Mean Return Confidence Interval Calculator
Enter equal-frequency percentage-point returns to calculate a two-sided Student-t confidence interval for the population mean. The page exposes sample standard deviation, standard error, critical value and every squared deviation, while withholding forecasts, strategy-validation claims, significance decisions and trading recommendations.
Enter one coherent return sample
Use signed percentage-point returns from one frequency, instrument, statement period and gross-or-net convention.
Two-sided confidence from 50% through 99.9%.
Enter 3 to 500 values without percent signs; separate with spaces, commas, semicolons or new lines.
Mean-return interval arithmetic
Entered Return Uncertainty Intervals 1.0.0.
On smaller screens, scroll horizontally to inspect the complete audit table.
| Observation | Return | Deviation from mean | Squared deviation |
|---|
How the mean return interval is calculated
s = √[Σ(r[i] − r̄)² ÷ (n − 1)]
SE = s ÷ √n
Interval = r̄ ± t1−α/2,n−1 × SE
The calculator first computes the arithmetic mean of the entered percentage-point returns. Sample variance divides the squared-deviation sum by n minus one, and the square root gives the sample standard deviation.
The standard error is the sample standard deviation divided by the square root of the observation count. The selected two-sided confidence level determines alpha and the positive Student-t critical value with n minus one degrees of freedom.
The margin of error is the critical value multiplied by the standard error. Subtracting and adding that margin to the sample mean gives the displayed lower and upper bounds without annualization or compounding.
Worked example from the audited fixture
The audited fixture has 10 returns, sample mean 0.49 percentage points and N − 1 sample standard deviation 0.73703611 percentage points.
- At 95% confidence, degrees of freedom are 9, the Student-t critical value is 2.26215716 and the standard error is 0.23307128 percentage points.
- The margin is 0.52724387 percentage points, so the interval is −0.03724387 to 1.01724387 percentage points. This describes estimator uncertainty under the assumptions; it is not a forecast range for the next trade.
Reproduce it: select “Load audited example” above. The governed engine retains full precision and rounds only the visible interface.
How to interpret the result
- The sample mean is the center of the two-sided interval because this version uses the symmetric Student-t construction.
- A wider interval records more sampling uncertainty under the entered sample size and dispersion; it is not automatically evidence of a weak strategy.
- Increasing confidence while holding the sample fixed increases the critical value and widens the interval.
- Adding genuinely comparable observations can reduce standard error, but repeated or selected observations do not create independent evidence.
- Whether zero lies inside the interval can be observed arithmetically, but this page does not turn that observation into a reject, accept or strategy-validation verdict.
Assumptions and limits
- Enter 3 to 500 equal-frequency percentage-point returns under one consistent gross-or-net and preprocessing convention.
- The model assumes independent observations and an approximately normal population for the exact small-sample Student-t interpretation.
- Serial correlation, volatility clustering, heavy tails, structural breaks and selection bias can make the nominal interval misleading.
- The calculator does not test normality, independence, stationarity, data completeness or cost completeness.
- No annualization, compounding, multiple-testing adjustment or future-return probability is calculated.
- No significance label, performance forecast, strategy 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 estimates a two-sided interval for the population mean behind one entered equal-frequency return sample under the stated Student-t assumptions.
- Version 1.0.0 uses N minus one sample standard deviation, then divides it by the square root of n to calculate standard error.
- Population standard deviation is unknown and estimated from the entered sample, so the critical value uses Student-t with n minus one degrees of freedom.
- They are bounds from a repeated-sampling construction under assumptions, not a fixed probability statement about these realized bounds or a range for the next return.
- No. The page reports estimator uncertainty only and makes no accept, reject, edge or strategy-validation decision.
- No. Enter observations from one equal-frequency, consistently preprocessed and consistently costed sample.
- No. The mean, standard error, margin and bounds remain in entered per-period percentage points.
- No. It creates no future-return probability, performance forecast, significance label, strategy grade, signal, position instruction or recommendation.
Sources and methodology
- NIST — Confidence Limits for the Mean — Published Student-t interval for a population mean when population standard deviation is unknown.
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 mean-return uncertainty 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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