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help for ^skewplot6^
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Skewness plots
--------------

    ^skewplot6^ varname [^if^ exp] [^in^ range]
    [^, skew^ graph_options ^by(^byvar^) mono miss^ing ]

    ^skewplot6^ varlist [^if^ exp] [^in^ range]
    [^, skew^ graph_options ^mono miss^ing ]


Description
-----------

^skewplot6^ produces by default a plot of the midsummary versus the spread 
for the variables in varlist, also known as the mid versus spread plot. 

With the ^skew^ option, it produces a plot of the skewness function versus 
the spread function. 

Such plots convey both the general character and the fine structure of the
symmetry or skewness of data sets, and can be used to compare distributions 
or to assess whether transformations are necessary or effective.

^skewplot6^ is a renamed clone of ^skewplot^ 2.0.0 which is for Stata 6 
or Stata 7. Stata 8 users should use ^skewplot^ 3.0.0 or later. 


Remarks
-------

Order n data values for a variable x and label them such that 
x_(1) <= ... <= x_(n). 

In a perfectly symmetric set of data, the midsummaries 

	(x_(1) + x_(n)) / 2, 
	(x_(2) + x_(n - 1)) / 2, 
	etc. 

would all be identical, and equal to the median. A plot of each 
midsummary 

	(x_(i) + x_(n - i + 1)) / 2

versus each difference or spread 

	x_(n - i + 1) - x_(i) 

would yield a horizontal straight line. Conversely, skewness in sets of 
data will be reflected by departures from horizontality.

Apart from the divisor of 2, this plot was suggested by J.W. Tukey 
(Wilk and Gnanadesikan 1968). See also Gnanadesikan (1977, Ch.6.2) or 
Fisher (1983). The form used here and the name `mid versus spread plot' 
are found in Hoaglin (1985). It is usual to plot only that half of the 
sample results for which spread is >= 0. 

The ^skew^ option produces an alternative form promoted by Benjamini and 
Krieger (1996, 1999). The identity 

	x_(n - i + 1) = median  

	              + (x_(n - i + 1) - x_(i)) / 2 

                      + (x_(i) + x_(n - i + 1) - 2 * median) / 2 

	              = median + spread function + skewness function 	

for x_(i) in the lower half of the sample leads to a plot of the skewness 
function versus the spread function, known as the skewness versus spread 
plot. Note that the skewness function is midsummary - median, and will be 
constant and zero for a perfectly symmetric distribution, and that the 
spread function is half the spread of the mid versus spread plot.

In addition, the ratio of the skewness and spread functions or

	x_(i) + x_(n - i + 1) - 2 * median
	----------------------------------
              x_(n - i + 1) - x_(i)

is a measure of skewness (in the traditional sense) originally suggested 
for quartiles by Bowley (1902) and generalised to this form by David and 
Johnson (1956). It varies between -1 and 1. A similar general measure was 
used by Parzen (1979). Graphically this measure is the slope of the line 
connecting (0,0) and each data point. 

See Benjamini and Krieger (1996, 1999) and Groeneveld (1998) for concise 
reviews tracing such ideas from late 19th century antecedents to recent 
work and further details on the interpretation of the skewness versus 
spread plot. 


Options
-------

^skew^ specifies the skewness versus spread plot, not the default 
    mid versus spread plot. 

graph_options are options allowed with ^graph, twoway^.

    Note that with ^by( )^ each group is treated graphically as if it
    were a separate variable, so long as the number of groups is not
    greater than the limit in Stata on the number of y variables on a
    scatter plot (20 in Stata 6.0).

    With more groups, all functions must be treated graphically as a
    single variable, by using the ^mono^ option, which enforces a
    monochrome treatment. The only ^connect^ line style appropriate is
    then ^c(L)^, and only one ^pen^ and point ^symbol^ may be used.

    By default ^ysc( )^ and ^xsc( )^ show the extremes observed for each 
    variable. With the ^skew^ option, the maximum value of spread (smax, 
    say) can be used in a second pass with ^ysc(-^smax^,^smax^)^, which are 
    the limits on possible values for skewness. For example, if the 
    maximum value observed is ^12.34^, setting ^ysc(-12.34,12.34)^ gives a 
    y axis stretched to show extremes possible for skewness, corresponding
    to limits for a Bowley-type measure of -1 and 1. 

^by(^byvar^)^ specifies that calculations are to be carried out
    separately for each group defined by ^byvar^. Any graph will,
    however, show results for all groups. ^by( )^ is only allowed 
    with a single varname.

^mono^ specifies a monochrome treatment, with a single ^pen^ colour,
    ^connect^ style and point ^symbol^. See above, under graph_options.

^missing^, used only with ^by( )^, permits the use of non-missing values
    of varname corresponding to missing values for the variable named by
    ^by( )^. The default is to ignore such values.


Examples
--------

 . ^skewplot6 mpg^
 . ^skewplot6 mpg, by(foreign) c(ll)^
 . ^skewplot6 mpg, by(rep78) c(L) sy([rep78]) mono skew^ 
 . ^skewplot6 length width height^


References
----------

Benjamini, Y. and Krieger, A.M. 1996. Concepts and measures for skewness
with data-analytic implications. Canadian Journal of Statistics 24: 
131-140.

Benjamini, Y. and Krieger, A.M. 1999. Skewness -- concepts and measures.  
In Kotz, S., Read, C.B. and Banks, D.L. (eds) Encyclopedia of Statistical
Sciences Update Volume 3. New York: John Wiley, 663-670. 

Bowley, A.L. 1902. Elements of statistics. London: P.S. King.
(2nd edition: see p.331.)

David, F.N. and Johnson, N.L. 1956. Some tests of significance with 
ordered variables. Journal, Royal Statistical Society B 18: 1-20. 

Fisher, N.I. 1983. Graphical methods in nonparametric statistics: a review
and annotated bibliography. International Statistical Review 51: 25-58.

Gnanadesikan, R. 1977. Methods for statistical data analysis of multivariate
observations. New York: John Wiley. 

Groeneveld, R. 1998. Skewness, Bowley's measures of. In Kotz, S., Read, 
C.B. and Banks, D.L. (eds) Encyclopedia of Statistical Sciences Update 
Volume 2. New York: John Wiley, 619-621. 

Hoaglin, D.C. 1985. Using quantiles to study shape. In Hoaglin, D.C., 
Mosteller, F. and Tukey, J.W. (eds) Exploring data tables, trends, and 
shapes. New York: John Wiley, 417-460. 

Parzen, E. 1979. Nonparametric statistical data modeling. Journal, American 
Statistical Association 74, 105-131. 

Wilk, M.B. and Gnanadesikan, R. 1968. Probability plotting methods for
the analysis of data. Biometrika 55: 1-17.


Author
------

         Nicholas J. Cox, University of Durham, U.K.
         n.j.cox@@durham.ac.uk

	 
Acknowledgments
---------------

         Richard Groeneveld tracked down the Bowley reference.


Also see
--------

On-line: help for @graph@, @symplot@
 Manual: [R] graph, [R] diagplots
 

