Percentage

Use a named numerator and denominator

The basic formula is:

Percentage = numerator divided by denominator times 100.

If a response occurs in 8 of 10 eligible opportunities, the percentage is 80%. Label it “8 of 10 eligible opportunities, 80%.” A standalone 80% leaves the reader to guess what was counted.

The numerator and denominator must use the same unit. Successful opportunities divided by all eligible opportunities is coherent. Responses divided by minutes produces a rate, rather than a percentage.

Define one eligible opportunity

An opportunity needs a start, available response, observation window, and outcome rule. For a help request, eligibility might require a problem that the person can detect, access to their communication system, an available partner, and enough time to respond.

If the device is unavailable or the partner never presents the opportunity, record a system or implementation issue. Do not score it automatically as a client error. Predefine whether a prompted response counts in the main numerator, a separate numerator, or another measure.

Equal percentages can carry unequal evidence

Eight of ten and 40 of 50 both equal 80%. The second result contains five times as many opportunities. One success in one opportunity equals 100%, yet it offers little information about consistency.

Always show raw counts. For graphs, consider placing the denominator beside each point or using a separate panel for opportunity count. Set a minimum exposure or maturity rule before interpreting trends.

A fictional example with two settings

Luca is a fictional 10-year-old who wants adults to honor an accessible request for help during model-building activities. The team defines a recognizable request through speech, gesture, sign, or AAC within 30 seconds of an eligible problem.

At a weekend club, Luca requests help in 8 of 10 opportunities, or 80%. At home, Luca does so in 40 of 50 opportunities, also 80%. The percentages match. The exposure, partners, materials, and observation periods differ.

Adults respond within 15 seconds to 7 of 8 club requests and 31 of 40 home requests. Partner-response percentages are 87.5% and 77.5%. These partner denominators contain Luca’s requests, rather than every task opportunity. Keeping the measures separate shows where support may need review.

Percent correct and percent occurrence need labels

“Percent correct” should identify the task, scoring criteria, response window, and treatment of prompts. “Percent occurrence” should identify whether the unit is opportunities, intervals, trials, days, or people.

An interval-recording percentage describes positive intervals under that procedure. It does not become exact percentage of time. A task-analysis percentage can describe correctly completed steps, while steps may differ in difficulty or importance.

Averages of percentages can mislead

Suppose one session is 1 of 2, or 50%, and another is 18 of 20, or 90%. The simple average of session percentages is 70%. Pooling the actual opportunities gives 19 of 22, or 86.4%.

Either summary may answer a planned question. State whether each session receives equal weight or each opportunity receives equal weight. Choose the rule before seeing results and retain session-level counts.

Missing data need their own state

Zero percent means the numerator was zero while at least one eligible denominator unit was observed. It should not represent a canceled session, missing form, inaccessible response, or absent opportunity.

Report completed observations divided by planned observations. Track exclusions by reason and age unresolved records. A high percentage from a small completed subset can hide substantial missingness.

Cap the denominator at genuine exposure

Avoid inventing opportunities from a schedule alone. If ten trials were planned and only six were validly presented, the client-outcome denominator is six. Report implementation as 6 of 10 planned trials, or 60%, then report the client result across the six actual opportunities. This two-measure approach assigns delivery gaps to the system and keeps the client calculation interpretable.

Interpret change with context

A percentage can change because of skill, communication access, opportunity difficulty, prompting, partner behavior, health, schedule, observer scoring, or denominator composition. A graph supports description and further assessment. It does not establish causation by itself.

Pair percentages with the person’s report, raw opportunity counts, relevant dimensions such as latency or duration, and measures of environment or partner performance. Preserve essential supports while collecting data.

Check calculation and data quality

Useful controls include records with named numerators and denominators, eligible opportunities with complete evidence, manual recalculation samples, invalid denominators, missing-observation rate, and observer agreement on eligibility and scoring.

The BACB BCBA Test Content Outline, Sixth Edition covers occurrence, temporal dimensions, reliability, validity, graphs, interpretation, and representative procedure selection. It describes examination content and provides no universal percentage denominator.