Confidence Interval Calculator

Related Calculators

Interval type: Chooses whether you are building an interval for a mean or for a proportion.

Confidence level (%): The long-run capture rate of the method, such as 95%. Higher levels usually make the interval wider.

Point estimate: The single sample-based estimate before uncertainty is added. It is the sample mean in mean mode or the sample proportion in proportion mode.

Critical value: The cutoff number from a probability distribution used to build the interval. Mean mode uses a t value; proportion mode uses a z value.

Standard error: How much the estimate would usually vary from sample to sample.

Margin of error: How far the interval extends from the center estimate on each side, or half-width for Wilson output.

Lower bound / Upper bound: The two ends of the confidence interval.

Proportion method: The rule used to build a proportion interval. Wilson score is usually more reliable than Wald normal approximation for small samples or extreme proportions [1].

Mistake: Using Mean from sample statistics when your data are really a count of yes/no outcomes.
Fix: Switch Interval type to Proportion from successes and total and fill in Number of successes and Total sample size (n) .

Mistake: Typing variance into Sample standard deviation .
Fix: Enter the standard deviation itself, not the variance. If you only have variance, take its square root before using Sample standard deviation .

Mistake: Entering a decimal or negative value for Sample size (n) or Total sample size (n) .
Fix: Count fields must be whole numbers. Use an integer at least 2 for mean mode and at least 1 for proportion mode.

Mistake: Making Number of successes larger than Total sample size (n) .
Fix: Keep Number of successes between 0 and the total sample size, inclusive.

Mistake: Expecting the same Margin of error behavior from every proportion method.
Fix: In Proportion method , remember that Wald normal approximation is symmetric around the estimate, while Wilson can be slightly asymmetric, so the displayed Margin of error is based on half the interval width.

Mistake: Thinking a higher Confidence level (%) should make the interval narrower.
Fix: Higher confidence uses a larger Critical value , which usually increases Margin of error and makes the interval wider.

• This calculator builds a one-sample confidence interval only. It does not compare two groups.
• In mean mode, the result uses a t interval with sample size minus 1 degrees of freedom. It assumes the sample is reasonably representative and that the sampling method is appropriate for a mean interval.
• Very small mean samples can produce unstable intervals. The math still works for Sample size (n) of 2 or more, but results may change a lot from one sample to another.
• In proportion mode, Number of successes must be between 0 and Total sample size (n), and displayed bounds are limited to the valid proportion range from 0 to 1.
• The Wald normal approximation can be weak when the sample is small or the sample proportion is near 0 or 1. Wilson score is the safer default in those cases [1][2].
• Rounding is for display only. Small differences from textbooks or software can happen if they round earlier or use a different t-value routine.
• Results depend on the numbers you enter. If your sample is biased, the interval can still be misleading even when the calculations are correct.

Core idea

A confidence interval starts with a sample estimate, then adds and subtracts an uncertainty amount based on the chosen confidence level.

Mean mode

When Interval type is mean, the calculator uses the sample mean, sample standard deviation, and sample size.

SE = s / sqrt(n)

CI = x_bar +/- t* x SE

Here, x_bar is the sample mean, s is the sample standard deviation, n is sample size, and t* is the t critical value with degrees of freedom n - 1. A t value is used because the population standard deviation is usually unknown in beginner one-sample mean problems [4].

Proportion mode

When Interval type is proportion, the calculator first finds the sample proportion.

p_hat = x / n

SE = sqrt(p_hat x (1 - p_hat) / n)

Here, x is Number of successes and n is Total sample size (n).

Wald normal approximation

If Proportion method is Wald, the interval is built with the usual z formula for a sample proportion [2].

CI = p_hat +/- z* x SE

This method is simple, but it can behave poorly for small samples or when p_hat is close to 0 or 1 [1][2].

Wilson score method

If Proportion method is Wilson score, the calculator uses the Wilson interval, which usually has better coverage than Wald in harder cases [1].

den = 1 + z^2 / n

center = (p_hat + z^2 / (2n)) / den

half = z x sqrt(p_hat(1 - p_hat)/n + z^2/(4n^2)) / den

CI = [center - half, center + half]

Because Wilson is not always perfectly symmetric around p_hat, the displayed Margin of error is based on half the interval width rather than simple plus-or-minus distance from p_hat.

Critical values used

For proportion mode, the calculator uses fixed two-sided z critical values for the listed confidence levels. For mean mode, it uses a t critical value based on the selected confidence level and n - 1 degrees of freedom.

Other outputs

margin of error = (upper bound - lower bound) / 2

interval width = upper bound - lower bound

Point estimate is the sample mean or sample proportion. Lower bound and Upper bound are the two ends of the interval. Method note explains which method was used and warns when Wald may be unreliable.

Mini example

Suppose you use proportion mode with 42 successes out of 100 at 95% confidence. Then p_hat = 42/100 = 0.42 and SE = sqrt(0.42 x 0.58 / 100) = 0.04936. With Wilson score, the interval is about 0.3282 to 0.5178, so the interval width is about 0.1896 and the displayed half-width margin is about 0.0948.

Assumptions behind the result

The formulas assume your sample is appropriate for the method and that observations are counted consistently. If the sample is biased, too small, or not representative, the interval may not reflect the real population as well as the confidence level suggests.