Return Difference Confidence Interval Calculator
Estimate a two-sided confidence interval for mean percentage-point return A minus B. Choose independent Welch samples with unequal variances or paired observations aligned by row; the calculator exposes its standard error and degrees of freedom but makes no significance, superiority, edge, forecast or trading verdict.
Enter the two return samples
Choose whether the samples are independent or paired, then enter signed percentage-point returns under comparable frequency and cost conventions.
Two-sided confidence from 50% through 99.9%.
Choose from the actual data design, not the result you prefer.
Numbers without percent signs; separate with spaces, commas, semicolons or new lines. Maximum 500.
In paired mode, each B row must correspond meaningfully to the A row at the same index.
Mean-difference interval arithmetic
Entered Return Uncertainty Intervals 1.0.0.
On smaller screens, scroll horizontally to inspect the complete audit table.
| Row | Sample A | Sample B | Paired A − B | Difference deviation | Squared deviation |
|---|
How the return difference interval is calculated
df = (vA + vB)² ÷ [vA² ÷ (nA − 1) + vB² ÷ (nB − 1)]
Paired: d[i] = A[i] − B[i], SE = sd ÷ √n
Interval = (r̄A − r̄B) ± t × SE
Independent mode calculates N minus one sample variances separately and uses the Welch standard error. Welch–Satterthwaite degrees of freedom can be fractional and do not assume equal population variances.
Paired mode subtracts B from A at each aligned row, then applies a one-sample Student-t interval to the resulting difference series. The paired estimate uses n minus one degrees of freedom.
Both modes report percentage-point difference A minus B. Neither mode performs annualization, multiple-testing correction or a hypothesis-test decision, and the page never chooses the study design on the user’s behalf.
Worked example from the audited fixture
The audited Welch fixture contains 10 A returns and 8 independent B returns. Their sample means are 0.49 and 0.20 percentage points, so A minus B is 0.29 percentage points.
- The Welch standard error is 0.28164201, degrees of freedom are 15.08283597 and the 95% critical value is 2.13043043.
- The margin is 0.60001871 percentage points, producing an interval from −0.31001871 to 0.89001871. A separate paired fixture verifies the row-aligned mode; neither result is labelled superior or significant.
Reproduce it: select “Load audited example” above. The governed engine retains full precision and rounds only the visible interface.
How to interpret the result
- A positive center means the entered sample mean for A exceeds B by that many percentage points; it does not prove durable outperformance.
- Welch mode supports unequal counts and variances, but its observations must be independent within and between the entered samples.
- Paired mode can isolate within-pair differences when rows are meaningfully matched; arbitrary row pairing invalidates that interpretation.
- A narrower interval means less estimator uncertainty under the selected design and assumptions, not necessarily lower trading risk.
- The displayed bounds remain a conditional statistical reference. The page makes no accept/reject, significant/not-significant or strategy-superiority decision.
Assumptions and limits
- Each sample must contain 3 to 500 equal-frequency percentage-point returns with comparable preprocessing and cost treatment.
- Welch mode assumes independent observations and approximately normal sampling behavior for the mean difference.
- Paired mode additionally requires meaningful one-to-one row alignment; equal row counts alone do not establish a valid pairing.
- The calculator cannot detect overlapping trades, regime changes, data snooping, serial dependence, heavy tails or incomparable periods.
- One interval does not adjust for trying many strategies, parameters, pairs, timeframes or sample definitions.
- No significance label, superiority claim, validated edge, forecast, grade, signal, position instruction or recommendation is generated.
Which uncertainty interval answers which question?
Mean location, difference between means, standard deviation and empirical resampling uncertainty are related but not interchangeable. The comparison below keeps the estimator, evidence and assumptions visible so one interval is not presented as a universal strategy-validation result.
| Tool | Evidence entered | Parameter or quantity estimated | Main boundary |
|---|---|---|---|
| Mean Return Confidence Interval | One entered return sample | Population mean return interval | Student-t; independence and approximately normal mean behavior. |
| Return Difference Confidence Interval | Two independent or row-paired return samples | Population mean A minus B interval | Design must match Welch independence or meaningful pairing. |
| Volatility Confidence Interval | One entered return sample | Population standard-deviation interval | Chi-square; highly sensitive to normality and independence. |
| Bootstrap Expectancy Calculator | One entered outcome sample plus seed | Resampling-percentile interval for the sample mean | Empirical resampling is not an assumption-free population guarantee. |
Frequently asked questions
- It always reports entered mean return A minus entered mean return B in percentage points.
- Use Welch only when A and B are independent samples. It permits unequal counts and does not assume equal population variances.
- Use paired mode only when every A row is meaningfully matched with the B row at the same index, such as two measurements for the same period.
- Version 1.0.0 uses the Welch-Satterthwaite formula from the two sample-variance-over-count terms, so degrees of freedom can be fractional.
- Yes. Welch mode accepts different counts, provided each sample has at least three valid observations and the independence design is appropriate.
- No. Paired mode requires equal row counts because it calculates A minus B for each aligned row before applying a one-sample Student-t interval.
- No. The result is conditional on sample selection, design and assumptions and includes no multiple-testing adjustment or durable-superiority conclusion.
- No. It creates no p-value, accept-reject label, verified edge, forecast, strategy grade, signal, position instruction or recommendation.
Sources and methodology
- NIST — Difference of Means Confidence Limits — Published independent unequal-variance mean-difference interval and Welch–Satterthwaite degrees of freedom.
- NIST — Paired Observations — Published reduction of paired observations to a one-sample analysis of differences.
Version 1.0.0 was locked only after formulas and assumptions were checked against the cited NIST references. Canonical fixtures were independently recomputed with SciPy before being compared with the browser engine. The calculator performs arithmetic locally and does not upload the entered observations.
Continue the two-sample review
Verify the return evidence before estimating uncertainty
Reconcile the exact statement period, sampling frequency, timezone, realized P&L, spread, commission, financing, currency conversion and missing observations before deriving returns. A statistically correct interval cannot repair incomplete, selected or inconsistent source evidence.
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