Covariance measures the tendency of two random variables to move in
tandem ( http://mathforum.org/library/drmath/view/51438.html ) .
Another detailed definition is from:
http://www.public.asu.edu/~pythagor/covariance.htm .
A useful online statistics text is at:
http://davidmlane.com/hyperstat/
An example using Excel is at:
http://www.mste.uiuc.edu/malcz/Spreads/COINTRO.html
Given two discrete random variables X and Y, with means Xm and Ym
respectively, their covariance is defined as the mean of the
products (X-Xm)(Y-Ym). This value turns out to equal the mean of XY
minus Xm*Ym . (The mean of the product minus the product of the
means).
In simple inferential statistics it is more common to use the
correlation coefficient than the covariance directly; the correlation
coefficient scales the covariance so that the values are always
between -1 and 1. To get the correlation coefficient from the
covariance, we just renormalize the random variables so that they each
have standard deviation of 1.
That is, the correlation coeffiecient of X and Y is the covariance of
Z_X and Z_Y, where Z_X is the random variable (X-Xm)/Sx and
Sx is the standard deviation of X.
An example of where we might use the covariance is in determining if
their is a relationship between two variables that we are interested
in. For example, suppose the two variables are "team payroll for a
major league team" and "number of wins in the season for that team".
Then if the correlation coefficient were 1 we could decide that
winning percentage increased with team payroll; if the correlation
were -1 we could decide that winning percentage decreased with team
payroll; and if the correlation were 0 we could decide there was no
relationship. (Actually we would want to do additional statistical
tests).
Analysis of variance (ANOVA) is used to test hypotheses about
differences between two or more means. (This presentation is lifted
from: http://davidmlane.com/hyperstat/intro_ANOVA.html ) . ANOVA does
not have a simple one-sentence definition like "covariance"; there are
many types of ANOVA. ANOVA is more of a general technique.
The example given at : http://davidmlane.com/hyperstat/intro_ANOVA.htm
----------------------begin quotation-------
Consider a hypothetical experiment on the effect of the intensity of
distracting background noise on reading comprehension. Subjects were
randomly assigned to one of three groups. Subjects in Group 1 were
given 30 minutes to read a story without any background noise.
Subjects in Group 2 read the story with moderate background noise, and
subjects in Group 3 read the story in the presence of loud background
noise.
-------end quotation-------
The experimenter wants to determine whether noise impacts reading
comprehension; that is, whether the three means for no noise, moderate
noise, and loud noise are equal.
-----begin quotation---------------------
The experimental design therefore has one factor (noise intensity) and
this factor has three levels: no noise, moderate noise, and loud
noise. Analysis of variance can be used to provide a significance test
of the null hypothesis that these three population means are equal. If
the test is significant, then the null hypothesis can be rejected and
it can be concluded that background noise has an effect.
-------end quotation----------------
Linear regression is used to determine the relationship between two
random variables X and Y. We try to write Y=a + bX + e
where a and b are constants and e is called the "residual" which we
want to make as small as possible.
A good applet to play around with linear regressions is at
http://www.math.csusb.edu/faculty/stanton/m262/regress/regress.html
. Here you can input points for X and Y and it will compute the
regression line for you.
Linear regression might be used when, for example, we have the height
and the weight of a number of adults and want to infer a linear
relationship between them. Once we have this relationship, then if we
know the height, but not the weight, of a new adult, we can guess the
adult's weight. (Example taken from above URL).
HELPFUL LINKS:
Covariance definition and some basic statistical information: (
http://mathforum.org/library/drmath/view/51438.html ) .
A more detailed definition of covariance:
http://www.public.asu.edu/~pythagor/covariance.htm .
A useful online statistics text is at:
http://davidmlane.com/hyperstat/
A covariance example using Excel is at:
http://www.mste.uiuc.edu/malcz/Spreads/COINTRO.html
An ANOVA example is at :
http://davidmlane.com/hyperstat/intro_ANOVA.htm
A regression applet is at:
http://www.math.csusb.edu/faculty/stanton/m262/regress/regress.html
SEARCH TERMS:
"linear regression", "ANOVA", "statistics covariance"
[Note: I am not sure in your comment "give one real world sitatuation
where each could be used" whether you wanted a single situation in
which each of the three techniques could be used, or whether you
wanted real world situations illustrating each of them. I also was not
sure whether you wanted "real-world" to mean "actually occurring" or
"something that could occur but was made up". If I've misinterpreted
here, please clarify and I can modify the examples]. |
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