<< 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 is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 1

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Collaborative statistics for mt230. OpenStax CNX. Aug 18, 2011 Download for free at http://legacy.cnx.org/content/col11345/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics for mt230' conversation and receive update notifications?

Ask