Directional Statistics - Measures of Location and Spread

Measures of Location and Spread

Various measures of location and spread may be defined for both the population and a sample drawn from that population. The most common measure of location is the circular mean. The population circular mean is simply the first moment of the distribution while the sample mean is the first moment of the sample. The sample mean will serve as an unbiased estimator of the population mean.

When data is concentrated, the median and mode may be defined by analogy to the linear case, but for more dispersed or multi-modal data, these concepts are not useful.

The most common measures of circular spread are:

  • The circular variance. For the sample the circular variance is defined as:

\overline{\mathrm{Var}(z)}=1-\overline{R}\,
and for the population

\mathrm{Var}(z)=1-R\,
Both will have values between 0 and 1.
  • The circular standard deviation

S(z)=\sqrt{\ln(1/R^2)}=\sqrt{-2\ln(R)}\,

\overline{S}(z)=\sqrt{\ln(1/\overline{R}^2)}=\sqrt{-2\ln(\overline{R})}\,
with values between 0 and infinity. This definition of the standard deviation (rather than the square root of the variance) is useful because for a wrapped normal distribution, it is an estimator of the standard deviation of the underlying normal distribution. It will therefore allow the circular distribution to be standardized as in the linear case, for small values of the standard deviation. This also applies to the von Mises distribution which closely approximates the wrapped normal distribution.
  • The circular dispersion

\delta=\frac{1-R_2}{2R^2}

\overline{\delta}=\frac{1-\overline{R}_2}{2\overline{R}^2}
with values between 0 and infinity. This measure of spread is found useful in the statistical analysis of variance.

Read more about this topic:  Directional Statistics

Famous quotes containing the words measures of, measures and/or spread:

    To have the fear of God before our eyes, and, in our mutual dealings with each other, to govern our actions by the eternal measures of right and wrong:MThe first of these will comprehend the duties of religion;Mthe second, those of morality, which are so inseparably connected together, that you cannot divide these two tables ... without breaking and mutually destroying them both.
    Laurence Sterne (1713–1768)

    A lake is the landscape’s most beautiful and expressive feature. It is earth’s eye; looking into which the beholder measures the depth of his own nature. The fluviatile trees next the shore are the slender eyelashes which fringe it, and the wooded hills and cliffs around are its overhanging brows.
    Henry David Thoreau (1817–1862)

    There’s Margaret and Marjorie and Dorothy and Nan,
    A Daphne and a Mary who live in privacy;
    One’s had her fill of lovers, another’s had but one,
    Another boasts, “I pick and choose and have but two or three.”
    If head and limb have beauty and the instep’s high and light
    They can spread out what sail they please for all I have to say....
    William Butler Yeats (1865–1939)