Confidence Intervals

An estimate without uncertainty is incomplete; a confidence interval reports both a centre and a defensible range.

General structure

Many confidence intervals have form

where

The standard error measures how much the estimator would vary across repeated random samples.

Correct confidence interpretation

A $95\%$ confidence method is designed so that about $95\%$ of intervals produced over many repeated samples would contain the true parameter.

After one interval is computed, the parameter is fixed and the interval either contains it or does not. In introductory language, say “we are $95\%$ confident that the interval captures the population parameter,” not that $95\%$ of population values lie inside it.

One-proportion interval

For sample proportion $\hat p=x/n$, an approximate interval is

where $z^$ matches the confidence level. For $95\%$, $z^\approx1.96$.

The method requires appropriate randomization/representativeness, approximate independence, and enough expected successes and failures.

Worked example

Confidence level and width

Higher confidence requires a wider interval when the data and method remain fixed, because a larger critical value is used. Lower confidence produces a narrower interval but a lower long-run capture rate.

Confidence and precision trade off.

Sample size and width

Standard errors commonly shrink roughly like $1/\sqrt n$. Quadrupling sample size approximately halves margin of error; merely doubling sample size does not halve it.

Larger samples reduce random sampling error but do not repair selection, nonresponse, or measurement bias.

Confidence interval for a mean

For a population mean with unknown standard deviation, a common interval is

The $t$ critical value depends on confidence level and degrees of freedom. Conditions include a suitable random design and a sampling distribution of the mean that is approximately normal, supported by population shape or sample size.

Margin of error is not total error

The calculated margin generally reflects random sampling variability under the model. It does not automatically include bias from undercoverage, nonresponse, leading questions, data fabrication, or confounding.

A numerically narrow interval can still miss the target systematically.

Statistical versus practical significance

An interval can show whether a null value is plausible and also reveal effect-size precision. A very small effect may be statistically distinguishable but practically unimportant; a wide interval may include both meaningful benefit and harm.

Interpret values in context, not only whether a threshold is crossed.

Reporting an interval

State the parameter, confidence level, numerical bounds, units, population, method, and important conditions. Round bounds consistently without hiding meaningful uncertainty.

Avoid describing the interval as containing a certain percentage of individual observations.

Common mistakes

Saying there is a $95\%$ chance the fixed parameter lies in this already-computed interval. Confidence belongs to the repeated method.

Saying $95\%$ of data lie between the bounds. The interval estimates a parameter.

Believing higher confidence narrows the interval. It widens it.

Believing large $n$ removes bias. It reduces sampling variability only.

Reporting bounds without population or conditions. Context defines the parameter.

Quick self-check

• What population parameter is being estimated?
• Was the sample collected in a way that supports inference?
• Do method-specific independence, success/failure, or shape conditions hold?
• Are estimate, standard error, critical value, and margin of error correct?
• Is the confidence interpretation about the method and parameter—not individual data?
• Are bias, practical importance, units, and population scope reported?

Related topics

• Sampling Methods
• Bias in Data
• Normal Distribution
• Standard Error