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.
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:
$-1\le r\le 1$ .
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
$\mathrm{r=0}$ 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!
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.
The line of best fit is:
$\hat{y}=-173.51+\text{4.83x}$
The correlation coefficient is
$r=0.6631$
The coefficient of determination is
${r}^{2}$ =
${\mathrm{0.6631}}^{2}$ = 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 fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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
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?
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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