An ABA standardized mean difference calculator can put a phase-mean change on a within-case variability scale. The number is compact; the judgment behind it is not. This worksheet therefore carries two deliberately uncorrected views of the same adjacent phase pair: one divided by the baseline sample standard deviation and one by the pooled within-phase sample standard deviation, alongside the raw observations and a documented sensitivity check. The aim is to make the denominator choice inspectable, not to produce a single preferred score.

Clinicians & ABA Professionals / Data, Outcomes and Clinical Decision-Making.

What this calculator returns

For one ordered Phase A and Phase B contrast, the worksheet reports:

  1. the valid count, mean, sample variance, and sample standard deviation for each phase;
  2. the raw mean difference, mean B - mean A;
  3. an uncorrected within-case SMD using the baseline sample SD;
  4. an uncorrected within-case SMD using the pooled within-phase sample SD;
  5. a therapeutic-direction label kept separate from the raw mathematical sign;
  6. both estimates after one documented observation-state sensitivity, when such a review is warranted.

The baseline and pooled estimates are not competing scores. They answer the same mean-difference question under different variance assumptions. A reviewer should not choose the larger or more convenient value after seeing the output.

A useful ABA standardized mean difference calculator should therefore expose the denominator and raw phase summaries beside every reported coefficient.

This calculator does not apply the small-sample correction used by default in current statistical software. It also produces no standard error, p-value, confidence interval, trend adjustment, autocorrelation adjustment, or universal magnitude label. The output is not comparable to a between-group standardized mean difference because its denominator represents variation within one case.

First, decide whether the comparison makes sense

The worksheet needs two numeric phases measured under a common rule. Record the operational definition, unit, opportunity or exposure base, observation duration, and collection procedure. Confirm that a one-unit difference has the same meaning in both phases. Preserve phase boundaries, dates or sequence positions, and every missing, invalid, corrected, late, or disputed row.

At least two valid observations are needed in each phase to calculate both sample SDs. A zero baseline SD leaves the baseline-SD view undefined. The pooled view is undefined when the pooled SD is zero. A very small nonzero denominator can produce a very large standardized result, so report the denominator itself and compare the number with the raw graph.

Before calculating, decide what supplemental question the phase-level comparison is meant to inform. Trend, serial dependence, changing variability, floor or ceiling effects, altered exposure, and measurement changes do not disappear when a formula runs. A complete numeric entry is not evidence of independence, homogeneous variance, stable baseline, experimental control, or clinical relevance.

Declare whether higher or lower values represent the person's intended direction before inspecting the result. The raw mean difference always remains mean B - mean A. When lower is preferred, a separate therapeutic-direction view may multiply the raw standardized values by -1. Never rewrite the raw sign.

Copyable input and assumption record

FieldLocked entryResolved?Note or sourceOutcome and operational definitionyes / noUnit and opportunity or exposure baseObservation duration and procedurePhase A and Phase B boundaryHigher or lower values preferredSame measurement meaning across phasesBaseline trend and variability reviewedEqual-variance assumption supportMissing, invalid, corrected, or disputed rowsCalculation version, analyst, and reviewer

PhaseOrderDate or sequenceRaw valueStatusContext or correction noteA1valid / missing / invalid / disputedA2A3B1B2B3

Missing stays missing. Do not enter a zero unless zero was the observed value under the declared measurement rule. If a source value or phase boundary changes, append a new calculation version rather than overwriting the earlier record.

Calculate each phase summary

For a phase with q valid observations x1 through xq, calculate the mean:

mean = sum(x_i) / q

Then calculate the sample variance and sample SD:

sample variance = sum[(x_i - mean)^2] / (q - 1)

sample SD = sqrt(sample variance)

Use full precision internally. Display enough intermediate values that another reviewer can reproduce the answer. The phase table should show each deviation or, at minimum, the sum of squared deviations, divisor, variance, and SD.

The raw phase-mean difference is:

D = mean B - mean A

If higher is preferred, the therapeutic-direction difference equals D. If lower is preferred, it equals -D. That orientation aids review but does not change the underlying observations.

Two denominators, kept side by side

Let m and n be the valid Phase A and Phase B counts, and let sA and sB be their sample SDs.

Using baseline variability only:

uncorrected SMDA = D / sA

This denominator uses baseline variability only. It is unavailable if sA = 0 and can become unstable when sA is very small.

The pooled within-phase variance is:

sp^2 = [(m - 1)sA^2 + (n - 1)s_B^2] / (m + n - 2)

The pooled sample SD and uncorrected pooled view are:

sp = sqrt(sp^2)

uncorrected SMDp = D / sp

Pooling corresponds to an assumption that a common within-phase variance is meaningful. A large difference between sA and sB, a mean-variance relationship, a floor or ceiling, or a changed procedure may make that assumption difficult to defend. Showing a pooled result does not validate it.

The current SingleCaseES calculation guidance and SMD reference document both denominator options. SingleCaseES normally applies a small-sample correction and can report approximate uncertainty; neither feature is reproduced here. The mathematical definitions support the formulas, but they do not endorse this simplified worksheet or its use in a particular case.

Blank result ledger

OutputPrimary valueSensitivity valueReconciliation noteValid Phase A count, mMatches active rowsValid Phase B count, nMatches active rowsMean AMean BRaw difference, Dmean B - mean ASample variance ADivisor m - 1Sample SD A, s_ASample variance BDivisor n - 1Sample SD B, s_BPooled variance, s_p^2Weighted within-phase variancesPooled SD, s_pUncorrected SMD_AD / s_AUncorrected SMD_pD / s_pRaw directionincrease / decrease / no changePreferred-direction labelpreferred / nonpreferred / unresolved

Do not round the inputs or intermediate denominators before the final display. If a spreadsheet or program is used, record its version and confirm that it uses sample rather than population SD.

Priya's fictional worked example

Priya is a fictional BCBA reviewing independent responses per ten matched teaching opportunities. Higher values were prespecified as the preferred direction. Phase A is 8, 9, 10, 11, 12. Phase B is 11, 14, 15, 15, 20. Every value is synthetic.

For Phase A, m = 5, mean A = 10, and the squared deviations sum to 10:

s_A^2 = 10 / 4 = 2.5

s_A = sqrt(2.5) = 1.5811388301

For Phase B, n = 5, mean B = 15, and the squared deviations sum to 42:

s_B^2 = 42 / 4 = 10.5

s_B = sqrt(10.5) = 3.2403703492

The raw mean difference is D = 15 - 10 = 5. Dividing it by the baseline SD gives:

uncorrected SMD_A = 5 / 1.5811388301 = 3.1622776602

The pooled variance and SD are:

s_p^2 = [(4 x 2.5) + (4 x 10.5)] / 8 = 6.5

s_p = sqrt(6.5) = 2.5495097568

uncorrected SMD_p = 5 / 2.5495097568 = 1.9611613514

Phase B varies more than Phase A, so the pooled denominator is larger and the pooled estimate is smaller. That difference is useful precisely because it exposes the variance assumption. Neither estimate tells Priya which answer is “right,” and neither says anything by itself about clinical importance.

Documented observation sensitivity

The final Phase B value of 20 occurred during a recorded interruption. It remains in the primary dataset because unusual context does not automatically invalidate a measurement. Priya also runs a labeled sensitivity without that row so reviewers can see its influence.

With Phase B 11, 14, 15, 15, n = 4, mean B = 13.75, s_B = 1.8929694486, and D = 3.75. The pooled SD becomes 1.7217101133.

sensitivity SMD_A = 3.75 / 1.5811388301 = 2.3717082451

sensitivity SMD_p = 3.75 / 1.7217101133 = 2.1780670108

Analysis stateMean AMean BBaseline SDPooled SDUncorrected SMD_AUncorrected SMD_pLocked primary10151.58113883012.54950975683.16227766021.9611613514Exclude disputed B=201013.751.58113883011.72171011332.37170824512.1780670108

Retain both rows and the dispute reason. The sensitivity does not authorize deletion, identify the true effect, or permit selection of the preferred coefficient.

Bring the result back to the graph

Chapter 9 of the linked single-case effect-estimation methods guide explains why within-case SMDs depend on variance assumptions and can conflict with visual evidence when baseline variation is zero or very small. It also discusses independence, homogeneous distribution, trend, and autocorrelation concerns. These are model questions, not boxes that become true through data entry.

Phase means erase order. Priya's same summaries could arise from an immediate step, a gradual trend, a delayed response, or a changing measurement process. Review the full graph for level, trend, variability, immediacy, overlap, consistency, measurement quality, treatment integrity, phase-change logic, and repeated demonstrations.

The What Works Clearinghouse Version 5.0 handbook treats systematic visual analysis and design quality as central to single-case evidence review. That framework does not make this worksheet a causal test. One phase pair and one standardized number cannot establish replication, a functional relation, generality, or attribution to an intervention.

Client meaning also remains separate from standardization. Review whether the outcome, direction, benefits, burdens, adverse effects, feasibility, dignity, access, and requested changes matter to the person. Invite accessible client and caregiver input and preserve dissent. A numeric distance from an observed SD is not social validity or clinical importance.

The BACB Ethics Codes page provides the current professional source for certificants. Those codes do not approve this calculator or convert an effect-size estimate into ethical or clinical compliance. Qualified professionals remain responsible for interpretation and action within competence, licensure, supervision, consent, payer, regulator, legal, and organizational requirements.

Leave a traceable, appropriately protected record

Use synthetic or appropriately de-identified data for training and testing. When an authorized clinical workflow requires identifiable data, use the least identifying information needed, approved systems, role-appropriate access, and a retained version history. The HHS Privacy Rule summary and Security Rule summary describe obligations for regulated entities. They do not determine applicability, certify this worksheet, or replace an organization's risk analysis.

Close the record with the raw-data and graph versions, selected phase pair, measurement rule, preferred direction, both variance assumptions, full-precision results, sensitivity reason, disagreements, client input, analyst, qualified reviewer, decision owner, unresolved evidence, and next review date.

Related resources

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