What a Confidence Interval Tells You
A Confidence Interval Calculator gives you a practical way to describe uncertainty. Instead of reporting only a single number (like an average or a percentage), a confidence interval provides a range that is consistent with your data at a chosen confidence level. The range is created from a point estimate plus/minus a margin of error. In reporting and decision-making, this is often more useful than a point estimate alone because it shows how precise your measurement is.
Confidence intervals appear everywhere: survey reporting (approval ratings), product analytics (conversion rates), quality control (defect rates), finance (average returns), healthcare (mean blood pressure), education (average scores), and operations (mean wait times). When people ask “How sure are we?”, a confidence interval is one of the cleanest answers.
Core Ingredients: Estimate, Standard Error, and Critical Value
Almost every confidence interval in this calculator follows the same structure:
Confidence Interval = estimate ± (critical value × standard error)
The estimate is your sample statistic (x̄ for a mean, p̂ for a proportion, or a difference like x̄₁ − x̄₂). The standard error measures how much the estimate tends to vary from sample to sample. The critical value depends on your confidence level and the distribution used (z or t). Multiply critical value by standard error and you get the margin of error.
Mean Confidence Interval
Use the Mean (t / z) tab when you want a confidence interval for a population mean based on a sample mean. If the population standard deviation is not known (typical), you use a t interval with degrees of freedom (df = n − 1). If the population standard deviation is truly known, you can use a z interval.
x̄ ± (critical × s / √n)
The key idea is that precision improves as sample size grows. Because standard error is s/√n, doubling your sample size does not halve the interval width, but it does reduce it. This is why large samples give tighter confidence intervals, and why small samples often produce wide ranges.
Proportion Confidence Interval
Use the Proportion (Wilson / Wald) tab when your outcome is a yes/no result, such as “purchase” vs “no purchase,” “defective” vs “non-defective,” or “supports candidate A” vs “does not support.” You enter successes (x) and sample size (n), and the tool computes p̂ = x/n.
This calculator includes two popular approaches:
- Wilson interval (recommended): Often more accurate, especially for smaller samples or extreme proportions near 0% or 100%.
- Wald interval: The simple textbook method (p̂ ± z√(p̂(1−p̂)/n)), but it can perform poorly in edge cases.
If you are unsure which to pick, keep Wilson selected. It is generally a safer default for real-world reporting.
Difference Between Two Means (Welch)
The Two Means (Welch) tab computes a confidence interval for the difference between two independent means: (x̄₁ − x̄₂). Welch’s method does not assume equal variances, which makes it widely used in A/B experiments, lab comparisons, and operational studies where spreads differ.
(x̄₁ − x̄₂) ± tdf × √(s₁²/n₁ + s₂²/n₂)
If the entire interval is above zero, it suggests group 1 is higher than group 2 at that confidence level. If the interval crosses zero, the data is consistent with no difference (though it does not “prove” equality).
Difference Between Two Proportions
The Two Proportions tab computes an interval for (p̂₁ − p̂₂). This is commonly used for conversion rates or pass/fail rates across two variants. You enter successes and totals for each group, and the tool returns the difference plus a confidence interval around it.
Interpreting this interval is straightforward: if the interval is entirely positive, group 1’s proportion is likely higher; if entirely negative, lower; and if it crosses zero, the evidence is not decisive at that confidence level.
How to Choose a Confidence Level
Confidence level controls how wide your interval is. Higher confidence means a wider interval because you are demanding more “coverage.” In practice:
- 90%: narrower intervals, faster conclusions, higher risk of missing the true value.
- 95%: common default for reporting and general analysis.
- 99%: wider intervals, more conservative decisions, useful for higher-stakes contexts.
Common Reporting Examples
A confidence interval is usually reported as a range. Here are typical ways it shows up in real work:
- Survey proportion: “Support is 48% ± 10% at 95% confidence” (or “95% CI: 38% to 58%”).
- Mean time: “Average handling time is 50 seconds (95% CI: 46 to 54).”
- Two-group difference: “Variant B improves conversion by 3.0 percentage points (95% CI: 0.6 to 5.4).”
Important Notes and Limitations
Confidence intervals rely on assumptions. These modes use standard approximations that work well in many practical scenarios, but results can be misleading if your data is heavily skewed, if observations are not independent, or if sampling is complex (clusters, weighting, stratification). For those cases, you may need specialized methods (bootstrap intervals, exact binomial intervals, survey-weighted intervals, or Bayesian credible intervals).
For proportions, small samples and extreme success rates are exactly where “simple” methods break down. That’s why this Confidence Interval Calculator offers Wilson intervals. For means, t intervals are a safer default when σ is unknown.
FAQ
Confidence Interval Calculator FAQs
Quick answers about confidence levels, margin of error, and choosing the right CI method.
A confidence interval is a range of plausible values for a population parameter (like a mean or proportion). It combines a point estimate with a margin of error based on a chosen confidence level.
95% is the most common default. Use 90% when you want a tighter interval and can accept more uncertainty. Use 99% for higher assurance (wider intervals).
Margin of error is the “plus/minus” amount around the estimate. It equals the critical value multiplied by the standard error.
Use a t interval when you estimate the standard deviation from the sample (most real cases). Use a z interval only when the population standard deviation is known or when your process is well-characterized.
Wilson is usually more accurate and behaves better for small samples or proportions near 0% or 100%. Wald is the simplest but can be unreliable in edge cases.
Welch’s method estimates the confidence interval for the difference between two means without assuming equal variances. It is a common, robust default.
Yes. For proportions, enter “successes” and “sample size”. The calculator converts them into a proportion estimate and returns the confidence interval.
It’s excellent for planning, reporting, and quick checks. For complex survey designs, weighted samples, clustering, or strict regulatory work, use specialized methods or consult an expert.