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help for circsummarize
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Summary statistics for circular data

circsummarize varlist [weight] [if exp] [in range] [, ci level(#)
rayleigh kuiper detail ]

circsummarize varname [weight] [if exp] [in range] [, by(byvar) ci
level(#) rayleigh kuiper detail ]

fweights and aweights are allowed; see help weights.  If weights are
specified, they are used in calculating the mean direction and the vector
strength (and hence the Rayleigh test and confidence limits if produced,
although this may not be justifiable). Weights are not used in
calculating the circular range or the Kuiper test.

The abbreviation circsu (only) is allowed.

Description

circsummarize produces summary statistics for circular variables with
scales between 0 and 360 degrees. These are by default

the mean direction (in degrees),

the vector strength or mean resultant length (i.e.  length of
resultant / number of observations)

and the circular range (length of smallest arc including all
observations) (in degrees).

Suppose we have n measurements of a circular variable t.  Then calculate
in turn
C = SUM cos t
S = SUM sin t
arctan(S / C), the mean direction
R = sqrt(C * C + S * S), the length of the resultant vector
R / n, the vector strength or mean resultant length.

Options

by() specifies that results are to be shown for each group defined by
values of byvar. This option is only available if a single varname is
specified.

ci produces a confidence interval for the vector mean suitable for large
samples (at least 25 values, say). This is based on an assumption
that sampling variation follows a normal distribution.  If we
calculate further

m_2 = (1 / n) SUM cos 2(t - vector mean of t),

then the circular standard error CSE is estimated by

sqrt[ n * (1 - m_2) / (2 * R * R)

and the confidence interval is estimated as

(vector mean - arcsin(z * CSE), vector mean + arcsin(z * CSE)),

where z is the appropriate percentage point, given level(), from the
Normal distribution with mean 0 and standard deviation 1.

level() specifies the confidence level in percent for the confidence
interval. See help for ci.

rayleigh produces the Rayleigh test, which tests a null hypothesis of
uniformity against an alternative hypothesis of unimodality.  The
resulting P-value is shown.

kuiper produces the Kuiper test, which tests a null hypothesis of
uniformity against any alternative. The resulting P-value is shown.
A numerical approximation due to Stephens (1970, p.118) is used that
gives accuracy to 3 decimal places for P < 0.447.

detail is a synonym for ci rayleigh kuiper.

Examples

. circsummarize axisasp wallasp

References

Fisher, N.I. 1993. Statistical analysis of circular data. Cambridge:
Cambridge University Press.

Stephens, M.A. 1970. Use of the Kolmogorov-Smirnov, Cramér-von Mises and
related statistics without extensive tables.  Journal, Royal
Statistical Society Series B 32: 115-22.

Author

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

Also see

On-line:  help for ci

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