220 High-Function Business Intelligence in e-business
Standard deviation is the root mean square distance of the data from the mean.
The definition of standard deviation is the square root of the variance.
The relationship between population standard deviation (SD
pop
) and sample
standard deviation (SD
samp
) is as follows:
Where:
n is the population size.
A.1.3 Covariance
This is not related to the variance function.
Covariance is a measure of the linear association between two variables. The
The covariance value depends upon the units of measurement of the variables
involved, and therefore unusable directly. A more useful measure of the linear
relationship can be gained via correlation.
Covariance is defined as follows:
Important: When the data values represent the entire set of values, then the
Population Standard Deviation is computed.
When the data values represent a sample (subset) of the entire set of values,
then the
Sample Standard Deviation is computed.
StdDev SQRT VAR X
()()
X
i
X
–()
2
i
1=
n
n
⁄==
SD
pop
n
1–()
n
-----------------
SD
samp
×=
Cov X Y
,()
X
i
X
–()(
i
1=
n
Y
i
Y
–())
n
⁄×=
Appendix A. Introduction to statistics and analytic concepts 221
Where:
X
i
is the i
th
observation on variable X.
Y
j
is the j
th
observation on variable Y.
X is the average of all values of X.
Y
is the average of all values of Y.
i starts at 1 and continues up to n observations.
The Greek letter Sigma represents the summary of the enclosed equation.
The meaning of covariance given Table A-2.
Table A-2 Covariance meaning
The relationship between population covariance (Covar
pop
) and sample
covariance (Covar
samp
) is as follows:
Where:
n is the population size.
Covariance Value Meaning
Greater than zero (positive) The variables are directly linearly related.
As one increases so does the other.
Zero There is no linear relationship between the
two variables.
Less than zero (negative) The variables are inversely linearly
related. As one increases the other
decreases.
Important: When the data values represent the entire set of values, then the
Population Covariance is computed.
When the data values represent a sample (subset) of the entire set of values,
then the
Sample Covariance is computed.
Covar
pop
n
1–()
n
-----------------
Covar
samp
×=
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