The circle
Compass bearings, time of day, and phase repeat after one full turn. 0° and 360° describe the same point.
INTRODUCTION / THE GEOMETRY OF DATA
What is
directional statistics?
The statistics of angles, directions, and orientations. The sample space might be a circle, a sphere, or a related geometric space.
01 / SAMPLE SPACES
Start with where the data live.
Spheres & hyperspheres
Unit vectors describe directions in three or more dimensions. Their values lie on a sphere or hypersphere.
Unoriented axes
A line without an arrow is unchanged when reversed. For axial data, opposite directions represent the same observation.
Other manifolds
Rotations and shapes lead to other geometric spaces. This connects directional statistics to the wider study of data on manifolds.
02 / CIRCULAR SUMMARIES
Average the vectors,
not the angle labels.
359° and 1° have a circular mean of 0°. Working with unit vectors keeps nearby directions together, even when their labels lie on opposite sides of the scale.
Mean cosine & sine
C̄ = n⁻¹ Σ cos θᵢ
S̄ = n⁻¹ Σ sin θᵢ
Mean direction
μ̂ = atan2(S̄, C̄)
Mean resultant length
R̄ = √(C̄² + S̄²)
R̄ lies between 0 and 1. A value near 1 indicates strong alignment. At R̄ = 0, the mean direction is undefined; opposing clusters can cancel, so a small R̄ does not by itself imply uniformity. Express μ̂ in the chosen angular range.
ON A STRAIGHT SCALE
The numerical average is 180°.
(359° + 1°) / 2 = 180°
The calculation is correct on a line, but its result points away from both observations.
ON THE CIRCLE
The mean direction is 0°.
03 / SEE THE DISTRIBUTION
Meet the
rose diagram.
A histogram, arranged around a circle.
Divide the circle into angular bins, then count the observations in each bin. Each sector shows the frequency of directions in that range.
- Read the bearing. Here, north is 0° and angles increase clockwise. Mathematical plots may use a different convention.
- Compare the sectors. These bins are equally wide. Sector area represents count, so radius grows with √count. Some plots use radius directly; always check the scale.
- Look beyond the mean. Several clusters can be hidden by a single average. Try the two opposing directions to see why.
For the single-direction sample, use the concentration slider to set the mean resultant length R̄ from 0 to 1. Rotation changes its direction without changing R̄. The opposing and evenly spaced examples keep R̄ = 0 to illustrate cancellation.
The samples are illustrative, not research observations. Changing the bin width or origin can change a rose diagram’s appearance.
ANGULAR OBSERVATIONSS¹ / 001
Frequency Mean direction Observations
Observations120
Mean direction · μ67.5°
Resultant length · R̄0.65
Set the sample’s mean resultant length; the number of observations stays fixed.
0° at north; angles increase clockwise.
View the counts and scale
Radius ∝ √count. Outer grid ring: 29 observations. Inner rings: 25%, 50%, and 75% of that count. The count scale adapts to the largest bin. At R̄ = 1 the observation marks overlap.
| Bearing | Count |
|---|---|
| 0.0–22.5° | 6 |
| 22.5–45.0° | 13 |
| 45.0–67.5° | 29 |
| 67.5–90.0° | 29 |
| 90.0–112.5° | 13 |
| 112.5–135.0° | 6 |
| 135.0–157.5° | 4 |
| 157.5–180.0° | 3 |
| 180.0–202.5° | 2 |
| 202.5–225.0° | 1 |
| 225.0–247.5° | 2 |
| 247.5–270.0° | 2 |
| 270.0–292.5° | 1 |
| 292.5–315.0° | 2 |
| 315.0–337.5° | 3 |
| 337.5–360.0° | 4 |
READING & DEFINITIONS
Rose-diagram construction and scaling: R circular: rose.diag ↗. Compass and mathematical conventions: R spatstat: rose ↗. Circular-mean definition: SciPy: circmean ↗.