1 - People

Transcript 1 - People

Summarizing Bivariate Data

Terms, Scatterplots and Correlation

Terms

multivariate

data set consists of measurements or observations on each of two or more variables.

The classroom data set introduced in the slides for Chapter 1 is a multivariate data set. The data set includes observations on the variables: age, weight, height, gender, vision (correction method), and smoke (status). Age, weight and height are numerical variables while gender, vision and smoke are categorical variables. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.

Terms

bivariate

data set consists of measurements or observations on each of two variables.

Scatterplots

A scatterplot is a plot of pairs of observed values (both quantitative) of two different variables.

When one of the variables is considered to be a response variable (y) and the other an explanatory variable (x). The explanatory variable is usually plotted on the x axis.

Example

A sample of one-way Greyhound bus fares

Destination City

from Rochester, NY to cities less than 750 miles

Albany, NY Baltimore, MD

was taken by going to Greyhound’s website.

Buffalo, NY Chicago, IL Cleveland, OH

The following table

Montreal, QU

gives the destination city,

New York City, NY

the distance and the one-

Ottawa, ON Philadelphia, PA

way fare. Distance

Potsdam, NY

should be the x axis and

Syracuse, NY

the Fare should be the y

Toronto, ON Washington, DC

axis.

Distance 240 430 69 607 257 480 340 467 335 239 95 178 496 Standard One-Way Fare 39 81 17 96 61 70.5

65 82 67 47 20 35 87

Example Scatterplot

$100 $90 $80 Greyhound Bus Fares Vs. Distance $70 $60 $50 $40 $30 $20 $10 50 150 250 350 450 Distance from Rochester, NY (miles) 550 650

Comments

The axes need not intersect at (0,0). For each of the axes, the scale should be chosen so that the minimum and maximum values on the scale are convenient and the values to be plotted are between the two values.

Notice that for this example, 1. The x axis (distance) runs from 50 to 650 miles where the data points are between 69 and 607.

2. The y axis (fare) runs from $10 to $100 where the data points are between $17 and $96.

Further Comments

It is possible that two points might have the same x value with different y values. Notice that Potsdam (239) and Albany (240) come very close to having the same x value but the y values are $8 apart. Clearly, the value of y in not determined

solely

by the x value (there are factors other than distance that affect the fare.

In this example, the y value tends to increase as x increases. We say that there is a positive relationship between the variables distance and fare.

It appears that the y value (fare) could be predicted reasonably well from the x value (distance) by finding a line that is close to the points in the plot.

Association

Positive Association - Two variables are positively associated when above-average values of one tend to accompany above average values of the other and below average values tend similarly to occur together. (i.e., Generally speaking, the y values tend to increase as the x values increase.) Negative Association - Two variables are negatively associated when above-average values of one accompany below-average values of the other, and vice versa. (i.e., Generally speaking, the y values tend to decrease as the x values increase.) Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.

The Pearson Correlation Coefficient

A measure of the strength of the linear relationship between the two variables is called the Pearson correlation coefficient. The Pearson sample correlation coefficient is defined by r   z z x y       x   Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.

Example Calculation

s x  325.615

 164.2125

x s y=59.0385

 25.506

240 430 69 607 257 480 340 467 335 239 95 178 496

39 81 17 96 61 70.5

65 82 67 47 20 35 87 x-x s x -0.5214

0.6357

-1.5627

1.7135

-0.4178

0.9402

0.0876

0.8610

0.0571

-0.5275

-1.4044

-0.8989

1.0376

y-y s y -0.7856

0.8610

-1.6481

1.4491

0.0769

0.4494

0.2337

0.9002

0.3121

-0.4720

-1.5305

-0.9424

1.0962

  x-x s x      y-y   s y 0.4096

0.5473

2.5755

2.4831

-0.0321

0.4225

0.0205

0.7751

0.0178

0.2489

2.1494

0.8472

1.1374

11.6021

r



11.601



0.9668

Some Correlation Pictures

Properties of r

The value of r does not depend on the unit of measurement for each variable.

The value of r does not depend on which of the two variables is labeled x.

The value of r is between –1 and +1.

The correlation coefficient is a) –1 only when all the points lie on a downward-sloping line, and b) +1 only when all the points lie on an upward-sloping line.

The value of r is a measure of the extent to which x and y are linearly related.

An Interesting Example

Consider the following bivariate data set: x 1.2

2.5

6.5

13.1

24.2

34.1

20.8

37.5

y 23.3

21.5

12.2

3.9

4.0

18.0

1.7

26.1

An Interesting Example

Computing the Pearson correlation coefficient, we find that r = 0.001

x  x x y s X y  y s y   x  s X x     y  s y y   1.2

2.5

6.5

13.1

24.2

34.1

20.8

37.5

23.3

21.5

12.2

3.9

4.0

18.0

1.7

26.1

-1.167

-1.074

-0.788

-0.314

0.481

1.191

0.237

1.434

0.973

0.788

-0.168

-1.022

-1.012

0.428

-1.249

1.261

-1.136

-0.847

0.133

0.322

-0.487

0.510

-0.296

1.810

0.007

r = 0.001

x  17.488, s x  13.951, y  13.838, s y  9.721

r  1    x  s X x     y s  y y    1 7 (0.007)  0.001

An Interesting Example

With a sample Pearson correlation coefficient, r = 0.001, one would note that there seems to be little or no linearity to the relationship between x and y.

Be careful that you do not infer that there is no relationship between x and y.

An Interesting Example

Note (below) that there appears to be an almost perfect quadratic relationship between x and y when the scatterplot is drawn.

30.0

25.0

20.0

15.0

10.0

5.0

0.0

0 5 10

Scatterplot

15 20

25 30 35 30.0

25.0

20.0

15.0

10.0

5.0

0.0

Scatterplot

5 10 15 20

1 - People

Transcript 1 - People

Summarizing Bivariate Data

Terms, Scatterplots and Correlation

Terms

Terms

Scatterplots

Example

Example Scatterplot

Comments

Further Comments

Association

The Pearson Correlation Coefficient

Example Calculation

r

11.601

0.9668

Some Correlation Pictures

Some Correlation Pictures

Some Correlation Pictures

Some Correlation Pictures

Some Correlation Pictures

Some Correlation Pictures

Properties of r

An Interesting Example

An Interesting Example

An Interesting Example

An Interesting Example

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