Confidence Interval Calculator
Compute a confidence interval for a population mean or proportion. Enter your sample statistics, pick a confidence level, and get the margin of error and the lower and upper limits.
Compute a Confidence Interval
Confidence Interval Formula
Where:
- \\(\\bar{x}\) — sample mean (point estimate for the population mean)
- \\(\\hat{p}\) — sample proportion, entered as a decimal (e.g. 0.55 for 55%)
- \\(s\) — sample standard deviation
- \\(n\) — sample size (number of observations)
- \\(t_{\\alpha/2,\\,n-1}\) — critical value from Student’s t distribution with \\(n-1\) degrees of freedom (used when \\(s\) estimates an unknown \\(\\sigma\))
- \\(z_{\\alpha/2}\) — critical value from the standard normal distribution
- \\(\\alpha = 1 - \\text{confidence level}\) — significance level
The formula builds the interval as the point estimate plus or minus the margin of error, where the margin of error is a critical value multiplied by the standard error \\(SE = s/\\sqrt{n}\) (mean) or \\(SE = \\sqrt{\\hat{p}(1-\\hat{p})/n}\) (proportion).
Confidence Interval – Calculation Example
Example: a sample of 25 measurements has a mean of 100 and a standard deviation of 15. Find the 95% confidence interval for the population mean. Because \\(n < 30\) and the population SD is unknown, the t distribution with \\(n - 1 = 24\) degrees of freedom is used.
\\[ SE = \\frac{s}{\\sqrt{n}} = \\frac{15}{\\sqrt{25}} = 3 \]
\\[ t_{0.025,\\,24} \\approx 2.0639, \\quad ME = 2.0639 \\times 3 \\approx 6.1917 \]
\\[ \\text{CI} = 100 \\pm 6.1917 = (93.8083,\\; 106.1917) \]
Result: the 95% confidence interval is approximately (93.81, 106.19), with a margin of error of ±6.19. You can reproduce this with the calculator’s default values.
How to Use the Confidence Interval Calculator
- Choose the Estimate type: “Sample mean” for averages of measured data, or “Sample proportion” for percentages or counts of successes.
- Enter the Sample mean (x̄). For a proportion, type the value as a decimal between 0 and 1, e.g. 0.55 for 55%.
- If you selected a mean, enter the Sample standard deviation (s). This field is hidden for proportions.
- Enter the Sample size (n) — a whole number of at least 2.
- Select the Confidence level: 80%, 90%, 95%, 98%, 99% or 99.9%. 95% is the most common choice.
- Pick the Critical value method. The default “Automatic” mode uses Student’s t when \\(n < 30\) and the normal z value when \\(n \\ge 30\). Choose “Normal z” only when the population standard deviation \\(\\sigma\) is known.
- Read the results: Margin of error, Lower confidence limit, Upper confidence limit, Standard error, the Critical value used, and a details block with the full interval and interpretation.
What a Confidence Interval Means
A confidence interval is a range of plausible values for an unknown population parameter, such as a mean or a proportion, based on a single sample. The sample mean or sample proportion is only a point estimate — it is unlikely to hit the true value exactly. The interval surrounds that estimate with a margin of error that reflects how much the estimate could plausibly shift if you drew a different sample.
The confidence level describes the long-run behavior of the procedure, not the probability for one specific interval. If you repeated the sampling many times and computed an interval each time, about 95% of those intervals (at the 95% level) would contain the true population value. Once an interval is computed, it either contains the parameter or it does not; the uncertainty is about the method, not the single number pair.
The width of the interval carries most of the practical information. A narrow interval means the estimate is precise; a wide interval means the data leave considerable doubt. The width grows with the standard deviation and with the critical value (higher confidence → wider interval), and shrinks as the sample size grows — quadrupling \\(n\) roughly halves the margin of error, because \\(SE\) is divided by \\(\\sqrt{n}\).
Reference: Common Critical Values (Two-Tailed)
| Confidence level | z critical value | t (df = 24, for comparison) |
|---|---|---|
| 80% | 1.2816 | 1.3178 |
| 90% | 1.6449 | 1.7109 |
| 95% | 1.9600 | 2.0639 |
| 98% | 2.3263 | 2.4922 |
| 99% | 2.5758 | 2.7969 |
| 99.9% | 3.2905 | 3.7454 |
Assumptions, Limitations and Tips
- Random sampling. The interval is only valid for the population actually sampled. Non-random or convenience samples invalidate the coverage guarantee.
- t vs. z. When the population SD is unknown and estimated by \\(s\), Student’s t is the correct distribution; it accounts for the extra uncertainty in estimating \\(\\sigma\) and gives a wider, more honest interval, especially for small \\(n\). Use z only when \\(\\sigma\) is genuinely known or when \\(n\) is large enough that the difference is negligible.
- Normality. For small samples the mean-based interval assumes the underlying data are approximately normal. For \\(n \\ge 30\), the central limit theorem makes this assumption largely unnecessary unless the data are extremely skewed.
- Proportions. The Wald interval used for proportions relies on a normal approximation. It works well when \\(n\\hat{p} \\ge 5\) and \\(n(1-\\hat{p}) \\ge 5\); the calculator warns you when this condition fails. For small samples or proportions near 0 or 1, a Wilson score interval is more reliable.
- Higher confidence costs width. Moving from 95% to 99% widens the interval noticeably. Choose the level that matches how serious an error would be in your application, not simply the most impressive number.
- Don’t confuse precision with confidence. A 99% interval from 10 observations may be so wide it is practically useless. Increasing \\(n\), not the confidence level, is the way to get a narrow interval.
Frequently Asked Questions
How do I calculate a confidence interval by hand, step by step?
Follow five steps: (1) compute the point estimate — the sample mean \(\bar{x}\) or proportion \(\hat{p}\); (2) compute the standard error, \(s/\sqrt{n}\) or \(\sqrt{\hat{p}(1-\hat{p})/n}\); (3) look up the critical value — \(t_{\alpha/2,\,n-1}\) from a t-table, or \(z_{\alpha/2}\) = 1.96 at 95%; (4) multiply the critical value by the SE to get the margin of error; (5) add and subtract the ME from the estimate. Always round only at the very end.
What sample size do I need to get a specific margin of error?
Solve the margin-of-error formula for \(n\). For a mean: \(n = (z\,s / ME)^2\). For a proportion, the most conservative choice assumes \(\hat{p} = 0.5\): \(n = z^2 \cdot 0.25 / ME^2\). For example, a 95% interval for a proportion with ME = 0.03 needs \(1.96^2 \times 0.25 / 0.03^2 \approx 1068\) observations. Always round up to the next whole number, since a smaller sample cannot meet the target.
How do I find the critical value without a calculator or table?
Memorize the common two-tailed z values: 80% → 1.28, 90% → 1.64, 95% → 1.96, 98% → 2.33, 99% → 2.58. For Student's t with small samples, interpolate a t-table using degrees of freedom \(n-1\), or use the approximation \(t \approx z + (z^3+z)/(4(n-1))\) for a quick estimate. Note that 1.96 is for two-tailed intervals only; a one-tailed 95% z value is 1.64.
Does a 95% confidence interval mean there is a 95% probability the true value is inside it?
Not exactly, and this is the most common misinterpretation. The 95% describes the long-run success rate of the method: if you drew many samples and built an interval from each, about 95% of those intervals would capture the true parameter. Once a specific interval is computed, the true value is either in it or not — there is no repeated randomness left. Statisticians avoid attaching a probability to a single finished interval.
Can I enter a percentage like 55 instead of 0.55 for a proportion?
No — the calculator expects the proportion as a decimal strictly between 0 and 1, so enter 0.55 for 55%. The output, however, shows both forms: the limits are printed as decimals and as percentages. If you entered 55, the standard error would be thousands of times too large and the interval meaningless. A quick check: proportions must always fall between 0 and 1.
My data are skewed or contain outliers. Is the interval still valid?
For small samples (roughly \(n < 30\)), the t-based interval assumes the underlying data are approximately normal, and strong skew or outliers can distort it — consider a bootstrap interval or a median-based method instead. For larger samples, the central limit theorem makes the interval for the mean reasonably robust, though extreme outliers still inflate the standard deviation and widen the interval. Trimming data after seeing it generally invalidates the confidence guarantee.
What are the most common mistakes when computing confidence intervals?
The top errors: using z instead of t when the population SD is unknown and \(n\) is small (the interval is too narrow); dividing by \(n\) instead of \(\sqrt{n}\) when computing the standard error; using the standard deviation of the mean rather than of the individual observations; confusing the confidence level with the probability the parameter is in one interval; and applying the interval to a different population than the one sampled. Also double-check that \(n\) counts observations, not groups or percentages.