Confidence Interval — Formula, Calculation & Examples

A confidence interval is a range of plausible values for a population parameter, based on sample data. You reach for one whenever a sample is used to say something about a larger population, and you want to show the uncertainty around an estimate instead of reporting a single number.

In many intro statistics problems, you build it as

The margin of error depends on how much uncertainty is in the sample and how confident you want to be. Higher confidence gives a wider interval; more precise data gives a narrower one.

When This Method Applies

Use a confidence interval when a sample is used to estimate a fixed population quantity, such as a mean or a proportion, and you want a plausible range rather than a point guess. The general structure is

For a population mean, two common versions apply in different conditions:

Use that form when the population standard deviation $\sigma$ is known, or when a normal approximation with a $z$ critical value is justified.

Use that form when $\sigma$ is unknown and you estimate spread with the sample standard deviation $s$. For smaller samples, this is usually paired with the condition that the population is approximately normal. The same pattern appears for proportions and other parameters, but the standard error changes.

The Steps

1. Choose the target

Decide which population quantity you are estimating, such as a mean or a proportion.

2. Pick a confidence level

Use a level such as $90\%$, $95\%$, or $99\%$, then find the matching critical value for your method.

3. Compute the estimate and standard error

Calculate the sample statistic and the standard error that matches your setting.

4. Find the margin of error

Multiply the critical value by the standard error.

5. Build and interpret the interval

Subtract and add the margin of error to the estimate, then describe the result as a plausible range for the parameter.

Keep three levers in mind as you work: a larger confidence level widens the interval, a larger sample size usually narrows it, and more variability in the data widens it. The main tradeoff is that more confidence usually costs precision.

A Full Worked Example: A 95% Interval

Suppose a sample of $64$ observations has mean $\bar{x} = 50$, and the population standard deviation is known to be $\sigma = 8$. Build a $95\%$ confidence interval for the population mean using a $z$ interval.

Start with

For a $95\%$ confidence level, use $z^* \approx 1.96$. Compute the standard error:

So the margin of error is

Build the interval:

which gives

A practical reading: if the model conditions are reasonable and the data come from this sampling process, values between $48.04$ and $51.96$ are plausible for the population mean.

Where Each Step Goes Wrong, And How To Check Yourself

The interpretation step is the one students most often misread. Saying there is a $95\%$ probability that the true parameter is in this specific interval is wrong in standard frequentist statistics: the parameter is fixed, and it is the procedure that has the long-run success rate. The safest reading is about the method. If the same sampling process were repeated many times and an interval were rebuilt the same way each time, about $95\%$ of those intervals would contain the true parameter.

The formula-choice step is the other trap. A $z$ interval, a $t$ interval, and a proportion interval do not use the same standard error, so check your conditions before picking one. A final self-check: confirm you are describing uncertainty in an estimate, not the spread of raw data. A confidence interval is about the parameter, not about where most individual observations fall.

To test your own command of the steps, try $\bar{x} = 72$, $\sigma = 10$, and $n = 100$ for a $95\%$ interval, then change only the sample size and watch the margin of error respond. That is one of the fastest ways to see why larger samples usually produce tighter intervals.

Why It Matters In Practice

Confidence intervals show up in polling, experiments, quality control, medicine, economics, and everyday data reporting. The interval is most informative when you compare it with a target or another estimate: a narrow interval supports a more precise conclusion than a wide one.