<< Chapter < Page Chapter >> Page >
Linear Regression and Correlation: The Correlation Coefficient and Coefficient of Determination is a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean with contributions from Roberta Bloom. The name has been changed from Correlation Coefficient.

The correlation coefficient r

Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? Use the correlation coefficient as another indicator(besides the scatterplot) of the strength of the relationship between x and y .

The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is a numerical measure of the strength of association between the independent variable x and the dependent variable y.

The correlation coefficient is calculated as

r = n Σ x y - ( Σ x ) ( Σ y ) [ n Σ x 2 - ( Σ x ) 2 ] [ n Σ y 2 - ( Σ y ) 2 ]

where n = the number of data points.

If you suspect a linear relationship between x and y , then r can measure how strong the linear relationship is.

    What the value of r tells us:

  • The value of r is always between -1 and +1: -1 r 1 .
  • The size of the correlation r indicates the strength of the linear relationship between x and y . Values of r close to -1 or to +1 indicate a stronger linear relationship between x and y .
  • If r=0 there is absolutely no linear relationship between x and y (no linear correlation) .
  • If r = 1 , there is perfect positive correlation. If r = -1 , there is perfect negative correlation. In both these cases, all of the original data points lie on a straight line. Of course,in the real world, this will not generally happen.

    What the sign of r tells us

  • A positive value of r means that when x increases, y tends to increase and when x decreases, y tends to decrease (positive correlation) .
  • A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase (negative correlation) .
  • The sign of r is the same as the sign of the slope, b , of the best fit line.
Strong correlation does not suggest that x causes y or y causes x . We say "correlation does not imply causation." For example, every person who learned math in the 17th century is dead. However, learning math does not necessarily causedeath!

Positive correlation

Scatterplot of points ascending from the lower left to the upper right.
A scatter plot showing data with a positive correlation. 0 r 1

Negative correlation

Scatterplot of points descending from the upper left to the lower right.
A scatter plot showing data with a negative correlation. -1 r 0

Zero correlation

Scatterplot of points in a horizontal configuration.
A scatter plot showing data with zero correlation. r =0

The formula for r looks formidable. However, computer spreadsheets, statistical software, and many calculators can quickly calculate r . The correlation coefficient r is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).

The coefficient of determination

r 2 is called the coefficient of determination. r 2 is the square of the correlation coefficient , but is usually stated as a percent, rather than in decimal form. r 2 has an interpretation in the context of the data:

  • r 2 , when expressed as a percent, represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression (best fit) line.
  • 1- r 2 , when expressed as a percent, represents the percent of variation in y that is NOT explained by variation in x using the regression line. This can be seen as the scattering of the observed data points about the regression line.

    Consider the third exam/final exam example Introduced in the previous section

  • The line of best fit is: y ^ = -173.51 + 4.83x
  • The correlation coefficient is r = 0.6631
  • The coefficient of determination is r 2 = 0.6631 2 = 0.4397
  • Interpretation of r 2 in the context of this example:
  • Approximately 44% of the variation (0.4397 is approximately 0.44) in the final exam grades can be explained by the variation in the grades on the third exam, using the best fit regression line.
  • Therefore approximately 56% of the variation (1 - 0.44 = 0.56) in the final exam grades can NOT be explained by the variation in the grades on the third exam, using the best fit regression line. (This is seen as the scattering of the points about the line.)

**With contributions from Roberta Bloom.

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 1

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Collaborative statistics (custom online version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11476/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics (custom online version modified by t. short)' conversation and receive update notifications?

Ask