Data consisting of directions or times in which the measurement scale is cyclic (after 23.59 comes 00.00, after 359° comes 0°, after 31 December comes 1 January). Special techniques are required for summarizing and modelling all types of cyclic data. For example, the histogram is replaced by the circular histogram or the rose diagram, and the principal distribution used to model the data is not the normal distribution but the von Mises distribution.
For observations θ1, θ2,…, θn, the variability of the data is measured by the concentration, R̄, defined by . Thus nR̄ =R, where R is the length of the resultant vector. The circular mean, θ̂, is defined only when R̄ ≠0 and is then the angle (0°≤θ̄ <360°) such that .
One way of representing cyclic data is to regard each observation as a move of length 1 unit in the stated direction. The complete sequence of such moves, taken in any order, will end at a finishing point that is a distance nR̄ from the start. The direction of this finishing point from the start will be the angle θ̄.
Cyclic data. Each data item is represented by a unit vector. The diagram shows the first two such vectors, and also the last two. The resultant vector, with length nR̄, connects the start of the first unit vector to the end of the last unit vector. The direction of the resultant vector is θ.
Subjects: Probability and Statistics.