Confidence Interval Calculator

A confidence interval (CI) is a range of plausible values for an unknown population parameter, constructed from sample data. A 95% confidence interval means: if you repeated the sampling procedure many times, about 95% of the constructed intervals would contain the true parameter.

Important: the 95% refers to the procedure, not to any single computed interval. Once an interval is constructed from data, it either does or doesn't contain the true parameter — but we don't know which.

Core structure: every confidence interval has the form

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

The estimate is the sample statistic ($\bar{x}$ or $\hat{p}$). The margin of error is a critical value times the standard error of the estimate.

Confidence intervals appear in:

• Election polling ('52% support, $\pm 3\%$ margin of error')
• Medical studies (effect size CIs)
• Quality control (mean defect rates)
• Any time you want to quantify uncertainty in an estimate, not just report a point value.

CI for a Population Mean (Z-Interval)

When the population standard deviation $\sigma$ is known and the sampling distribution is approximately normal (large $n$ or normal population):

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

where $z^*$ is the critical value for the chosen confidence level.

CI for a Population Mean (T-Interval)

When $\sigma$ is unknown (you only have $s$, the sample standard deviation) — much more common in practice:

$\bar{x} \pm t^*_{n-1} \cdot \frac{s}{\sqrt{n}}$

The critical value $t^*$ comes from the t-distribution with $n - 1$ degrees of freedom. For large $n$ ($\geq 30$), $t^* \approx z^*$ and the two intervals are very similar.

CI for a Population Proportion

For a sample proportion $\hat{p} = x/n$ (where $x$ is the number of successes):

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

Valid when $n\hat{p} \geq 10$ and $n(1 - \hat{p}) \geq 10$ (success-failure condition).

Critical Values

Margin of Error

$\text{ME} = (\text{critical value}) \times (\text{standard error})$

Increasing the sample size $n$ decreases the standard error (and therefore the margin of error) by a factor of $\sqrt{n}$. Quadrupling $n$ halves the margin of error.

Choosing Confidence Level

• Higher confidence = wider interval. A 99% CI is wider than a 95% CI, which is wider than a 90% CI.
• 95% is the default in most academic and professional contexts.
• 99% when stakes are higher (medical, safety); 90% when a tighter point estimate matters more than coverage.

• Misinterpreting the 95%: 'There is a 95% probability the true mean is in this interval' is wrong (frequentist). The correct statement is about the procedure: 95% of similarly constructed intervals contain the true parameter.
• Using z when t is appropriate: with unknown $\sigma$, use $t^*$. Using $z^*$ understates uncertainty, especially for small $n$.
• Forgetting $\sqrt{n}$ in the standard error: $\sigma/\sqrt{n}$, not $\sigma/n$.
• Wrong critical value direction: $z^* = 1.96$ for 95% (two-tailed), not 95th-percentile $z = 1.645$. The two-tailed critical value cuts off $\alpha/2$ in each tail.
• Skipping the success-failure condition for proportions: if $n\hat{p}$ or $n(1-\hat{p}) < 10$, the normal approximation breaks down — use an exact (Clopper-Pearson) or score-based interval.
• Conflating CI with prediction interval: a 95% CI estimates the mean with 95% coverage. A prediction interval estimates a single future observation — much wider.