Confidence Interval Calculator

StatisticsLast updated: August 17, 2026

A confidence interval (CI) is an interval estimate built around a point estimate computed from a sample (a mean or a proportion) that shows the plausible range for the true value in the population. A point estimate is a single number and hides the uncertainty that comes from sampling variation; a confidence interval reports that uncertainty openly. Saying "the mean is 24.5, 95% CI [23.18, 25.82]" instead of "the mean is 24.5" also tells the reader how precise the estimate is.

This calculator covers the two common scenarios. In mean mode, the population mean of a numeric variable is estimated from the sample mean using the t distribution; in proportion mode, the population percentage of a binary (yes/no) characteristic is computed with the normal approximation (the Wald method). You do not need raw data; the summary statistics from SPSS, Excel, or a published article (n, mean, standard deviation, or n and the number of events) are enough. APA 7 and most academic journals explicitly recommend reporting a 95% confidence interval alongside effect estimates.

The t distribution is used for the mean of a numeric variable; the normal approximation is used for yes/no proportions.
Number of observations or participants.
The sample standard deviation with the n−1 denominator.

Confidence Levels and Critical Values

Two things set the width of a confidence interval: the critical value of the chosen confidence level and the sample size. The z values below come from the standard normal distribution and are what the calculator uses in proportion mode.

90% confidence level
z = 1.645
α = 0.10; produces the narrowest interval and is preferred for rough, exploratory analysis
95% confidence level
z = 1.960
α = 0.05; the academic standard. APA 7 recommends reporting a 95% confidence interval alongside effect estimates.
99% confidence level
z = 2.576
α = 0.01; used in medicine and in high-stakes decisions. On the same data it produces an interval about 31% wider than the 95% interval.
Critical value in mean mode
t (df = n − 1)
Because the population standard deviation is unknown, t is used instead of z; for 95%, t = 2.064 at n = 25 and 1.984 at n = 100. As the sample grows, the t values approach z.
Margin of error vs. sample size
proportional to 1 / √n
As the sample grows the interval narrows, but the returns diminish: halving the interval requires roughly four times the sample.

Critical values derived from the standard normal distribution; they do not change over timeKaynak: Standard normal (z) distribution tables; the confidence interval reporting recommendation of the APA Publication Manual (7th edition)

Formula

Mean: CI = x̄ ± t(α/2; n−1) · s / √n
Proportion (Wald): CI = p̂ ± z(α/2) · √( p̂ (1 − p̂) / n )
Margin of error = critical value × standard error
Critical values (95%): t → depends on df (2.064 for n = 25); z → 1.960

The reading "the population value lies in this interval with 95% probability" is technically wrong: what is random is not the population value but the interval itself, which changes from sample to sample. The 95% is the long-run coverage rate of the method that produces the intervals.

How to Calculate

  1. Choose the calculation mode: the mean of a numeric variable, or a yes/no proportion?
  2. Set the confidence level — 95% is the standard choice in academic work.
  3. Enter the summary statistics: n, x̄, and s in mean mode; n and the number of events in proportion mode.
  4. Read the lower and upper bounds, the margin of error, and the critical value in the results section.
  5. Copy the statistically correct sentence from the interpretation line into your report.

Worked Examples

95% CI for a scale score mean

For a sample of 25 people with a mean of 24.5 and a standard deviation of 3.2, the standard error is 0.64, the t critical value is 2.064, and the margin of error is ±1.32; the 95% confidence interval is [23.18, 25.82].

Sample mean (x̄): 24.50 · 95% confidence interval: [23.18; 25.82] · Lower bound: 23.18

95% CI for a survey proportion

If 220 of 400 participants said "yes", the sample proportion is 55%; at the 95% confidence level the margin of error is ±4.88 points and the confidence interval runs from 50.12% to 59.88%.

Sample proportion (p̂): 55.00% · 95% confidence interval: 50.12% – 59.88% · Lower bound: 50.12%

A wider interval at the 99% level

For a sample of 100 observations with a mean of 50 and a standard deviation of 8, the 99% confidence interval is [47.90, 52.10] (t critical 2.626, margin of error ±2.10). As the confidence level rises, the critical value grows and the interval widens.

Sample mean (x̄): 50.00 · 99% confidence interval: [47.90; 52.10] · Lower bound: 47.90

Frequently Asked Questions

What does a 95% confidence interval actually mean?
The 95% expresses the long-run success rate of the method used: if samples of the same size were drawn from the same population over and over and an interval computed for each, about 95% of those intervals would contain the true population value. For the single interval in front of you, saying "the parameter is inside it with 95% probability" is not technically correct; the confidence belongs to the method, not to the interval.
How does a confidence interval relate to the p-value?
They are two presentations of the same information: if a 95% confidence interval excludes the null value (0 for a difference, or μ₀ for a mean), the two-tailed test gives p < 0.05; if it includes it, p ≥ 0.05. Because the confidence interval also tells you about the size of the effect and the precision of the estimate, reporting it alongside the p-value is recommended.
Why is the t distribution used for a mean rather than z?
The population standard deviation is almost never known, so the sample standard deviation is used in the calculation; the t distribution accounts for that extra uncertainty. In small samples the t distribution has heavier tails than the normal and produces wider, more realistic intervals. As the sample grows (n > 30), the t values approach the z values.
What can I do to narrow a confidence interval?
The width depends on three things: sample size, variability in the data, and the confidence level. The most effective route is to increase the sample; because the margin of error shrinks in proportion to √n, halving the interval requires roughly four times the sample. Lowering the confidence level narrows the interval too, but it raises the risk of being wrong.
When does the normal (Wald) approximation cause trouble in proportion mode?
The Wald interval falls below the intended coverage rate in small samples and for proportions close to 0 or 1, and it can even cross the 0–1 bounds. As a rule of thumb, both the number of events and the number of non-events should be at least 5 (preferably 10); this calculator warns you when that is violated. In extreme cases, methods such as the Wilson score interval should be preferred.
Which of the 90%, 95%, and 99% levels should I choose?
In the social sciences and in most academic work, 95% is the standard choice. 99% is used in medicine and high-stakes decisions to reduce the risk of non-coverage, but it produces a wider interval; 90% can be preferred for rough exploratory analysis. Always state in your report which level you chose.