Rayleigh Test of Uniformity

The Rayleigh test checks whether circular data are uniformly distributed around the circle against a unimodal alternative, using the length of the mean resultant vector.

레일리 검정은 원형 자료가 원 둘레에 고르게 퍼져 있는지를 평균 결과벡터의 길이로 검정하며, 대립가설은 한 방향으로 모인 분포입니다.

Do these events cluster at a particular time of day, or are they spread evenly around the clock?

사건이 특정 시간대에 몰려 있는지, 하루 내내 고르게 일어나는지 검정하고 싶어요.

The Rayleigh test computes Z = R²/n from the resultant length R of the unit vectors and compares it with the distribution expected under uniform directions. A small p value means the directions concentrate around one preferred direction, which is then reported as the mean direction.

When to use Rayleigh Test of Uniformity

• You want to know whether angles or times show a preferred direction
• The alternative of interest is a single cluster
• Data are angles, times of day or seasonal dates

When to use something else

• Directions may cluster in two opposite modes; use Rao's spacing test or double the angles first
• The data are linear rather than circular; use a one-sample test for linear data

Assumptions and requirements

• Observations are independent
• The alternative distribution is unimodal (for example von Mises)
• At least about 10 observations for the p-value approximation

What the analysis reports

• Rayleigh Z statistic
• Resultant length and mean resultant length
• p value (finite-sample approximation)
• Mean direction

How to report it

A Rayleigh test rejected circular uniformity, Z = [value], p = [value], R̄ = [value], n = [value]; mean direction = [value]°.

Related methods

• Circular Descriptive Statistics
• Watson-Williams Test

Run Rayleigh Test of Uniformity on Quorant