BPS - 3rd Ed. Chapter 41 Scatterplots and Correlation.

BPS - 3rd Ed. Chapter 4 1

Chapter 4

Scatterplots and Correlation


Variable (X) and Variable (Y)

Prior chapters one variable at a time This chapter relationship between two

variables One variable is an “outcome”: response

variable (Y) The other variable is a “predictor”:

explanatory variable (X) Are X and Y related? X Y?


Question

A study investigates whether the there is a relationship between gross domestic product and life expectancy:

Which is the explanatory variable (X)?

Which is the response variable (Y)?

All other variables that may influence life expectancy are “lurking” and may confound the relation between X and Y. Are there lurking variables in this analysis?


This chapter considers the case in which both X and Y are quantitative variables

Bivariate data points (xi, yi) are plotted on graph paper to form a scatterplot

Scatterplot


X = percent of students taking SAT

Y = mean SAT verbal score

What is the relationship between X and Y?

Example of a scatterplot


Interpreting scatterplots Form

Can data be described by straight line? [Linearity] Direction

Does the line slope upward or downward Positive association = above-average values of Y

accompany above-average values of X (and vice versa) Negative association = above-average values of Y

accompany below-average values of X (and vice versa)

StrengthDo data point adhere to imaginary line?


Form [discuss]

0

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$0 $10 $20 $30 $40 $50 $60 $70

Income

Hea

th S

tatu

s M

easu

re

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Age

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Age

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uca

tion

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Physical Health Score

Men

tal H

ealt

h S

core


Strength and direction

Direction: positive, negative or flat

Strength: How closely does a non-horizontal straight line fit the points of a scatterplot?Close fitting strong

Loose fitting weak


Strength cannot be reliably judged visually

These two scatterplots are of the same data (they have the exact same correlation)

The second scatter plot looks like a stronger correlation, but this is an artifact of the axis scaling


Correlation coefficient (r)

Let r denote the correlation coefficient r is always between -1 and +1, inclusive Sign of r denotes direction of association Special values for r :

r = +1 all points on upward sloping line r = -1 all points on downward sloping line r = 0 no line or horizontal line The closer r is to +1 or –1, the better the fit of

points to the line


Examples of Correlations Husband’s versus Wife’s ages

r = .94 Husband’s versus Wife’s heights

r = .36 Professional Golfer’s Putting Success:

Distance of putt in feet versus percent success

r = -.94


Correlation Coefficient r Data on variables X and Y for n

individuals:x1, x2, … , xn and y1, y2, … , yn

Each variable has a mean and std dev:2) ch. (see and )) yx

s ,y (s ,x (

n

1i y

i

x

i

s

yy

s

xx

1-n

1r


Correlation coefficient r

y

iY

x

iX

s

yyz

s

xxz

The formula for r can be understood by converting data points to standardized scores:

n

1i1-n

1r YX zz where


Illustrative example (gdp_life.sav)

Per Capita Gross Domestic Productand Average Life Expectancy for

Countries in Western Europe

Does GDP predict life expectancy?



Country Per Capita GDP (X) Life Expectancy (Y)

Austria 21.4 77.48

Belgium 23.2 77.53

Finland 20.0 77.32

France 22.7 78.63

Germany 20.8 77.17

Ireland 18.6 76.39

Italy 21.5 78.51

Netherlands 22.0 78.15

Switzerland 23.8 78.99

United Kingdom 21.2 77.37


Illustrative example (gdp_life.sav) Scatterplot

GDP

24232221201918

LIF

E_

EX

P

79.5

79.0

78.5

78.0

77.5

77.0

76.5

76.0


Illustrative example (gdp_life.sav)x y

21.4 77.48 -0.078 -0.345 0.027

23.2 77.53 1.097 -0.282 -0.309

20.0 77.32 -0.992 -0.546 0.542

22.7 78.63 0.770 1.102 0.849

20.8 77.17 -0.470 -0.735 0.345

18.6 76.39 -1.906 -1.716 3.271

21.5 78.51 -0.013 0.951 -0.012

22.0 78.15 0.313 0.498 0.156

23.8 78.99 1.489 1.555 2.315

21.2 77.37 -0.209 -0.483 0.101

= 21.52 = 77.754sum = 7.285

sx =1.532 sy =0.795

yi /syy xi /sxx

x y

y

i

x

i

s

y-y

s

x-x



0.809

(7.285)110

1

n

1i y

i

x

i

s

yy

s

xx

1-n

1r


Interpretation of r

Direction of association: positive or negative

Strength of association: the closer |r| is to 1, the stronger the correlation. Here are guidelines:

0.0 |r| < 0.3 weak correlation

0.3 |r| < 0.7 moderate correlation

0.7 |r| < 1.0 strong correlation

|r| = 1.0 perfect correlation


Interpretation of r

For GDP / life expectancy example, r = 0.809. This indicates a strong positive correlation

GDP

24232221201918

LIF

E_

EX

P

79.5

79.0

78.5

78.0

77.5

77.0

76.5

76.0


Problems with Correlations

Not all relations are linear Outliers can have large influence on r Lurking variables confound relations


Not all Relationships are Linear Miles per Gallon versus Speed

r 0 (flat line) But there is a non-

linear relationy = - 0.013x + 26.9

r = - 0.06

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speed

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Not all Relationships are Linear Miles per Gallon versus Speed

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speed

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allo

n Curved relationship.

r was misleading.


Outliers and Correlation

The outlier in the above graph decreases r

If we remove the outlier strong relation


Exercise 4.15: Calories and sodium content of hot dogs

(a) What are the lowest and highest calorie counts? …lowest and highest sodium levels?

(b) Positive or negative association?

(c) Any outliers? If we ignore outlier,is relation still linear? Does the correlation become stronger?


Exercise 4.13: IQ and school grades

(a) Positive or negative association?

(b) Is form linear? Does it appear strong?

(c) What is the IQ and GPA for the outlier on the bottom there?

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BPS - 3rd Ed. Chapter 41 Scatterplots and Correlation.

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