Medelvinkel

Medelvinkel is a term used in Swedish to describe the average orientation or mean direction of a set of angles measured on a circle. In circular statistics, the medelvinkel is preferred over the ordinary arithmetic mean because angles wrap around at 360 degrees (or 2π radians), which can make a naive average misleading.

To compute the medelvinkel, convert each angle θi to a point on the unit circle by taking its cosine and sine. Let C be the average of the cosines and S be the average of the sines: C = (1/n) Σ cos(θi), S = (1/n) Σ sin(θi). The medelvinkel μ is then given by μ = atan2(S, C), which returns the angle whose cosine and sine are C and S. The length of the mean resultant vector, R = sqrt(C^2 + S^2), indicates how concentrated the angles are around μ; a small R suggests widely dispersed data and a less reliable mean direction. Circular variance can be defined as 1 − R.

Edge cases include when the data are uniformly distributed around the circle, yielding a small R and an unstable μ, or when there are conflicting clusters. In such cases other measures, such as the circular median or clustering-based summaries, may be more informative.

Applications of the medelvinkel appear across disciplines, including meteorology, biology (animal orientation), navigation, robotics, and quality control, wherever directional data are analyzed. See also circular statistics, mean resultant length, and circular variance.