Volatility Confidence Interval Calculator
Enter equal-frequency percentage-point returns to estimate a two-sided chi-square confidence interval for population standard deviation. The calculator shows N − 1 sample volatility, both chi-square critical values and every squared deviation, while clearly requiring approximately normal independent observations and withholding forecasts or risk guarantees.
Enter one return sample
Use signed percentage-point returns from one frequency, period, instrument and gross-or-net convention; the result stays in per-period percentage points.
Two-sided confidence from 50% through 99.9%.
Enter 3 to 500 values without percent signs; separate with spaces, commas, semicolons or new lines.
Volatility 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 volatility interval is calculated
Lower = s × √[(n − 1) ÷ χ²1−α/2,n−1]
Upper = s × √[(n − 1) ÷ χ²α/2,n−1]
The calculator mean-centers the entered per-period returns and computes sample standard deviation with the n minus one denominator. That sample estimate is shown separately from the interval bounds.
For the selected two-sided confidence level, the model obtains lower-tail and upper-tail chi-square critical values with n minus one degrees of freedom. Because those critical values appear in opposite denominators, the standard-deviation interval is generally asymmetric.
The result remains in per-period percentage points. Version 1.0.0 performs no square-root-of-time annualization, volatility forecast, risk limit, position sizing or conversion to price or cash units.
Worked example from the audited fixture
The audited fixture has 10 percentage-point returns, sample standard deviation 0.73703611 and 9 degrees of freedom.
- At 95% confidence, the lower-tail chi-square critical value is 2.70038950 and the upper-tail critical value is 19.02276780.
- The population standard-deviation interval is 0.50695938 to 1.34554062 percentage points. Its asymmetry comes from the chi-square distribution and is not evidence of a directional return forecast.
Reproduce it: select “Load audited example” above. The governed engine retains full precision and rounds only the visible interface.
How to interpret the result
- Sample standard deviation is the point estimate; the bounds show parameter uncertainty under the stated normal and independence assumptions.
- The interval is asymmetric because the chi-square sampling distribution is asymmetric, especially with few observations.
- Higher confidence widens the interval when the entered observations remain unchanged.
- More genuinely comparable independent observations generally narrow sampling uncertainty, but they cannot remove structural or measurement uncertainty.
- The upper bound is not a guaranteed worst-case volatility and the lower bound is not a safe floor for future risk.
Assumptions and limits
- Enter 3 to 500 equal-frequency percentage-point returns under one consistent sampling and cost convention.
- The chi-square construction requires independent observations from an approximately normal population for exact nominal coverage.
- FX returns often exhibit heavy tails, serial dependence, volatility clustering and structural breaks that violate those assumptions.
- The calculator does not test the assumptions, estimate conditional volatility or account for bid-ask spread, slippage and missing observations.
- No annualization, horizon scaling, VaR, expected shortfall, stress scenario or future-volatility probability is calculated.
- No risk guarantee, 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 estimates a two-sided interval for population standard deviation from one entered equal-frequency percentage-point return sample.
- Version 1.0.0 calculates N minus one sample standard deviation and uses n minus one degrees of freedom in the chi-square construction.
- The chi-square distribution is asymmetric, especially with small degrees of freedom, so the lower and upper standard-deviation distances need not match.
- Exact nominal coverage requires independent observations from an approximately normal population; the calculator states but cannot test those assumptions.
- Heavy tails, volatility clustering and structural breaks can make a normal-population chi-square interval materially misleading.
- No. It is a conditional parameter-confidence bound, not a guaranteed maximum, stress scenario or future-volatility ceiling.
- No. Sample standard deviation and both bounds remain in the per-period percentage-point unit entered.
- No. It produces no future-volatility probability, risk guarantee, VaR, forecast, grade, signal, position instruction or recommendation.
Sources and methodology
- NIST — Confidence Interval for a Standard Deviation — Published chi-square interval and the normal-population assumption.
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 volatility 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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