases, y increases

x = SAT scorey = GPA

GP

AScatter Plots and Types of Correlation

4.003.753.50

3.002.752.502.252.00

1.501.75

3.25

300 350 400 450 500 550 600 650 700 750 800

Math SAT

Negative Correlation–as x increases, y decreases

x = hours of trainingy = number of accidents

Scatter Plots and Types of Correlation

60

50

40

30

20

10

0

0 2 4 6 8 10 12 14 16 18 20

Hours of Training

Acc

iden

ts

No linear correlation

x = height y = IQ

Scatter Plots and Types of Correlation

160

150140

130120

110

100

90

80

60 64 68 72 76 80

Height

IQ

Correlation Coefficient Interpretation

Coefficient

Range

Strength of

Relationship

0.00 - 0.20 Very Low

0.20 - 0.40 Low

0.40 - 0.60 Moderate

0.60 - 0.80 High Moderate

0.80 - 1.00 Very High

Direction

• Positive relationship

Height

Weight

r = +.80

Direction

• Negative relationship

Exam score

TV

watching per

week

r = -.80

Interpreting correlations - Summary

• Absolute size shows strength of relationship

• The higher the absolute number, the stronger the relationship – A correlation of -.80 is reflects as powerful a

relationship as one of +.80

• A correlation of 0.00 means no relationship– E.g., Can’t predict GPA from ID number

• All correlations range from -1.00 to +1.00

Strength of relationship

• Perfect Correlation

Exam score

TV

watching per

week

r = -1.0

Strength of relationship

• Strong Correlation

Exam score

Quality of B

reakfast

r = + 0.8

Strength of relationship

• Moderate Correlation

Weight

Shoe S

ize

r = + 0.4

Strength of relationship

• Weak Correlation (negative)

Weight

Shoe S

ize

r = - 0.2

Strength of relationship

• No Correlation (horizontal line)

Height

IQ

r = 0.0

One more example

Amount ofStudy Time

Exam Grade

Social Security Number

# of classes missed

+.80

-.60.00

More examples

• Positive relationshipsPositive relationships:– water consumption and

temperature.– study time and grades.– time spent in jail to

severity of offense.– What else??

• Negative relationshipsNegative relationships:– alcohol consumption

and driving ability.– # of hateful remarks

and # of friends.– What else??

Why used: 1) Prediction; 2) Validity (does something measure what it’s suppose to measure; 3) Reliability(does something produce a consistent score).

*** Easier to do than experiments ***

Pearson correlation coefficient

• r = the Pearson coefficient

• r measures the amount that the two variables (X and Y) vary together (i.e., covary) taking into account how much they vary apart

• Pearson’s r is the most common correlation coefficient; there are others.

Computing the Pearson correlation coefficient

• To put it another way:

• Or

separately vary Y and X which todegree

ther vary togeY and X which todegreer

separately Y and X ofy variabilit

Y and X ofity covariabilr

Sum of Products of Deviations

• Measuring X and Y individually (the denominator):– compute the sums of squares for each variable

• Measuring X and Y together: Sum of Products– Definitional formula

– Computational formula

• n is the number of (X, Y) pairs

))(( YYXXSP

n

YXXYSP

Correlation Coefficent:

• the equation for Pearson’s r:

• expanded form:YX SSSS

SPr

nY

YnX

X

nYX

XYr

22

22

Limitations of Pearson’s r

1. Correlation does not mean causation!!• Third Variable problem – there’s always the

possibility of a third factor causing the relationship.

• E.g., Moderate, positive relationship between viewing violent TV and engaging in aggressive behaviors.

Possibilities

Viewing violent television

Tendency to engagein aggressive behaviors

Viewing violent television

Tendency to engagein aggressive behaviors

A third factor;EX. genetic tendencyto like violence Viewing violent

television

Tendency to engagein aggressive behaviors

Limitations of Pearson’s r

1. Correlation does not mean causation

2. Restriction of range– Restricted range of measured values can lead

to inaccurate conclusions about the data

Limitations of Pearson’s r

3. Outliers (extreme scores) Scores with extreme X and/or Y value can drastically

effect Pearson’s r

4. Ambiguity of the strength of the relationship Pearson r does not give a directly interpretable strength

of the relationship between X and Y

5. Interval or ratio data.

Coefficient of Determination

• r2 = percentage of variance in Y accounted for by X

• Calculated by squaring r (Pearson correlational coefficient)

• Ranges from 0 to 1 (positive only)

• This number is a meaningful proportion (unlike the Pearson’s r).

Coefficient of Determination: An example

• Example: – What percentage of variance is accounted for in

Y by X with a Pearson r = 0.50?– The r2 = (0.50)2 = 0.25 = 25%

• The number is always positive