Confidence Interval Calculator

Calculate confidence intervals for means and proportions.

Interval Type

Sample Mean (x̄)

Sample Size (n)

Population Std Dev (σ)

Confidence Level

What is a Confidence Interval?

A confidence interval provides a range of plausible values for an unknown population parameter based on sample data. Instead of giving a single estimate, it quantifies the uncertainty in your estimate.

The interval has the form: point estimate ± margin of error

Confidence Level

The confidence level (commonly 90%, 95%, or 99%) represents how often the interval would contain the true parameter if we repeated the sampling process many times.

A 95% confidence interval does NOT mean there’s a 95% probability the parameter is in the interval. It means 95% of similarly constructed intervals would contain the parameter.

Types of Confidence Intervals

Z-Interval ($ \sigma $ known)

When the population standard deviation is known:

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

T-Interval ($ \sigma $ unknown)

When using the sample standard deviation:

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

The t-distribution has heavier tails, giving wider intervals to account for the extra uncertainty.

Proportion Interval

For estimating population proportions:

$$\hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Critical Values

Confidence	$ z^* $	$ t^* $ $df=29$
90%	1.645	1.699
95%	1.960	2.045
99%	2.576	2.756

Factors Affecting Interval Width

Sample Size

Larger samples give narrower intervals ($ \sqrt{n} $ in the denominator).

Confidence Level

Higher confidence requires wider intervals.

Variability

More variable data gives wider intervals.

Interpreting the Interval

Correct: “We are 95% confident that the true population mean is between 45 and 55.”
Incorrect: “There is a 95% probability that the population mean is between 45 and 55.”

The parameter is fixed (not random). The interval either contains it or doesn’t.