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Circular statistics (directions / angles)

Circular statistics summarise directional data with the mean direction, resultant length and circular SD, plus the Rayleigh test of uniformity and, across groups, the Watson-Williams comparison.

What is Circular statistics (directions / angles)?

Angles live on a circle, so ordinary means fail (the arithmetic mean of 1° and 359° is 180° — the circular mean is 0°). Each angle becomes a unit vector; the vector mean's direction is the mean direction and its length R ∈ [0,1] measures concentration (1 = all angles identical).

The Rayleigh test uses R to test uniformity against a unimodal alternative. Watson-Williams is the circular one-way ANOVA analogue (assumes concentrated von Mises samples with similar κ).

When should I use Circular statistics (directions / angles)?

  • Wind/current directions, animal orientation, gait phase.
  • Time-of-day or day-of-year data mapped to angles (24 h → 360°).

What data does it need?

One numeric angle column (degrees or radians) + optional group column.

What does it report?

Rose plot (angular histogram), mean direction, R, circular SD, Rayleigh z + p, optional Watson-Williams F + p.

What does it assume?

  • Rayleigh targets unimodal departures — it has little power against multimodal (e.g. bimodal opposite) patterns.
  • Watson-Williams: von Mises-ish samples, concentration reasonably high and similar across groups.

How do I interpret the result?

Small Rayleigh p = a preferred direction exists; R tells you how concentrated. R near 0 with large n usually means genuinely diffuse directions.

See also

References

  • Fisher (1993). Statistical Analysis of Circular Data.
  • Jammalamadaka & SenGupta (2001). Topics in Circular Statistics.