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help for ^circsumm^
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Summary statistics for circular data
------------------------------------

    ^circsumm^ varname [weight] [^if^ exp] [^in^ range]
    [ ^, ci lev^el^(^#^) r^ayleigh ^k^uiper ^d^etail ]

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

^circsumm^ produces summary statistics for circular variables with
scales between 0 and 360 degrees. These are
    the mean direction (in degrees)
    the length of the resultant vector
    the vector strength or mean resultant length
        (length of resultant / number of observations)
    the circular range (length of smallest arc including all
        observations) (in degrees)


Remarks
-------

Suppose we have n measurements of a circular variable y. Then calculate

    C = SUM cos y

    S = SUM sin y

    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
-------

^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(y - vector mean of y)

    then the circular standard error CSE is estimated by

    SQRT[ n * (1 - m_2) / (2 * R * R) ]

    and the confidence interval is estimated as from

    vector mean - arcsin(z * CSE)

    to

    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. The default is 95, meaning that z is
    approximately 1.96.  See help for ^ci^.

^rayleigh^ produces the Rayleigh test, which tests a null hypothesis of
    uniformity against an alternative hypothesis of unimodality (P-value
    shown)

^kuiper^ produces the Kuiper test, which tests a null hypothesis of
    uniformity against any alternative (V statistic and P-value shown).
    A numerical approximation due to Stephens (1970, p.118) is used that
    gives accuracy to 3 decimal places for P < 0.447 (V > 1.26).

^detail^ is a synonym for ^ci rayleigh kuiper^.


Note on weights
---------------

aweights, fweights and iweights are allowed.

If weights are specified, they are used in calculating the mean
direction, the resultant vector length and the vector strength, and
hence the Rayleigh test if produced. They are not used in calculating
the circular range or the Kuiper test.

The resultant vector length, expressed in the units of the weights, is
the vector strength * the mean weight.


Examples
--------

        . ^circsumm wallasp^
        . ^circsumm wallasp, r k^
        . ^circsumm wallasp if grading==2^


Saved values
------------

     S_1   number of observations
     S_2   mean direction
     S_3   vector length
     S_4   vector strength
     S_5   circular range
     S_6   P-value for Rayleigh test  (if ^rayleigh^ option)
     S_7   Vn statistic for Kuiper test (if ^kuiper^ option)
     S_8   V statistic for Kuiper test  (if ^kuiper^ option)
     S_9   P-value for Kuiper test (if ^kuiper^ option)
     S_10  vector length in weight units (if weights used)
     S_11  lower confidence limit (if ^ci^ option) (degrees)
     S_12  upper confidence limit (if ^ci^ option) (degrees)
     S_13  circular standard error (if ^ci^ option) (radians)


References
----------

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

Stephens, M.A. 1970. Use of the Kolmogorov-Smirnov, Cramer-von Mises and
related statistics without extensive tables. Journal of the 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 @circcent@, @circcomp@, @ci@