Within-phase median level calculation sorts valid observations by value and selects the middle value. With an even number, average the two middle values. Keep chronological order in the graph and use sorting only for the calculation. Report the phase, valid observations, missing states, median, range, trend, and variability because one level summary cannot describe the full data path.

What the median answers

The within-phase median is a compact description of level. It answers: what value sits in the middle after the valid observations in this phase are ranked? It is often useful when one unusually high or low point would pull the mean away from most observations. It does not show when values occurred, whether the phase was rising or falling, or how widely values varied. Those features stay visible on the raw graph.

A within-phase median level calculation is most useful when the phase boundary and observation denominator are already trustworthy.

Calculate the median for one defined phase and one measurement system at a time. Mixing baseline and intervention, combining different targets, or pooling observations collected with different definitions creates a statistic for a cohort that never actually existed.

Lock the phase before sorting

Start with the chronological record. Write down the phase label, start and end dates, target definition, unit, observation opportunity, setting, and the rule used to decide whether a point is valid. Confirm that a phase change was entered where the intervention or measurement procedure actually changed.

Resolve data states before calculating. A blank should remain missing rather than becoming zero. A session with no opportunity to observe the target may be structurally unavailable rather than a zero occurrence. A duplicate timestamp needs source review. A late entry can remain valid when its provenance is clear, but its entry date should not replace its service date. Keep excluded values and reasons in the audit trail.

The denominator is the number of valid phase observations, written as \(n\). Report coverage when missingness could affect interpretation, such as “9 valid observations from 12 planned sessions.” The median calculation uses the nine valid values; the graph and note should still show the three missing sessions.

Calculate it step by step

  1. Copy the valid phase values without changing the original record.
  2. Sort the copy from lowest to highest.
  3. Count the values.
  4. If \(n\) is odd, select the value in position \((n+1)/2\).
  5. If \(n\) is even, average the values in positions \(n/2\) and \(n/2+1\).
  6. Link every sorted value back to its date or session identifier.

For an even phase, show the two middle values. If the ordered values are 2, 4, 7, and 11, the median is \((4+7)/2=5.5\). A value of 5.5 need not have occurred. It is the midpoint of the two central ranks.

Ties require no special weighting. They remain separate observations if each represents a valid, distinct measurement opportunity. Repeated values can make the median stable, but they do not establish stable responding across time.

Worked example: Inez's phase

Inez has five valid observations in one phase: 2, 5, 6, 8, and 20. There are no imputed values. The ordered list is also 2, 5, 6, 8, 20. Because \(n=5\), the middle position is \((5+1)/2=3\). The third value is 6, so the within-phase median is 6.

The arithmetic mean is \((2+5+6+8+20)/5=41/5=8.2\). The value 20 raises the mean, while the median remains anchored at the third rank. That difference is useful context, not evidence that one summary is automatically correct. The range is 2 to 20, and the chronological graph is still needed to determine whether 20 was an isolated point, the start of a trend, or part of a change in conditions.

Inez's worksheet records five valid observations out of five scheduled observations, median 6, mean 8.2, range 2 to 20, no missing values, and the original session order. Another reviewer can reproduce the calculation without reconstructing the phase from a screenshot.

Read the median beside the graph

On the graph, retain each observation, phase-change line, date or session scale, and any condition annotations. A horizontal median marker may help a reader see level, provided it is labeled as a summary rather than a fitted trend. Do not connect the sorted values as though they occurred in that order.

A concise report could say:

Across five valid observations in phase A, the median was 6 (range 2 to 20; no missing observations). The last observation was substantially higher than the earlier values, so level was interpreted with the chronological trend and variability rather than from the median alone.

Compare phases only after confirming the same outcome definition, unit, observation opportunity, and phase membership. A change from a median of 6 to 9 describes a level difference. It does not by itself show immediacy, replication, maintenance, social importance, or experimental control.

Watch for edge cases

  • A short phase can produce a median that shifts sharply when one point is added.
  • Strong monotonic trend can make a middle rank a poor description of the value expected at the phase boundary.
  • A bimodal phase can have a median in a sparsely observed middle region.
  • Floor or ceiling effects can create many ties and hide meaningful changes in independence, latency, prompting, or context.
  • Irregular observation opportunities can make identical values represent different exposure. Recalculate a rate or percentage first when the measurement definition requires a denominator.
  • A measurement-system change within the phase calls for a new phase or an explicit comparability analysis, not a blended median.

Missing values should rarely be imputed for routine visual analysis. If a research protocol uses imputation, name the method, run a sensitivity check, and retain the observed-only result. Never substitute the phase median for missing observations and then calculate the median again.

Interpret within single-case limits

The WWC Version 5.0 handbook treats level as one element of visual analysis alongside trend, variability, immediacy, overlap, consistency, and demonstrations of effect. Its standards concern education research evidence. They do not authorize a treatment change for an individual client.

The responsible clinician should integrate the median with the raw data, implementation fidelity, contextual events, health information within scope, client experience, and repeated phase comparisons. A median has no causal interpretation on its own. Even a large level change can coincide with maturation, schedule changes, new medication, altered observation opportunities, or another uncontrolled event.

Review the result accessibly

Show Inez the graph and summary in a communication format she can use. Explain “median” as the middle ranked observation and point to the actual five values. Ask whether the outcome is important, whether the phase conditions were acceptable, and whether the apparent change matches her experience. Keep AAC, interpreters, mobility supports, breaks, food, water, bathroom access, prescribed care, and emergency help available throughout review.

Record Inez's words, selections, gestures, device output, assent, dissent, and requested changes according to the applicable consent and documentation rules. A favorable median cannot make an unwanted goal appropriate or justify withholding access and health supports.

Clinician release checklist

Before using the result, verify:

  • one clearly bounded phase and one stable measurement definition;
  • valid observations, exclusions, duplicates, and missing-state reasons;
  • the correct odd or even formula and visible intermediate values;
  • coverage, range, ties, trend, variability, and contextual annotations;
  • the raw chronological graph and an optional labeled median marker;
  • client-accessible review and documented feedback;
  • a qualified interpretation that states uncertainty and avoids causal overreach; and
  • the calculation owner, version, decision, and next review date.

The BACB ethics resources, CASP practice-guideline summary, and ASHA AAC guidance provide professional and access context. The median remains a descriptive aid. It cannot determine medical necessity, authorization, diagnosis, treatment intensity, safety, or payer coverage.

Related resources

Sources