Confidence Intervals: What the Range Means
A confidence interval is a range calculated from sample data to estimate an unknown population value. As NIST explains, it expresses uncertainty around an estimate; the confidence level describes how reliably the interval procedure covers the population value across repeated sampling.
What a 95% confidence interval means
In frequentist statistics, the population parameter is fixed but unknown, while samples and their interval endpoints vary. NIST defines 95% confidence as long-run coverage: about 95% of intervals constructed through repeated use of the same sampling and calculation procedure contain the true parameter, provided the method’s conditions hold.
Once an interval has been calculated, the true parameter is either inside or outside its bounds. The 95% label belongs to the procedure; it does not assign a 95% probability to that particular observed range.
Identify what the interval estimates
Confidence intervals can estimate population means, proportions, differences between groups, and other parameters. Penn State’s introductory lesson distinguishes each population parameter from the sample statistic used to estimate it. An interval for a mean and an interval for a difference between means answer different questions.
The main elements are:
- Point estimate: the sample statistic used to estimate the population parameter.
- Confidence level: the intended coverage of the interval method.
- Confidence limits: the lower and upper endpoints of the reported range.
NIST distinguishes two-sided intervals from one-sided bounds: a two-sided interval bounds the parameter from both directions, while a one-sided bound limits it from only one direction. The confidence percentage alone does not specify which was used.
How confidence intervals are calculated
Many common symmetric intervals follow Penn State’s margin-of-error formulation:
confidence interval = point estimate ± margin of error
margin of error = critical value × standard error
The standard error measures variation in a statistic across samples. For a symmetric interval, the total width is twice the margin of error. This form is useful for understanding the calculation, but not every confidence interval is symmetric.
A confidence interval for a mean
For a population mean with an unknown population standard deviation, Penn State gives the one-sample t interval:
x̄ ± t* × s / √n
Here, x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and t* is the critical value from a t distribution with n − 1 degrees of freedom. The method assumes suitable sampling conditions and a normal population, or a sufficiently accurate large-sample approximation.
When the population standard deviation σ is known, NIST gives the normal-model mean interval as x̄ ± z* × σ / √n. The appropriate critical value depends on the confidence level and whether the interval is one-sided or two-sided.
A documented 95% interval calculation
In NIST’s worked calculation, 195 observations have a sample mean of 9.261460 and a sample standard deviation of 0.022789. Using a t critical value of 1.9723, NIST reports a 95% confidence interval of 9.258242 to 9.264679. These are bounds for the population mean, rather than limits for individual observations.
A confidence interval for a proportion
A common normal approximation for a binomial proportion is p̂ ± z* × √(p̂ × (1 − p̂) / n), where p̂ is the observed sample proportion. NIST’s proportion guidance explains that symmetric normal approximations can be inadequate with very small samples or very few events, and can even produce impossible negative lower bounds.
NIST also documents Wilson score and exact binomial intervals. Exact binomial intervals need not be symmetric around the observed proportion. The method therefore matters alongside the reported confidence level.
What makes an interval wider or narrower?
For normal-based mean intervals, NIST’s precision formula connects the margin of error to the confidence multiplier, population standard deviation, and square root of the sample size. Holding the other inputs fixed, greater variability increases the margin, while a larger sample decreases it.
With a known standard deviation and fixed confidence level, the margin scales as 1 / √n. Consequently, halving the target margin requires four times the sample size under that calculation. This relationship concerns precision under the stated assumptions.
Higher confidence also increases width for the same data and method. Penn State lists these standard normal critical values for two-sided intervals:
| Confidence level | Normal critical value, z* |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
These normal multipliers do not replace the sample-size-dependent t multiplier in a t interval.
Reading an interval for a difference between groups
For an estimated difference, zero represents no difference. As Wolfe and Hanley explain, interpreting a 95% interval relative to zero connects to the corresponding two-sided test at the 5% significance level. This connection requires matching methods and assumptions.
| Position of the difference interval | Interpretation under the matching method |
|---|---|
| Entirely above zero | Evidence of a positive difference at that level. |
| Entirely below zero | Evidence of a negative difference at that level. |
| Includes zero | The corresponding test does not reject no difference at that level. |
An interval containing zero also contains other values. Its inclusion of zero therefore does not establish that the true difference is exactly zero.
Separate intervals for the two groups cannot substitute for an interval for their difference. Wolfe and Hanley show that separate 95% intervals can overlap even when the difference is statistically significant. Comparing groups requires the uncertainty of the difference itself.
The calculation must also match the design. NIST distinguishes paired from unpaired comparisons and equal from unequal variance methods. Measurements linked in pairs need a calculation that accounts for that pairing.
Statistical significance and practical importance
Excluding zero does not establish that a difference is large enough to matter. Russell Poldrack’s discussion of hypothesis testing explains that very small effects can become statistically significant with sufficiently large samples. Practical importance concerns the magnitude of an effect, rather than its significance label.
Interpret the estimate and both endpoints against the magnitude relevant to the research question. A range entirely above zero can still include effects below a meaningful threshold; significance alone does not resolve that comparison.
What a narrow interval cannot establish
Precision does not establish freedom from bias. The U.S. Census Bureau’s reliability guidance distinguishes sampling variation from systematic errors caused by nonresponse, missing population coverage, or recording mistakes. Standard errors do not measure all these sources of error.
A narrow interval can therefore coexist with a biased estimate. Increasing sample size reduces sampling uncertainty under the model; it does not, by itself, correct systematic problems in how observations were selected or recorded.
Coverage also changes meaning when several intervals are considered together. Individual 95% intervals do not automatically provide 95% coverage for the entire set. For prespecified contrasts, NIST describes Bonferroni’s method, which adjusts individual confidence levels to maintain an overall coverage level. Individual and simultaneous confidence claims must be distinguished.
Confidence, prediction, and credible intervals
These intervals have different targets or probability interpretations:
| Interval | What it describes |
|---|---|
| Confidence interval | A population parameter estimated by a procedure with stated long-run coverage. |
| Prediction interval | A new observation, including its additional random variation. |
| Bayesian credible interval | A parameter range containing a stated share of posterior probability. |
A confidence interval for a mean does not describe where 95% of individual observations fall. NIST’s prediction guidance explains that prediction intervals include both uncertainty in the modeled mean and variation in a new measurement. For the same model and level, they are wider than the corresponding confidence interval for the modeled mean.
A 95% credible interval contains 95% of the posterior probability under the specified Bayesian model. As Poldrack’s Bayesian explanation describes, that posterior combines the data’s likelihood with a prior. This conditional probability interpretation differs from frequentist coverage across repeated samples.
Reporting a confidence interval
Report the population and parameter, the point estimate and units, the confidence level, both endpoints, and the calculation method. For a group comparison, report the difference and its interval, including how the range relates to zero and any threshold for practical importance.