# Confidence Intervals > A confidence interval gives a range of plausible values for a statistic. Learn how to calculate them for returns and win rates and how to read them in backtests. Source: https://learn.tradelabsai.com/math/confidence-intervals/ Track: Math and Statistics · Level: Intermediate · Updated: 2026-10-03 Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice. Cite as: TradeLabs Learn, "Confidence Intervals", https://learn.tradelabsai.com/math/confidence-intervals/ A backtest reports a single number for average return, win rate or Sharpe ratio. But that number is an estimate from a limited sample, and the true value could be higher or lower. A confidence interval expresses this uncertainty as a range of plausible values. Reading results as ranges rather than single numbers is one of the simplest ways to avoid overconfidence in trading research. ## The basic formula For a mean, using the normal approximation: ``` confidence interval = estimate ± z × standard error ``` | Confidence level | z value | |---|---| | 90% | 1.645 | | 95% | 1.96 | | 99% | 2.576 | For small samples, the t distribution replaces z, giving slightly wider intervals. See [Student's t-Distribution](https://learn.tradelabsai.com/math/students-t-distribution/) and [Sampling and Standard Error](https://learn.tradelabsai.com/math/sampling-and-standard-error/). ## Worked example: average return **Example: A strategy's average trade** A strategy has 100 trades averaging +0.40% with a standard deviation of 2.5%. Standard error = 2.5% / √100 = 0.25%. 95% confidence interval = 0.40% ± 1.96 × 0.25% = 0.40% ± 0.49% = about minus 0.09% to +0.89%. The interval includes zero, so the data cannot rule out the possibility that the strategy has no edge at all. With 400 trades and the same numbers, the interval would be 0.40% ± 0.245%, or about +0.16% to +0.65%, excluding zero. See [Statistical Significance in Trading](https://learn.tradelabsai.com/math/statistical-significance/). ## Worked example: win rate ``` CI for a proportion = p ± z × √(p × (1 - p) / n) ``` A 58% win rate on 50 trades: SE = √(0.58 × 0.42 / 50) ≈ 7.0%. The 95% interval is about 44% to 72%. A win rate that looks strong could plausibly be below 50%. For small samples or extreme proportions, the Wilson interval is more accurate. ## What a confidence interval means A 95% confidence interval means that if you repeated the sampling process many times, about 95% of the intervals built this way would contain the true value. It does not mean there is a 95% probability that this particular interval contains the true value, though in practice many people use it that way. Bayesian credible intervals do allow that interpretation. See [Bayesian Statistics](https://learn.tradelabsai.com/math/bayesian-statistics/). ## Confidence intervals for the Sharpe ratio **Example: How uncertain is a Sharpe ratio?** A strategy shows an annual Sharpe ratio of 1.2 over 4 years. An approximate standard error is √((1 + 0.5 × 1.2²) / 4) = √(1.72 / 4) ≈ 0.66. The 95% interval is about 1.2 ± 1.29, from roughly minus 0.1 to 2.5. Four years is not enough to be confident the strategy is good. See [Sharpe Ratio](https://learn.tradelabsai.com/portfolio/sharpe-ratio/). ## When normal intervals are unreliable - **Skewed or fat tailed returns:** normal intervals can be too narrow. Bootstrapping is often better. See [Bootstrap and Permutation Tests](https://learn.tradelabsai.com/math/bootstrap-and-permutation-tests/). - **Autocorrelated returns:** standard errors are understated; use adjusted methods. See [Autocorrelation and Partial Autocorrelation](https://learn.tradelabsai.com/math/autocorrelation/). - **Small samples:** use the t distribution or bootstrap. - **Data mining:** if you tested many strategies and picked the best, its interval is too optimistic. See [P-Hacking and Multiple Testing](https://learn.tradelabsai.com/research/p-hacking-and-multiple-testing/). ## Using intervals in decisions 1. **Report intervals with every key statistic** in research. 2. **Look at the lower bound:** would the strategy still be worth trading at the pessimistic end? 3. **Compare strategies by overlap:** heavily overlapping intervals mean you cannot tell which is better. 4. **Size positions with the uncertainty in mind:** a wide interval suggests smaller size. See [Position Sizing](https://learn.tradelabsai.com/risk/position-sizing/). ## A note on wording When sharing research, write results as ranges, such as "average trade +0.4% (95% interval minus 0.1% to +0.9%)", rather than a single figure. Readers immediately see how much confidence the evidence supports. ## Frequently asked questions ### What is a confidence interval? A range of values, calculated from sample data, that is likely to contain the true value of a statistic at a chosen level of confidence, such as 95%. ### How do you calculate a 95% confidence interval for a mean? Take the sample mean and add and subtract about 1.96 times the standard error. ### Why are confidence intervals useful in trading? They show how uncertain backtest results are, helping traders avoid trusting statistics from small or noisy samples. Next, learn to test whether results are real in [Hypothesis Testing and P-Values](https://learn.tradelabsai.com/math/hypothesis-testing-and-p-values/). ## Continue learning - Next lesson: [Hypothesis Testing and P-Values](https://learn.tradelabsai.com/math/hypothesis-testing-and-p-values/) - Previous lesson: [Central Limit Theorem](https://learn.tradelabsai.com/math/central-limit-theorem/) - Related: [Central Limit Theorem](https://learn.tradelabsai.com/math/central-limit-theorem/): The central limit theorem says averages of many independent values tend toward a normal distribution. Learn what it means for trading statistics and when it fails. - Related: [Sampling and Standard Error](https://learn.tradelabsai.com/math/sampling-and-standard-error/): Standard error measures how much an estimate like a win rate or average return varies between samples. Learn the formulas and what they mean for backtests. - Related: [Hypothesis Testing and P-Values](https://learn.tradelabsai.com/math/hypothesis-testing-and-p-values/): Hypothesis tests check whether results are likely due to chance. Learn null hypotheses, test statistics and p values, a strategy test and how p values mislead. - Related: [Statistical Significance in Trading](https://learn.tradelabsai.com/math/statistical-significance/): Statistical significance helps judge whether trading results reflect a real edge or luck. Learn the t statistic rule of thumb, sample size and multiple testing. - Related: [Bootstrap and Permutation Tests](https://learn.tradelabsai.com/math/bootstrap-and-permutation-tests/): Bootstrap and permutation tests use resampling to measure uncertainty and test significance without strict assumptions. Learn how they work, examples and pitfalls. - Related: [Sharpe Ratio](https://learn.tradelabsai.com/portfolio/sharpe-ratio/): The Sharpe ratio measures return per unit of risk. Learn the formula, how to annualise it, what counts as a good Sharpe ratio, its limitations and common mistakes.