Is the plot itself a Rayleigh test?

No. A plot can include results from that test, but displaying observations and a resultant is descriptive.

Does r near zero prove uniformity?

No. Opposing groups can cancel.

Can daily times be plotted?

Yes, when the question genuinely concerns phase within a repeating daily cycle rather than elapsed duration.

AI Glossary / Statistics and probability / Intermediate

Raleigh plot

Definition

A Raleigh plot, in the circular-data sense intended here, displays observations as positions around a circle and often adds their mean direction and mean resultant length. The cited scientific publication spells this display “Rayleigh plot.” The supplied term is retained for traceability; both labels here refer to the same circular-data display. Typical observations include compass headings or phases within a repeating daily cycle. The plot helps reveal directional concentration, but its mean arrow does not fully describe the distribution.

What it means

Circular measurements wrap around: 359 degrees is close to 1 degree. Ordinary arithmetic averages can therefore produce misleading summaries. A circular display preserves this geometry and makes patterns near the chosen boundary easier to see.

The mean arrow combines two pieces of information. Its direction summarizes the average orientation, while its normalized length describes how strongly the unit vectors reinforce one another. A short arrow can arise from broad dispersion or from opposing, tightly concentrated groups. It does not by itself show that observations are uniformly distributed around the circle.

How it works

Choose a consistent reference direction and rotation convention. For times within a 24-hour cycle, convert time to angle using 360 degrees per 24 hours, or 15 degrees per hour. Plot each observation on the circumference; distinguish groups clearly if necessary.

Convert every angle to a unit vector. Average the cosine components to obtain C and the sine components to obtain S. The mean direction is atan2(S, C), expressed in the chosen angular range. The mean resultant length is r = sqrt(C squared + S squared), between zero and one. If the resultant is zero, its direction is undefined.

Draw the mean arrow from the center with length r relative to a unit-radius circle. State whether a figure instead uses an unnormalized vector sum or another scale. A Rayleigh uniformity test is a separate inferential procedure; its assumptions, sample size, and alternative matter. The visual summary alone supplies neither a p-value nor a confidence interval.

Three practical examples

1. A wildlife researcher plots the departure bearings of independently tracked animals and compares seasonal directional patterns.

2. A laboratory displays daily peak phases of biological rhythms, preserving the fact that late-night and early-morning peaks can be close together.

3. An operations analyst explores the time of day when independent equipment alerts occur, checking whether a single daily cluster or several clusters are present.

Think of it this way

Think of people pulling equally hard on ropes pointing in different directions. If everyone pulls nearly the same way, their combined pull is strong. If two equal groups pull in opposite directions, the combined pull disappears even though each group is highly organized. The mean arrow behaves similarly.

Worked example

Consider three original teaching observations with directions 350, 0, and 10 degrees. Their ordinary arithmetic mean is 120 degrees, which points far away from the cluster.

The sine components cancel. The average cosine is C = (cos 350 + cos 0 + cos 10)/3 = (1 + 2 cos 10)/3, approximately 0.989872. Thus S = 0, r is approximately 0.989872, and the circular mean direction is 0 degrees. The plot shows a long arrow toward the cluster crossing the angular boundary.

Now consider only two observations at 0 and 180 degrees. Their vectors cancel, so r is zero and the mean direction is undefined in exact arithmetic. Yet the two observations are plainly not a uniform spread around the circumference. This second construction demonstrates why the raw points remain essential. Neither tiny example is presented as a meaningful significance test or a population estimate with established precision.

Illustrative case study

In a fictional logistics company, an analyst studies alert times from separate sites. A circular plot shows one group near midnight and another near noon. The overall mean arrow is short, which initially prompts a manager to describe the timing as random.

The analyst retains the individual points and separates the sites by operating schedule. The two visible groups correspond to different maintenance cycles. The team investigates each schedule rather than treating a small resultant as evidence that timing has no structure. Repeated alerts from the same site are also recognized as dependent observations before any formal testing.

Common misconceptions and mistakes

Do not average clock times as ordinary numbers without considering the boundary. Do not interpret r as a percentage of correctly predicted observations, the probability of a hypothesis, or the strength of a causal relationship.

A large r says little about precision without sample size and sampling design. A small r does not exclude multiple directional clusters. Axial data, where opposite directions represent the same axis, need appropriate treatment rather than an unmodified directional average. Unequal weights, repeated measurements, missing phases, and selection of only strong rhythms can all change the interpretation.

Business use

Businesses can use circular plots to explore recurring timing patterns or physical directions before choosing models. The practical benefit is recognizing a cycle's boundary and identifying clusters hidden by linear summaries. Document time zones, daylight-saving adjustments, observation units, and the process that generated the measurements. Pair visual findings with operational context and independent validation before changing schedules.

Check your understanding

Two equally weighted observations occur at 90 and 270 degrees. What does their mean resultant show?

Answer: The vectors cancel, giving r = 0 and an undefined mean direction. It does not establish that the underlying observations are uniformly distributed.

Related terms

Phase (Fourier)
Circular data
circular mean
mean resultant length
directional statistics
Rayleigh test
phase
polar plot
multimodality

Learn next

Learn how circular means differ from linear means, then examine circular dispersion and methods for detecting multiple modes. Study the assumptions of the Rayleigh test separately from the construction of the display.

Sources

SciPy, circmean documentation: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.circmean.html

Herzog, Kiss and Mazuski, Measuring synchrony in the mammalian central circadian circuit: https://pmc.ncbi.nlm.nih.gov/articles/PMC5110928/

The scientific article's indexed methods passage supports the Rayleigh-plot terminology and phase-synchrony application; direct full-page retrieval encountered browser verification. The supplied source uses Raleigh for this circular-data display; the cited research uses Rayleigh. Both names are retained here to clarify the terminology. Calculations and fictional business examples are original.

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