Circular Descriptive Statistics

Circular descriptive statistics summarize angles, clock times or compass directions by the mean direction, mean resultant length, circular variance and circular standard deviation instead of arithmetic means.

원형 기술통계는 각도·시각·방위처럼 끝과 처음이 이어진 자료를 산술평균 대신 평균방향, 평균 결과벡터 길이, 원형분산, 원형표준편차로 요약하는 방법입니다.

What is the average time of day of these events when some happen just before and some just after midnight?

자정 전후로 일어난 사건들의 평균 발생 시각을 제대로 구하고 싶어요.

Circular statistics convert each value to a unit vector and average the vectors, so 23:00 and 01:00 average to midnight rather than noon. The length of the mean vector, R-bar, measures how concentrated the directions are, from 0 for evenly spread to 1 for identical directions.

When to use Circular Descriptive Statistics

• The variable is an angle, compass direction, time of day or day of year
• Values wrap around so that the largest and smallest are neighbours
• You need a mean direction and its confidence interval

When to use something else

• The variable is linear without wrap-around; use ordinary descriptive statistics
• You need to test whether directions are uniform; use the Rayleigh test

Assumptions and requirements

• Observations are independent
• The distribution is unimodal for the mean direction to be meaningful
• The period of the scale is known (360 degrees, 2π radians or 24 hours)

What the analysis reports

• Mean and median direction
• Mean resultant length and circular variance
• Circular standard deviation and angular deviation
• Confidence interval for the mean direction
• Concentration parameter kappa

How to report it

The mean direction was [value]°, 95% CI [[value]°, [value]°], with a mean resultant length of R̄ = [value] and a circular standard deviation of [value]° (n = [value]).

Related methods

• Rayleigh Test of Uniformity
• Watson-Williams Test

Run Circular Descriptive Statistics on Quorant