Confidence Interval — Formula, Calculation & Examples

A confidence interval is a range of plausible values for a population parameter, based on sample data. In many intro statistics problems, you build it as

$$\text{estimate} \pm \text{margin of error}$$

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.

What a confidence interval means in plain language

If you see a $95\%$ confidence interval, the safest interpretation is about the method, not a single finished interval. If the same sampling process were repeated many times and the interval were rebuilt the same way each time, about $95\%$ of those intervals would contain the true parameter.

So a confidence interval is a way to show uncertainty around an estimate. It gives a plausible range, not a guarantee.

Confidence interval formula

The general structure is

$$\text{estimate} \pm \text{critical value} \times \text{standard error}$$

For a population mean, two common versions are:

$$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$$

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

$$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$$

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 in many settings, but the standard error changes for means, proportions, and other parameters.

What changes the width of a confidence interval

Three levers matter most:

1. A larger confidence level makes the interval wider.
2. A larger sample size usually makes the interval narrower.
3. More variability in the data makes the interval wider.

That is the main tradeoff: more confidence usually costs you precision.

95% confidence interval example

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

$$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$$

For a $95\%$ confidence level, use $z^* \approx 1.96$.

Now compute the standard error:

$$\frac{\sigma}{\sqrt{n}} = \frac{8}{\sqrt{64}} = \frac{8}{8} = 1$$

So the margin of error is

$$1.96 \times 1 = 1.96$$

Build the interval:

$$50 \pm 1.96$$

which gives

$$(48.04,\ 51.96)$$

A practical reading is: 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.

Common mistakes with confidence intervals

One common mistake is saying there is a $95\%$ probability that the true parameter is in this specific interval. In standard frequentist statistics, the parameter is fixed and the interval procedure is what has the long-run success rate.

Another mistake is using the wrong formula without checking conditions. A $z$ interval, a $t$ interval, and a proportion interval do not use the same standard error.

Students also mix up a confidence interval for a parameter with the spread of raw data. A confidence interval is about uncertainty in an estimate, not about where most individual observations fall.

When confidence intervals are used

Confidence intervals show up in polling, experiments, quality control, medicine, economics, and everyday data reporting. They are useful whenever a sample is used to say something about a larger population.

In practice, the interval matters most when you compare it with a target or with another estimate. A narrow interval supports a more precise conclusion than a wide one.

Try a similar problem

Try your own version with $\bar{x} = 72$, $\sigma = 10$, and $n = 100$ for a $95\%$ confidence interval. Then change only the sample size and watch what happens to the margin of error. That is one of the fastest ways to build intuition for why larger samples usually produce tighter intervals.