Comparisons of Statistics
Professor Harvey A. Singer
School of Management
George Mason University
I
Inferences about the Data Distribution.
1
Skewness vs. Symmetry.
Skewness refers to the lack of symmetry of distribution of data about its center.
1.1
A simple skewness factor.
The quantity
x
x
x
ˆ

is a nondimensional measure of the lack of symmetry or skewness, where
x
is the
sample mean and
x
ˆ is the sample median.
It compares the offset between the sample
mean and median to the sample mean.
This skewness factor contains an algebraic sign.
The algebraic sign indicates the direction of skewing.
The magnitude of the skewing
factor indicates the severity of the skewing.
A skewness factor with a value of zero
indicates no skewing:
the mean and median coincide.
When the median is less in magnitude than the mean,
x
x
ˆ and the median lies to the left
of the mean on the number line.
As a result, the term in the numerator is positive and so
too is the skewness factor.
In this case, the distribution is positively skewed so that the
“hump” is on the left and the long tail is on the right; the distribution is rightskewed.
The interpretation is that most of the data is less in value than the sample mean.
More of
the sample data lie below (to the left) of the mean than above it (to the right).
When the median is greater in magnitude than the mean,
x
x
ˆ
<
and the median lies to the
right of the mean on a number line.
As a result, the term in the numerator is negative and
so too is the skewness factor.
In this case, the data distribution is negatively skewed so
that the “hump” is on the right and the long tail is on the left; the distribution is left
skewed. The interpretation is that most of the data is greater in value than the sample
mean.
More of the sample data lie above (to the right) of the mean than below it (to the
left).
© 1999 by Harvey A. Singer
1
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The higher the magnitude the more
severely skewed and the more asymmetric is the distribution.
In summary,
=
<

right
skewed
if
skew
no
symmetric
if
left
skewed
if
x
x
x
0
)
(
0
0
ˆ
1.2
An alternative skewness factor.
An alternative nondimensional measure of skewing is simply the ratio of the median to
mean, viz
x
x
ˆ
is another dimensionless measure of the skew or lack of symmetry of a distribution.
Here

=

<
skewed
left
if
skew
no
symmetric
if
skewed
right
if
x
x
1
)
(
1
1
ˆ
1.3
Pearson skewness factor.
Yet another dimensionless measure of skewness is the Pearson skewness factor, defined
as
(
29
s
x
x
ˆ
3

The same rules and interpretations apply as for the simple skewness factor in section 1.1
above.
2
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 Fall '08
 SINGER
 Standard Deviation, Harvey A. Singer

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