Percentage exceeding median calculation finds the baseline median, counts intervention observations beyond it in the predefined improvement direction, and divides by valid intervention observations. State whether improvement means higher or lower values and treat points equal to the median according to the declared rule. Report raw phases, ties, trend, missing states, and limits alongside the percentage.

What PEM measures

Percentage exceeding the median, commonly abbreviated PEM, asks how many intervention observations are better than the baseline median. It uses all valid intervention points in the denominator and a single baseline reference value. PEM is quick to audit and can help describe phase separation. It loses information about how far values moved and when the movement occurred.

A percentage exceeding median calculation is suitable only when baseline and intervention values share a stable measurement definition.

Define the desired direction before looking at the result. For a skill or independent response, higher values may represent improvement. For duration of distress or unsafe events, lower values may be preferred. The clinical meaning comes from the person's goal, measurement definition, context, and preferences rather than from the arithmetic sign alone.

Verify the phases and denominator

Keep baseline and intervention as separate, bounded cohorts. Confirm the target, unit, observation opportunity, phase-change date, and data-collection procedure. A baseline percentage and an intervention count cannot share one PEM calculation. A change in operational definition or denominator requires correction or a new phase.

Resolve missing, duplicate, late, and structurally unavailable observations before calculation. A blank is not zero. A session with no opportunity to observe the target may be missing by design. Retain excluded points with reasons and report coverage for each phase, such as five valid baseline points from six planned and six valid intervention points from eight planned.

The PEM denominator is the number of valid intervention observations. Missing intervention points stay outside the denominator, but their pattern belongs in the interpretation. Repeated cancellation during difficult sessions could make observed intervention values unrepresentative.

Calculate PEM step by step

  1. State whether improvement means higher or lower values.
  2. Sort valid baseline values and calculate the baseline median.
  3. Return to the intervention values in chronological order.
  4. Count intervention values strictly beyond the median in the improvement direction.
  5. Count ties with the median separately.
  6. Divide the improvement count by all valid intervention values and multiply by 100.

For higher-is-better data:

\[PEM=(B{above}/nB)\times100.\]

For lower-is-better data, use the number of intervention values below the baseline median. Under the strict rule used here, a value equal to the baseline median does not enter the numerator. It remains in the denominator. State the tie rule because alternate software or publications may handle boundary values differently.

Worked example: Kira's phase comparison

Kira's baseline values are 2, 4, 4, 6, and 8. The ordered values are unchanged, and the middle value is 4. The baseline median is therefore 4. Higher values were defined as improvement before intervention data were scored.

The intervention values are 3, 5, 4, 7, 9, and 6. Values 5, 7, 9, and 6 strictly exceed 4. The value 4 is a tie, and 3 is below the reference. There are four improvement values among six valid intervention observations:

\[PEM=(4/6)\times100=66.666\ldots\%=66.7\%.\]

The audit reconciles the phase as four improvements, one tie, and one nonimprovement, totaling six. No values were imputed. The denominator remains six even though the tied and lower observations do not contribute to the numerator.

Kira's baseline rises from 2 to 8. PEM does not remove that trend. Some intervention values would have been expected to exceed the median if the baseline trajectory continued unchanged, so the result needs careful visual interpretation.

Put the calculation on the graph and in the report

Display the raw chronological phases with the phase-change line. Add a horizontal reference at the baseline median and label it “baseline median = 4.” Marking intervention points above the line can support review, provided ties remain recognizable. Avoid graphing the percentage as though it were another session value.

A useful report sentence is:

Baseline contained five valid observations with a median of 4. Four of six valid intervention observations strictly exceeded that reference, one tied it, and one was below it, yielding PEM = 66.7%. Baseline showed an increasing trend, so PEM was interpreted with the raw graph and other visual-analysis features.

Include planned and valid observation counts, missing reasons, improvement direction, reference median, numerator, denominator, ties, rounding, and calculation version. This makes the result reproducible and prevents “66.7% improvement” from being mistaken for a 66.7% change in magnitude.

Test edge cases before using PEM

  • With a very short baseline, one added value may move the median enough to change several classifications.
  • Many baseline ties can place the reference at a floor or ceiling and compress the result.
  • Strong baseline trend can inflate or depress PEM because the metric uses level without projecting trend.
  • An intervention outlier counts the same as a value barely beyond the median; PEM ignores distance.
  • Small intervention phases produce coarse percentages. With three valid points, each observation changes PEM by 33.3 percentage points.
  • A reversal in the desired direction requires a new calculation rule. Never switch direction after seeing the values.
  • Nonrandom missingness can bias the denominator even when the arithmetic is correct.

Universal cutoffs such as “effective” or “ineffective” can mislead. Phase size, trend, variability, design quality, and the cost of a wrong decision affect interpretation. Research comparing decision accuracy across overlap measures supports retaining these method and design limits.

Integrate visual and causal reasoning

The WWC Version 5.0 handbook emphasizes level, trend, variability, immediacy, overlap, consistency, and repeated demonstrations in single-case visual analysis. PEM addresses a narrow part of overlap. It cannot show an immediate effect, stable implementation, maintenance, generalization, or a replicated functional relation.

A PEM of 66.7% may accompany a clinically important change, a trivial change, or no convincing intervention effect. Review magnitude, phase timing, fidelity, context, adverse effects, health factors within scope, and other services. Qualified clinicians interpret the whole record. The metric cannot decide diagnosis, medical necessity, intensity, authorization, discharge, or safety.

Review the result with Kira

Show Kira the raw graph, the median line, and the six intervention observations in an accessible format. Explain that four points were above the middle baseline value. Ask whether the outcome and observed differences matter to her and whether intervention conditions were acceptable. Record her communication, assent or dissent when applicable, and requested changes.

Keep AAC, interpreters, mobility supports, breaks, food, water, bathroom access, prescribed care, and emergency help available. The ASHA AAC guidance supports continued access to communication systems during assessment and review.

Clinician release checklist

Verify the target and desired direction, baseline and intervention boundaries, valid-point coverage, missingness, baseline ordered list and median, strict tie rule, intervention numerator and denominator, arithmetic, graph annotation, trend and variability, client review, qualified interpretation, limitations, owner, and next review date. The BACB ethics resources support qualified, documented clinical review. Preserve the raw phases and calculation sheet with the clinical record under applicable privacy and access controls.

Related resources

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