Mean Confidence Interval Calculator
Use sample data or summaries to estimate uncertainty around a mean return.
Student’s t · unknown population SD · independent-sample assumption
Results
Review or adjust your inputs, then calculate to see your results and interpretation.
Understand this calculation
The interval centers on the sample mean. Its margin is the t critical value times the standard error. More observations generally narrow it; greater confidence or sample SD generally widens it.
See calculation method and assumptions
Formula
Mean Confidence Interval
Two-sided t interval for the mean
Standard error and margin of error
Add and subtract the t critical value at n−1 degrees of freedom times the sample standard error; population SD is unknown.
Standard error is s divided by √n. Margin of error is the t critical value times standard error.
Variables, rates and periods
- n / x̄ / s
- n is sample size, x̄ the sample mean, and s the sample SD using n−1 in the variance denominator.
- 1−α / df
- Confidence is 1−α, and df=n−1. This tool uses two-sided Student's t.
- t* / SE / ME
- t* is the quantile for the degrees of freedom and confidence level; SE=s/√n and ME=t*×SE.
- Units
- Mean return and bounds use %. SD, standard error, and margins use percentage points; variance uses squared percentage points. No annualization.
Assumptions of the t method
- Observations should form an independent, representative sample.
- Approximate population normality is especially important with small samples.
- Degrees of freedom are n−1 and at least two observations are required.
- A larger sample does not automatically fix serial dependence or sampling bias.
SD and SE are different
- Sample SD describes dispersion of individual observations.
- SE, s/√n, describes sampling variation of the mean under the assumptions.
- The interval estimates a population mean, not the range of individual monthly returns.
Estimate a population mean interval from data or sample summaries using Student’s t when population SD is unknown.
Worked example
- Enter a 2% mean, a four-percentage-point sample SD and n=12, or supply sample observations.
- Choose 95% confidence; with unknown population SD use t with 11 degrees of freedom.
- Read the SE, margin and bounds, and check whether independence and distribution assumptions are reasonable.
Frequently asked questions
What does a 95% confidence interval mean?
Across repeated independent samples and intervals built by the same method, about 95% of intervals cover the fixed population mean when the assumptions hold. It is not a 95% probability statement about this one computed interval.
Why not always use 1.96?
When population SD is unknown and estimated with sample SD, the t critical value depends on degrees of freedom and confidence. Small-sample t critical values generally exceed their normal counterparts.
Can I directly use daily stock returns?
You can compute it, but must check assumptions. Serial dependence, volatility clustering, heavy tails or unrepresentative samples can invalidate the nominal coverage of a simple mean t interval.