Skewness plots
skewplot varname [if exp] [in range] [, skew by(byvar) missing scatter_options]
skewplot varlist [if exp] [in range] [, skew scatter_options]
Description
skewplot 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.
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 or quasi-range
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 or 1997, 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.
by(byvar) specifies that calculations are to be carried out separately for each group defined by byvar. by() is allowed only with a single varname.
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.
scatter_options refers to options of graph twoway scatter.
Examples
. webuse citytemp . describe . skewplot *dd . skewplot *dd, skew . skewplot cooldd, by(region) . skewplot cooldd, by(region) ms(i i i i) c(l l l l) . skewplot temp*
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 (2nd edition 1997). 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 n.j.cox@durham.ac.uk
Acknowledgments
Richard Groeneveld tracked down the Bowley reference.
Also see
On-line: graph, symplot Manual: [G] graph, [R] diagnostic plots