Looking at data: relationships

- Correlation

Lecture Unit 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 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.

The correlation coefficient "r"

Correlation can only be used to describe quantitative variables. Categorical variables don’t have means and standard deviations.

Time to swim: x = 35, sx = 0.7

Pulse rate: y = 140 sy = 9.5

Example: calculating correlation (x1, y1), (x2, y2), (x3, y3) (1, 3) (1.5, 6) (2.5, 8)

1 1.67 3 5.67 1.5 1.67 6 5.67 2.5 1.67 8 5.67

(3 1)(.76)(2.52)

1.67, 5.67, .76, 2.52

.9538

x yx y s s

r

Properties of Correlation r is a measure of the strength of the linear relationship between x

and y. No units [like demand elasticity in economics (-infinity, 0)] -1 < r < 1

Values of r and scatterplots

r near +1r near -1

r near 0

x x

y

y

r near 0

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

Properties (cont.) r has no unitr = -0.75

r = -0.75

z-score plot is the same for both plots

Properties (cont.)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.

Properties of Correlation (cont.) r = -1 only if y = a + bx with slope b<0

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

y = 1 + 2x

y = 11 - x

Properties (cont.) High correlation does not imply cause and effectCARROTS: 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 !!!

Properties (cont.) Cause and Effect 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?

Properties (cont.) Cause and Effect

r measures the strength of the linear relationship between x and y; it does not indicate cause and effect

correlation

r = .935

x = fouls committed by player;

y = points scored by same player

(x, y) = (fouls, points)

01020304050607080

0 5 10 15 20 25 30

Fouls

Po

ints

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

The correlation is due to a third “lurking” variable – playing time

Properties (cont.) r does not distinguish x & y

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

"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 = -0.75 r = -0.75

Correlations are calculated using

means and standard deviations,

and thus are NOT resistant to

outliers.

Outliers

Just moving one point away from the

general trend here decreases the

correlation from -0.91 to -0.75

PROPERTIES (Summary) r is a measure of the strength of the linear relationship between x and y.

No units [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 does not imply causation

r does not distinguish between x and y

r can be affected by outliers

Correlation: Fuel Consumption vs Car Weight

FUEL CONSUMPTION vs CAR WEIGHT

2

3

4

5

6

7

1.5 2.5 3.5 4.5

WEIGHT (1000 lbs)

FU

EL

CO

NS

UM

P.

(gal

/100

mile

s)

r = .9766

SAT Score vs Proportion of Seniors Taking SAT

88-89 SAT vs % Seniors Taking SAT

825

875

925

975

1025

1075

0 20 40 60 80

% Seniors that Took SAT

88-8

9 S

AT

Sta

te A

vg.

88-89 SAT

IWND

SC

NC

DC

r = -.868

Extra Slides

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!

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].

When variability in one

or both variables

decreases, the

correlation coefficient

gets stronger

( closer to +1 or -1).

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.

Correlation only describes linear relationships

1) What is the explanatory variable?

Describe the form, direction and strength

of the relationship?

Estimate r.

(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?

Review examples

ageman = agewoman + 2

equation for a straight line

r = 1

r = 0.94

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?