Average directed angles without crossing the wrong boundary

Identify the arrows behind the labels

A fictional studio collects orientation labels from a set of rotating product illustrations. The report is meant to describe the supplied frames, not to prescribe where a camera should move. Before calculating a summary, put a directed arrow on each reference frame and record which visible feature defines its direction. A symmetrical silhouette alone may not establish which end is the front.

Use equally weighted directed angles 330°, 30°, 90°: zero points right and angles increase counterclockwise. This is a 360° direction, not an undirected axis or an animation path.

Keep the input labels attached to their frames. If a value was manually copied from a different application, ask how that application described orientation. Do not repair an unfamiliar convention by changing the number until the picture looks right. The receiving analyst needs a consistent interpretation before a summary can answer the intended question.

Ask the report owner why a summary is needed. A selected hero frame, an inventory of supplied views and a descriptive statistic are different deliverables. The first is an editorial choice, while the second may require retaining every frame. Do not replace those tasks with a single angle merely because a spreadsheet has an empty summary cell.

Calculate the supplied three-observation comparison

Average the unit vectors (cos θ, sin θ). Their sum is (√3,1), so the mean vector is (√3/3,1/3), with length 2/3 and direction atan2(1,√3)=30°. Ordinary numeric averaging gives 150°.

SciPy 1.17.0 circmean with low=0, high=360 returns approximately 30°, matching independent trigonometry. Reference documentation is 1.18.0. No R execution is claimed.

Draw the reference arrows around one clearly marked circle, then put the two proposed summary arrows in a separate layer. Keep the underlying observations visible. The reviewer should be able to see what each candidate purports to summarize rather than receiving a polished icon detached from the evidence. This drawing is a diagnostic teaching aid, not a new production asset.

Record the executed calculation with the frame identifiers and input convention. If the original observations are later corrected, mark the report as a new version instead of quietly editing the illustration alone. Otherwise the visual and the recorded values may disagree while both appear current. A useful report lets another person reconstruct why its summary changed.

Retain a result that cannot be summarized

For 0° and 180°, exact vectors (1,0) and (−1,0) cancel: no resultant direction exists. Floating SciPy execution returns 90° here; its documentation warns that a zero-resultant return is implementation-defined. A numeric return does not establish a meaningful direction.

Design an honest unavailable-result state for the report. It should preserve the individual observations and explain that the selected summary does not supply a direction for this case. Do not silently substitute the previous batch’s result or choose a convenient default icon. Either would make missing interpretive information look like a new observation.

The analyst should define appropriate numerical handling for the actual workload, including its uncertainty and precision. This tiny exercise does not prescribe a universal cutoff or certify every implementation. If a report requires a decision despite an unresolved summary, return to the original task: the owner may need to choose a particular frame or show the observations separately rather than demand that the statistic invent an answer.

Make the visual report say what was actually done

Label the final illustration as a description of the supplied set. Do not call it the most popular view, a recommended camera pose or an optimum composition unless separate evidence supports that statement. A descriptive calculation has no knowledge of the product feature the campaign needs to communicate or the layout in which an image will appear.

For the fictional handoff, provide the reference frames, the comparison drawing and the unavailable-result example together. Ask a fresh reviewer to explain the distinction without being coached. If the reviewer thinks the fallback icon is an actual measured direction, revise the reporting state. If they think the summary selects an approved hero image, revise the caption.

The finished worksheet should help the team keep a convenient number in its proper role. No real orientation record, equipment setting or campaign artwork changes in this exercise. The practical outcome is a report that preserves its inputs, uses a named interpretation and can acknowledge when the requested summary is not meaningful.

Make the reasoning concrete.

A studio dashboard produces an attractive summary arrow for every batch, including one for which the directional summary is undefined. The designer assumes that every returned number represents a supported observation.

The fictional dashboard specification is revised to preserve the unresolved case instead of manufacturing a direction.

Take the questions into your next brief.

Use these prompts in your own notes. Nothing is submitted from this page.

1. Reference

   Which visible feature establishes each directed observation?

2. Convention

   What does the input label mean in its source application?

3. Purpose

   Does the owner need a statistic, an inventory or a chosen frame?

4. Unavailable

   How will the report preserve a case the selected summary cannot describe?

5. Caption

   Could the reader mistake the description for a production recommendation?

Watch for the wrong conclusion.

• Using a default icon as if it were a newly observed direction.
• Losing the reference frames after producing a summary.
• Calling a descriptive angle the best creative viewpoint.

Sources & scope

All business examples and numerical fixtures are original and fictional. Sources support the identified mathematical or implementation distinctions. Qualified analysts own model specification and inference. No causal result, model fit, actual customer data or operational change is claimed.

• SciPy: circmean →Primary documentation · Supports: Supports the circular mean and return-value boundary.
• circular: Mean Direction →Primary documentation · Supports: Supports the unit-vector definition.