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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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
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