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ES9

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Weight

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Regression PlotY = 2.31464 + 1.28722X

r = 0.559

Chapters 4 ~ Scatterplots & Correlation

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ES9 Chapter Goals

• To be able to present bivariate data in tabular and graphic form

• To gain an understanding of the distinction between the basic purposes of correlation analysis and regression analysis

• To become familiar with the ideas of descriptive presentation

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Three combinations of variable types:

1. Both variables are qualitative (attribute)

2. One variable is qualitative (attribute) and the other is quantitative (numerical)

3. Both variables are quantitative (both numerical)

Bivariate DataBivariate Data: Consists of the values of two different response variables that are obtained from the same population of interest

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ES9 Two Quantitative Variables1. Expressed as ordered pairs: (x, y)2. x: input variable, independent variable

y: output variable, dependent variable

Scatter Diagram: A plot of all the ordered pairs of bivariate data on a coordinate axis system. The input variable x is plotted on the horizontal axis, and the output variable y is plotted on the vertical axis.

Note: Use scales so that the range of the y-values is equal to or slightly less than the range of the x-values. This creates a window that is approximately square.

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Example: In a study involving children’s fear related to being hospitalized, the age and the score each child made

on the Child Medical Fear Scale (CMFS) are given in the table below:

Age (x ) 8 9 9 10 11 9 8 9 8 11CMFS (y ) 31 25 40 27 35 29 25 34 44 19

Age (x ) 7 6 6 8 9 12 15 13 10 10CMFS (y ) 28 47 42 37 35 16 12 23 26 36

Example

Construct a scatter diagram for this data

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ES9

• age = input variable, CMFS = output variable

Solution

Child Medical Fear Scale

1514131211109876

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CMFS

Age

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ES9 Linear Correlation• Measures the strength of a linear relationship

between two variables

– As x increases, no definite shift in y: no correlation

– As x increases, a definite shift in y: correlation

– Positive correlation: x increases, y increases

– Negative correlation: x increases, y decreases

– If the ordered pairs follow a straight-line path: linear correlation

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• As x increases, there is no definite shift in y:

Example: No Correlation

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• As x increases, y also increases:

Example: Positive Correlation

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• As x increases, y decreases:

Example: Negative Correlation

Output

Input

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ES9 Please Note Perfect positive correlation: all the points lie along a line

with positive slope

Perfect negative correlation: all the points lie along a line with negative slope

If the points lie along a horizontal or vertical line: no correlation

If the points exhibit some other nonlinear pattern: no linear relationship, no correlation

Need some way to measure correlation

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ES9 Bivariate DataCoefficient of Linear Correlation: r, measures the strength of the linear relationship between two variables

rx x y yn s sx y

( )( )( )1

Pearson’s Product Moment Formula:

1 1r

Notes: r = +1: perfect positive correlation r = -1 : perfect negative correlation

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ES9 Alternate Formula for r

SS “sum of squ ares for ( )x x” x

xn

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SS “sum of squ ares for ( )y y” y

yn

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SS “sum of squares for ( )xy xy” xyx yn

r xyx y

SSSS SS

( )( ) ( )

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ES9

Example: The table below presents the weight (in thousands of pounds) x and the gasoline mileage (miles per gallon) y for ten different automobiles. Find the linear

correlation coefficient:

Example

x y x2 y2 xy

x y x2 y2 xy

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ES9 Completing the Calculation for r

SS( ) ( ) .y y

yn

22 2

10665 30910

1116 9

SS( ) . ( . )( ) .xy xyx yn

1010 9 34 1 30910

42 79

r xyx y

SSSS SS

( )( ) ( )

.( . )( . )

0.42 797 449 1116 9

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SS( ) . ( . ) .x x

xn

22 2

123 73 34 110

7 449

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ES9 Please Note r is usually rounded to the nearest hundredth

r close to 0: little or no linear correlation

As the magnitude of r increases, towards -1 or +1, there is an increasingly stronger linear correlation between

the two variables

Method of estimating r based on the scatter diagram. Window should be approximately square. Useful for checking calculations.