Is there a relationship between the lengths of body

parts?

The invalid assumption that correlation implies cause is probably among the two or three most serious and common errors of human reasoning.

--Stephen Jay Gould, The Mismeasure of Man

Linear Correlation & Regression

Essentials: Correlation(The invalid assumption that correlation implies cause is probably among the two or three most serious and common errors of human reasoning. --Stephen Jay Gould,

The Mismeasure of Man.)

Correlation – potential relationships, not causality.

Know the steps one might employ before obtaining a correlation.

Know the characteristics of the Pearson Product Moment Correlation Coefficient (for us the correlation).

Be able to calculate a correlation and determine if it is statistically significant.

Be able to create a scatter plot of the paired data being studied.

Be able to determine the directionality of a correlation and its strength via formula and observation of plotted data.

Correlation Correlation – A correlation exists between two

variables when one of them is related to the other in some way.

Paired Data – A measurement on two variables for each unit in a population or sample.

Scatterplot – a graph in which the paired (x,y) data are plotted with a horizontal x-axis (independent variable) and a vertical y-axis (dependent variable). Each individual pair is plotted as a single point.

ANATOMY OF A SCATTER PLOT

A scatterplot graphs the relationship between paired (x, y) quantitative data values. If it is believed that there is a causal relationship, the independent variable (x) is placed on the x-axis, while the dependent variable (y) is placed on the y-axis.

The data presented in this scatterplot represent the time and distance of eight balsa wood airplane flights. Making the assumption that time in air might affect overall distance, the time variable was placed on the x-axis. The distance variable is presented on the y-axis. Each dot on the graph corresponds to one (x,y) pair from the data set.

Time and Distance Relationship for Straight Flights

of Starfire Balsa Wood Airplanes

Data Collected: Fall 2004

Time (sec.)

987654321

Dis

t (c

m)

5000

4000

3000

2000

1000

0

Time (sec.) Distance (cm.)2.75 6268.59 27706.42 45805.22 29332.9 15033.02 19734.31 22351.68 1250

Building a Scatterplot:

1) Identify two quantitative variables that appear to have a relationship. If there appears to be a causal relationship, the values of the independent variable (x) are recorded on the x-axis and the values of the dependent variable (y) are recorded via the y-axis.

2) Create a graph with the x-axis containing a scale appropriate to the x variable and a label, which identifies the measurement scale, e.g. seconds. On the y-axis place the scale for the y variable and include a label.

3) Obtain a listing of the paired data values. (The data for this scatterplot are noted below.)

4) Using the (x,y) coordinates, place a mark on the graph for each set of paired values.

5) Add a title and other useful information

Data used for this scatterplot

Title.

Y-axis variable and measurement scale.

X-axis variable and measurement scale.

Data points for the paired variables.

e.g. (8.59, 27.70)

Tar and Nicotine Amounts

In 29 Brands of Cigarettes

NICOTINE

1.61.41.21.0.8.6.4.20.0

TAR

20

10

0

Scatter plot

Paired Data For Six Dining Parties

x x

yy y

x( a ) P o s i t iv e ( b ) S t r o n g

p o s i t iv e( c ) P e r f e c t

p o s i t iv e

x x

yy y

x( a ) P o s i t iv e ( b ) S t r o n g

p o s i t iv e( c ) P e r f e c t

p o s i t iv e

Positive Linear Correlation

x x

yy y

x(d ) N e g a t iv e (e ) S tro n g

n e g a t iv e( f) P e r fe c t

n e g a t iv e

x x

yy y

x(d ) N e g a t iv e (e ) S tro n g

n e g a t iv e( f) P e r fe c t

n e g a t iv e

Negative Linear Correlation

x x

yy

(g ) N o C o rre la tio n (h ) N o n lin e a r C o rre la tio n

x x

yy

(g ) N o C o rre la tio n (h ) N o n lin e a r C o rre la tio n

No Linear Correlation

The Linear Correlation Coefficient

Denoted r when considering a sample, and

(rho) when considering a population. The Linear Correlation Coefficient is a measure

of direction and magnitude between the paired x and y values in a sample. Its value is obtained using the following formula:

2222 )()(

))((

yynxxn

yxxynr

Facts About r

The value of r is always between –1 and 1. The sign (-/+) of r reflects the direction of

the correlation. If r is negative, then there exists a negative

association between the two variables. That is, as one increases, the other decreases.

If r is positive, then there exists a positive relationship between the two variables. That is, as one increases, the other increases.

The magnitude of the correlation indicates the strength of the association. Values closer to –1 and 1 signify a stronger association A value of –1 is a perfect negative correlation. A value of 1 is a perfect positive correlation.

Facts About r (cont.)

The value of r does not change if all values of either variable are converted to a different scale.

The value of r is not affected by the choice of x and y. That is, if x and y are interchanged, the value of r will not change.

Facts About r (cont.)

Does a Correlation Actually Exist?

The answer to this can be somewhat subjective. How strong does a correlation need to be? Start by asking the following:

Does it make sense to look at this relationship? Does a scatter plot present a relationship (either

positive or negative)? If yes to both, calculate r.

We Begin With a Hypothesis

In linear correlation, the null hypothesis states that no linear correlation exists. In other words,

In notation The alternative hypothesis states that a

linear correlation does exist. In other words

In notation

0:0 H

0:1 H

0

0

We Test The Hypothesis

Based on the sample data, a value for r is obtained. This is called the test statistic.

The absolute value of the test statistic is then compared to the appropriate value in a table of critical values of r.

456789

101112131415161718192025303540455060708090100

n

.999

.959

.917

.875

.834

.798

.765

.735

.708

.684

.661

.641

.623

.606

.590

.575

.561

.505

.463

.430

.402

.378

.361

.330

.305

.286

.269

.256

.950

.878

.811

.754

.707

.666

.632

.602

.576

.553

.532

.514

.497

.482

.468

.456

.444

.396

.361

.335

.312

.294

.279

.254

.236

.220

.207

.196

= .05 = .01

456789

101112131415161718192025303540455060708090100

n

.999

.959

.917

.875

.834

.798

.765

.735

.708

.684

.661

.641

.623

.606

.590

.575

.561

.505

.463

.430

.402

.378

.361

.330

.305

.286

.269

.256

.950

.878

.811

.754

.707

.666

.632

.602

.576

.553

.532

.514

.497

.482

.468

.456

.444

.396

.361

.335

.312

.294

.279

.254

.236

.220

.207

.196

= .05 = .01Table of CriticalValues for r

Conclusion If the absolute value of r exceeds the table

value, we reject the null hypothesis which states that no significant linear correlation exists.

If the absolute value of r does not exceed the table value, we fail to reject the null hypothesis.

Recall Linear Correlation

Association between 2 quantitative variables.

Paired data (bivariate data). Scatter plot. Positive/Negative. Correlation coefficient, r.