Interactive module guide

TOST Equivalence Test Calculator — Two Means

This TOST equivalence calculator shows two means are practically equivalent by rejecting both one-sided tests against pre-declared bounds.

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The TOST procedure (two one-sided tests) turns “show there is no meaningful difference” into a testable claim.

You declare, in advance, the largest difference that would still count as practically negligible—the equivalence bounds Δ_L and Δ_U—and then run two one-sided Welch t tests: one rejecting differences at or below Δ_L, one rejecting differences at or above Δ_U.

Only if both reject is equivalence established, and the reported p-value is the larger of the two.

Equivalently: the (1−2α) confidence interval for the mean difference must lie entirely inside the bounds.

A conventional non-significant t test can never justify “no difference”; TOST is the tool that can.

Use the TOST equivalence calculator when the goal is to demonstrate practical equality; declare the equivalence bounds on substantive grounds before looking at the data.

When to use this TOST equivalence test calculator

Use it when

Use it whenever the scientific claim is sameness: a generic matches the brand, a cheaper process matches the standard, a refactored system matches legacy behavior, an online course matches the classroom.
Use it after a redesign to show a guardrail metric did not move by more than a tolerable amount—an increasingly common A/B-testing companion to the usual superiority test.
Use it to discipline conclusions from “failed” significance tests: where colleagues would write “no significant difference, therefore no difference”, TOST states exactly what magnitude of difference the data can actually rule out.

Choose another method when

Avoid inventing bounds after seeing the data; bounds chosen to make the interval fit are circular, and the procedure’s validity rests on their pre-specification.
Avoid running it with bounds that do not bracket zero or that no stakeholder would defend as negligible; the conclusion inherits every weakness of the bounds.
Avoid reading a failed TOST as evidence of a difference: non-equivalence and difference are separate claims, and small samples routinely fail both the equivalence and the superiority test at once.

Interactive tool

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Example preview

Equivalence within half a unit

Inputs: x̄₁=10.2 (s=1.1, n=30) vs x̄₂=10.05 (s=1.3, n=32); bounds ±0.5.
Representative result: TOST p≈0.128 and the 90% interval [−0.36, 0.66] leaves the bounds, so equivalence is not established.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

2:28 min

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Read the complete transcript

TOST Equivalence Testing. A non-significant difference test does not prove sameness. TOST asks a different question: can the data exclude every difference large enough to matter? Declare a lower and upper equivalence bound before seeing the result. They live on the raw mean-difference scale and must bracket zero. This page accepts two independent means, positive standard deviations, integer sample sizes of at least two, valid bounds, and a pre-specified alpha. Compute mean one minus mean two, a Welch standard error, and Satterthwaite degrees of freedom. Equal variances are not required. The lower test asks whether the difference is safely above the lower bound. The upper test asks whether it is safely below the upper bound. Both one-sided tests must reject. The reported TOST p-value is the larger, less favorable p-value, so one passing side is never enough. The equivalent interval rule uses one minus two alpha coverage: ninety percent when alpha is point zero five. The entire interval must fit inside both bounds.

Keep three claims separate. Inside both bounds supports equivalence; crossing a bound is inconclusive; lying beyond a bound can support a relevant difference. Now compare two production lines. Means are ten point two and ten point zero five, standard deviations one point one and one point three, with sizes thirty and thirty-two. The observed difference is point one five. Welch standard error is point three zero five, with about fifty-nine point four degrees of freedom. Against the lower bound minus point five, t is two point one three and p is point zero one nine. That danger is ruled out. Against the upper bound plus point five, t is minus one point one four seven and p is point one two eight. That side does not reject. TOST p is therefore point one two eight. The ninety-percent interval runs from minus point three six to plus point six six, crossing the upper bound, so equivalence is not established. More precise data can place the interval inside pre-declared bounds, while common shifts change nothing. Never widen bounds after looking, call failure a difference, or use this independent-means page for paired data. Run TOST free at Distri Scope dot com.

How to read the result

Read the interval against the bounds first: entirely inside means equivalence, straddling a bound means inconclusive, entirely outside on one side would even indicate a relevant difference.

The reported p-value is conservative by construction—it is the worse of the two directions, so a small TOST p means both dangers (too low and too high) are ruled out.

The statistic display shows both one-sided t values; typically one is comfortably large and the other is the binding constraint.

How to use the TOST equivalence test calculator

Enter summary statistics for both samples (mean, SD, size) plus the equivalence bounds and α; raw data are not needed.

The bounds are on the raw difference scale (units of the measurement), must satisfy Δ_L < 0 < Δ_U, and default to ±0.5 as a placeholder to replace with substantively justified values.

Standard errors are Welch-based with Satterthwaite degrees of freedom, so unequal variances need no special handling.

Formula, hypotheses, and assumptions

t_1 = \frac{(\bar{x}_1-\bar{x}_2) - \Delta_L}{SE}, \quad t_2 = \frac{(\bar{x}_1-\bar{x}_2) - \Delta_U}{SE}

Conditions to review

Independent samples with approximately normal means
Welch standard errors — no equal-variance assumption
Equivalence bounds chosen before seeing the data
The bounds represent a practically negligible difference

Calculator parameters

Sample 1 Mean: default 10.2.
Sample 1 Std Dev: default 1.1.
Sample 1 Size: default 30.
Sample 2 Mean: default 10.05.
Sample 2 Std Dev: default 1.3.
Sample 2 Size: default 32.
Equivalence Lower Bound (Δ𝗟): default -0.5.
Equivalence Upper Bound (Δ𝗨): default 0.5.
Significance Level (α): default 0.05.

What the method is doing

The two statistics are t₁ = (d − Δ_L)/SE tested rightward and t₂ = (d − Δ_U)/SE tested leftward; the TOST p-value is the larger one-sided p, and both must fall at or below α.

The companion interval uses 1 − 2α coverage (90% at α = 0.05) — that is not a typo but the exact interval whose inside-the-bounds reading matches the two tests.

The implementation is pinned against an independent scipy computation of both t statistics, the Welch df, the p-value, and the interval.

Worked example: bounds of half a unit

Two production lines are compared on a quality score. Line 1: x̄ = 10.2, s = 1.1, n = 30. Line 2: x̄ = 10.05, s = 1.3, n = 32. Differences within ±0.5 units are considered negligible by engineering judgment. Run TOST at α = 0.05.

1The observed difference is 0.15 with Welch SE ≈ 0.305 and Satterthwaite df ≈ 59.4.
2Against the lower bound: t₁ = (0.15 − (−0.5))/0.305 ≈ 2.130, one-sided p ≈ 0.019 — differences at or below −0.5 are rejected.
3Against the upper bound: t₂ = (0.15 − 0.5)/0.305 ≈ −1.147, one-sided p ≈ 0.128 — differences at or above +0.5 are NOT rejected.
4TOST p = max(0.019, 0.128) ≈ 0.128 > 0.05, and the 90% interval [−0.36, 0.66] pokes above the upper bound: equivalence is not established.

Interpretation

The data rule out the lines differing by half a unit in one direction but not the other, so the honest conclusion is “inconclusive at these bounds”, not “equivalent” and not “different”. The remedy is more data or wider (still defensible) bounds—decided before the next look, not after. The example also shows TOST’s asymmetry in action: the binding side is whichever bound sits closer to the observed difference.

Common mistakes

Do not conclude equivalence from a non-significant ordinary t test; absence of evidence is the fallacy TOST exists to prevent.
Do not tune bounds until the interval fits; pre-specification is the procedure’s entire epistemic content.
Do not forget that the interval has 1 − 2α coverage; quoting it as a 95% interval at α = 0.05 misstates the method.

Limits and independent validation

Only the two-independent-means case from summary statistics is implemented; paired TOST and proportion equivalence are not yet available.

Normality of the means is still assumed via the Welch t machinery.

No standardized-effect input: bounds are on the raw scale, which is usually where stakeholders can actually defend them.

Before using the result

Check the interval reading against the test reading — they must agree exactly; disagreement indicates an input error.
Recompute one t statistic by hand from the summary inputs.
Ask the bound question aloud: “would a difference of Δ_U actually matter to anyone?” If the answer is unclear, the analysis is not ready to run.

two sample t test →Run the superiority companion question on the same two samples.confidence interval calculator →Build the standalone interval whose bounds reading TOST formalizes.

← Repeated Measures ANOVA Power & Sample Size →

See every option in the Statistical Hypothesis Test Calculator or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.