04/20/23 Chapter 3 1

Chapter 3

Scatterplots and Correlation

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Both Ch 3 and Ch 4• Relationships between two quantitative

variablesX explanatory variable

Y response variable

• Illustrative Example: What is the relationship between “per capita gross domestic product” (X) and “life expectancy” (Y)?

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Data n = 10 “GDP & 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

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Scatterplot: Life_Exp vs. GDP

GDP

24232221201918

LIF

E_

EX

P79.5

79.0

78.5

78.0

77.5

77.0

76.5

76.0

This is the data point for Switzerland (23.8, 78.99)

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Interpreting Scatterplots1. Form: Straight? Curved?

2. Outliers: Deviations from overall pattern

3. Direction of association:– Positive association (upward)– Negative association (downward)– No association (flat)

4. Strength: Extent to which points adhere to predicted trend line (next slide)

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No association Moderate positive assn Strong positive assn

Weak negative assn. Strong negative assn.Very strong negative assn.

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Interpretation: life expectancy example

This is the data point for Switzerland (23.8, 78.99)

GDP

24232221201918

LIF

E_

EX

P

79.5

79.0

78.5

78.0

77.5

77.0

76.5

76.0

• Form: linear• Outliers:

none• Direction:

positive• Strength:

hard to tell by eye

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Example #2

Interpretation • Form: linear• Outliers: none• Direction: positive• Strength: looks strong

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Example #3

• Form: linear• Outliers: none• Direction: negative• Strength: weak(?)

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Example #4 (Age & Health)

• Form: linear(?)• Outliers: none• Direction: negative(?)• Strength: weak

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Example #5 (Physical & Mental Health)

• Form: U-shaped• Outliers: (?)• Direction: down then

up• Strength: (?)

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• These two figures display the same data set with different axis scaling but the bottom figure looks “stronger” (optical illusion)

• To overcome this difficulty: calculate correlation coefficient r

Strength is Difficult to Judge by Eye Alone

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Correlation Coefficient r • Notation: r ≡ Pearson’s correlation coefficient• Always between −1 and +1

r = +1 all points on upward sloping line

r = -1 all points on downward line

r = 0 no line or horizontal line

The closer r is to +1 or –1, the stronger the correlation

Positive or negative sign indicates direction of correlation

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Guidelines for interpreting “strength” via r

• 0.0 | r | < 0.3 “weak”

• 0.3 | r | < 0.7 “moderate”

• 0.7 | r | < 1.0 “strong”

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Examples

• Husband’s age / Wife’s age• r = .94 (strong positive correlation)

• Husband’s height / Wife’s height• r = .36 (moderate positive correlation)

• Distance of golf putt / percent success• r = -.94 (strong negative correlation)

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Calculating r by hand• Calculate mean and standard deviation of X

• Calculate mean and standard deviation of Y

• Turn all X values into “z scores”

• Turn all Y values into “z scores”

• Calculate r

i

Xx

x xz

s

iY

y

y yz

s

n

1i1-n

1r YX zz

What is a z score?

• z ≡ “standardized value”

• Tells you the number of units above or below the mean in standard deviation units

Examples: •A z score of 1 indicates the value is 1 standard deviation above the mean•A z score of –1 indicates the value is 1 standard deviation below the mean•A z score of 0 indicates the value is equal to the mean

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Calculating r by hand (Example)X Y ZX

ZY ZX ∙ ZY

1 82 6 0 0 03 4 1 -1 -1

1 0 1 2X Yz z

n

i 1

1

1

1

n

X Yi

r z zn

1

23 1

1

1 21

1xz

1

8 61

2yz

1 1

1 1 1x yz z

1

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4

16 2

31 [by calculator]x

x

s

118 6

32 [by calculator]y

y

s

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r by hand can be tedious!(Life expectancy data)

X Y ZXZY ZX ∙ ZX

21.4 77.48 -0.078 -0.345 0.02723.2 77.53 1.097 -0.282 -0.30920.0 77.32 -0.992 -0.546 0.54222.7 78.63 0.770 1.102 0.84920.8 77.17 -0.470 -0.735 0.34518.6 76.39 -1.906 -1.716 3.27121.5 78.51 -0.013 0.951 -0.01222.0 78.15 0.313 0.498 0.15623.8 78.99 1.489 1.555 2.31521.2 77.37 -0.209 -0.483 0.101

x-bar= 21.52 sx= 1.532

y-bar= 77.754 sy = 0.795 X Yz z

n

i 1

7.285

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Example: Calculating r

x yz z 1r

n -1

r = .809 strong positive correlation

1(7.285)

10 1

0.809

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Calculating rUse your calculator in 2-var mode!

TI two-variablecalculator

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Beware!

• r applies to linear relations only

• Outliers have large influences on r

• Association ≠ causation

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Nonlinear relation (mpg vs. speed)

0

5

10

15

20

25

30

35

0 50 100

speed

05

1015

2025

3035

0 50 100

speed

Strong non-linear relationshipsCan show r = 0r = 0

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Outliers Have Undue Influence

With the outlier, r 0

Without the outlier, r .8

x