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Looking at data: relationships

Correlation

Chapter 7

Objectives

Correlation

     The correlation coefficient “r” R does not distinguish x and y R has no units R ranges from -1 to +1 Influential points

The correlation coefficient "r"

The correlation coefficient is a measure of the direction and strength of the linear relationship between 2 quantitative variables. It is calculated using the mean and the standard deviation of both the x and y variables. Time to swim: x = 35, s x = 0.7

Pulse rate: y = 140 s y = 9.5

Correlation can only be used to describe

quantitative

variables. Categorical variables don’t have means and standard deviations.

PROPERTIES

 scaleless [like demand elasticity in economics (-infinity, 0)]  -1 <

r

< 1 

r =

-1 only if y = a + bx with slope b<0 

r

= +1 only if y = a + bx with slope b>0

Correlation: Fuel Consumption vs Car Weight

7 6 5 4 3 2 1.5

FUEL CONSUMPTION vs CAR WEIGHT r = .9766

2.5

3.5

WEIGHT (1000 lbs)

4.5

SAT Score vs Proportion of Seniors Taking SAT

88-89 SAT vs % Seniors Taking SAT

1075 1025 975 925 875 825 0 IW

ND SC

20 40

NC

60

% Seniors that Took SAT DC

80

r = -.868

88-89 SAT

"r" ranges from -1 to +1

"r" quantifies the

strength

and

direction

of a linear relationship between 2 quantitative variables.

Strength:

how closely the points follow a straight line.

Direction

: is positive when individuals with higher X values tend to have higher values of Y.

Part of the calculation involves finding z, the standardized score we used when working with the normal distribution.

You DON'T want to do this by hand.

Make sure you learn how to use your calculator!

Example: calculating correlation

  (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ) (1, 3) (1.5, 6) (2.5, 8)

r x



1.67,

y



5.67,

s x



.76,

s y



2.52

      

.9538

  

Standardization:

Allows us to compare correlations between data sets where variables are measured in different units or when variables are different. For instance, we might want to compare the correlation between [swim time and pulse], with the correlation between [swim time and breathing rate].

“r” does not distinguish x & y

The correlation coefficient, r, treats x and y symmetrically.

r = -0.75

r = -0.75

"Time to swim" is the explanatory variable here, and belongs on the x axis. However, in either plot r is the same (r=-0.75).

"r" has no unit

Changing the units of variables does not change the correlation coefficient "r", because we get rid of all our units when we standardize (get z-scores).

z-score plot is the same for both plots r = -0.75

r = -0.75

When variability in one or both variables decreases, the correlation coefficient gets stronger (  closer to +1 or -1).

Correlation only describes linear relationships

No matter how strong the association, r does not describe curved relationships.

Note: You can sometimes transform a non-linear association to a linear form, for instance by taking the logarithm. You can then calculate a correlation using the transformed data.

High correlation does not imply cause and effect

CARROTS: Hidden terror in the produce department at your neighborhood grocery



Everyone who ate carrots in 1920, if they are still alive, has severely wrinkled skin!!!



Everyone who ate carrots in 1865 is now dead!!!



45 of 50 17 yr olds arrested in Raleigh for juvenile delinquency had eaten carrots in the 2 weeks prior to their arrest !!!

More Correlations

 There is a strong positive correlation between the monetary damage caused by structural fires and the number of firemen present at the fire. (More firemen-more damage)  Improper training? Will no firemen present result in the least amount of damage?



(1,2) (24,75) (1,0) (18,59) (9,9) (3,7)(5,35) (20,46) (1,0) (3,2) (22,57)

x = fouls committed by player; r measures the strength of y = points scored by same player the

linear relationship

between x and y; it does not indicate cause and effect  Example

(x, y) = (fouls, points)

r = .935

80 70 60 50 40 30 20 10 0 0 5 10 15

Fouls

20 25 30

Influential points

Correlations are calculated using means and standard deviations, and thus are NOT resistant to outliers.

Just moving one point away from the general trend here decreases the correlation from -0.91 to -0.75

Review examples

1) What is the explanatory variable?

Describe the form, direction and strength of the relationship?

Estimate r.

r = 1 r = 0.94

(in 1000’s)

2) If women always marry men 2 years older than themselves, what is the correlation of the ages between husband and wife? age man = age woman + 2

equation for a straight line

Thought quiz on correlation

1.

Why is there no distinction between explanatory and response variable in correlation?

2.

Why do both variables have to be quantitative?

3.

How does changing the units of one variable affect a correlation?

4.

What is the effect of outliers on correlations?

5.

Why doesn’t a tight fit to a horizontal line imply a strong correlation?