Confidence Interval Calculator

A confidence interval (CI) is a range of values that estimates an unknown population parameter with a specified level of confidence. Rather than providing a single point estimate (like a sample mean or proportion), a confidence interval acknowledges uncertainty and gives a plausible range where the true parameter is likely to fall. The "confidence level" (typically 90%, 95%, or 99%) represents the long-run frequency with which such intervals would contain the true parameter if we repeated the sampling process many times.

It's crucial to understand that a 95% confidence interval does NOT mean "there's a 95% probability the true parameter is in this interval." Once the interval is calculated, the parameter either is or isn't in it—the probability is either 0 or 1. What 95% means is: if we constructed confidence intervals from 100 random samples using the same method, approximately 95 of those intervals would contain the true parameter. This is a statement about the long-run performance of the method, not about a single interval.

Confidence Interval for a Population Mean (σ Known - Z-Interval)

When the population standard deviation σ is known and the sample size is large (n ≥ 30), or the population is normally distributed for any n, use the z-interval formula:

Where x̄ is the sample mean, z_α/2 is the critical z-value (e.g., 1.96 for 95% confidence), σ is the population standard deviation, and n is the sample size. The term σ / √n is the standard error of the mean (SE), which measures the variability of the sample mean across different samples.

Confidence Interval for a Population Mean (σ Unknown - t-Interval)

In practice, the population standard deviation σ is rarely known. When you only have the sample standard deviation s, use the t-interval formula:

Where t_{α/2, df} is the critical t-value with degrees of freedom df = n - 1. The t-distribution accounts for the additional uncertainty from estimating σ with s. For small samples, the t-distribution has heavier tails than the normal distribution, resulting in wider confidence intervals. As n increases, the t-distribution approaches the normal distribution.

Confidence Interval for a Population Proportion (Wald Method)

For a single proportion p̂ = x / n, the Wald (normal approximation) confidence interval is:

This formula works well when n is large and p̂ is not too close to 0 or 1. A common rule of thumb is np̂ ≥ 10 and n(1 - p̂) ≥ 10. However, the Wald interval has known problems with small samples or extreme proportions—it can produce intervals outside [0, 1] or have poor coverage (the actual confidence level is lower than the nominal level).

Wilson Score Interval (Better for Proportions)

The Wilson score interval is a more reliable alternative to the Wald interval, especially for small samples or extreme proportions. It adjusts the center point and width of the interval to account for the discrete binomial distribution, resulting in better coverage properties. The Wilson interval never extends outside [0, 1] and provides more accurate confidence levels, particularly when p̂ is near 0, 0.5, or 1, or when n is small. Many statisticians recommend using Wilson by default for proportion intervals.

Confidence Intervals for Differences

Two-Sample Mean Difference (Pooled t-Interval): When comparing means from two independent samples with equal variances, use the pooled standard deviation and the formula:

Where df = n₁ + n₂ - 2 and SE_pooled combines the sample variances weighted by sample sizes. The pooled method assumes equal population variances (homoscedasticity), which should be verified with an F-test or Levene's test before use.

Two-Sample Mean Difference (Welch's t-Interval): When the assumption of equal variances is violated or uncertain, use Welch's t-interval, which does not pool variances and uses adjusted degrees of freedom (Welch-Satterthwaite approximation). Welch's method is more robust and is the default in many statistical software packages because it doesn't require the equal variances assumption.

Two-Sample Proportion Difference: For comparing two proportions p̂₁ and p̂₂, the confidence interval for the difference is:

This uses the Wald approach for differences. More sophisticated methods like the Wilson interval or exact methods (e.g., score interval) can be used for better coverage, especially with small samples or extreme proportions.

1. Choose a Calculation Mode: Select the type of confidence interval you need based on your data and research question:
   • Mean (σ known) — Z-Interval: Use when the population standard deviation is known (rare in practice) and the population is approximately normal or n ≥ 30.
   • Mean (σ unknown) — t-Interval: The most common choice for estimating a population mean. Use the sample standard deviation s and the t-distribution with df = n - 1.
   • Proportion — 1-Sample: Estimate a single population proportion (e.g., conversion rate, defect rate, approval rating). Choose between Wald (simple but less accurate) and Wilson (better coverage).
   • Mean Difference — 2-Sample: Compare means from two independent groups. Choose pooled (assumes equal variances) or Welch (robust, doesn't assume equal variances).
   • Proportion Difference — 2-Sample: Compare two proportions (e.g., A/B test, treatment vs control). Available in Wald or Wilson variants.
2. Set Confidence Level: Choose your desired confidence level:
   • 90%: Narrower interval, less conservative, appropriate for preliminary analyses or when precision is more important than certainty.
   • 95%: The most common choice in scientific research, balancing precision and confidence.
   • 99%: Wider interval, more conservative, appropriate for high-stakes decisions or when false positives are costly.
   • Custom: Enter any value between 50% and 99.99% for specialized applications.
3. Set Decimal Precision: Choose how many decimal places to display in results (0-6). Higher precision is useful for scientific reporting or when dealing with very small proportions or large sample sizes. For most practical purposes, 2-4 decimals are sufficient.
4. Enter Required Inputs: The inputs depend on your selected mode:
   • For means (1-sample): Sample size n, sample mean x̄, and sample standard deviation s (or population σ for z-interval).
   • For proportions (1-sample): Sample size n and number of successes x (the calculator computes p̂ = x / n).
   • For two-sample means: For each group: n, x̄, and s. If using pooled, both samples should have similar variances.
   • For two-sample proportions: For each group: n and x (successes).
   Ensure your data meet the assumptions for the chosen method (see Tips section).
5. Click Calculate: Once all inputs are entered, click the Calculate button. The calculator will display:
   • Point Estimate: The sample statistic (x̄, p̂, or difference) that serves as the center of the interval.
   • Standard Error (SE): A measure of the variability of the point estimate across different samples. Smaller SE means more precise estimates.
   • Critical Value: The z or t value corresponding to your confidence level and degrees of freedom (for t).
   • Margin of Error (ME): The half-width of the confidence interval, calculated as Critical Value × SE.
   • Confidence Interval: The range [Point Estimate - ME, Point Estimate + ME], typically displayed as [Lower, Upper].
   • Error-Bar Chart: A visual representation showing the point estimate with error bars extending to the confidence limits.
6. Interpret and Report Results: Use the confidence interval to make inferences about the population parameter. If the interval for a difference excludes 0, it suggests a statistically significant difference at the chosen confidence level (e.g., 95% CI excludes 0 is equivalent to p < 0.05 in a two-tailed test). Report both the point estimate and the CI in your results: "The mean difference was 5.2 units (95% CI: [2.1, 8.3])." Consider the practical significance of the interval width and whether it's narrow enough for your purposes.

• σ Known vs Unknown (Z vs t): In practice, the population standard deviation σ is almost never known. Even in well-studied populations, σ is estimated from data. Therefore, the t-interval is the appropriate choice for nearly all real-world mean estimation problems. Only use z-intervals when you have strong justification that σ is truly known from extensive historical data or theoretical considerations. When in doubt, use t—it's more conservative and automatically adjusts for the uncertainty of estimating σ.
• Small Samples (n < 30) Require t-Distribution: For small samples, the t-distribution accounts for the additional variability from estimating σ with s. The smaller the sample, the wider the t-interval relative to a z-interval. Additionally, small samples require stronger assumptions about the underlying population distribution—specifically, that the population is approximately normal. Check this with a histogram or Q-Q plot. If the data are heavily skewed or have outliers, consider data transformations, non-parametric methods (e.g., bootstrap), or robust statistics.
• Proportions Near 0 or 1 / Small n: Use Wilson: The Wald interval for proportions can fail badly when p̂ is close to 0 or 1, or when n is small. It can produce intervals that extend below 0 or above 1 (impossible for proportions), and the actual coverage is often much lower than the nominal confidence level. The Wilson score interval corrects these problems by adjusting both the center and width of the interval. It's more computationally complex but is the recommended default for proportion intervals. Use Wilson unless you have a very large n (say, n > 1000) and p̂ is well away from the extremes.
• Two-Sample Means: Pooled vs Welch: The pooled t-interval assumes equal population variances (σ₁² = σ₂²) and combines sample variances to get a single estimate. This can be more powerful (narrower interval) if the assumption is true, but it can be misleading if variances differ. Welch's t-interval does not assume equal variances and uses adjusted degrees of freedom. Because the equal variances assumption is often violated in practice and can be hard to verify with small samples, Welch is the safer default choice. Most modern statistical software (R, Python, SPSS) uses Welch by default for two-sample t-tests and intervals.
• Report Both ME and CI: The margin of error (ME) is the half-width of the confidence interval and represents the precision of your estimate. Reporting both the ME and the full CI provides different useful information: ME tells you "plus or minus how much," while the CI shows the plausible range. For example: "The mean customer satisfaction score was 7.2 ± 0.4 (ME), with a 95% CI of [6.8, 7.6]." This communicates both the point estimate precision and the range clearly.
• Check Assumptions Before Trusting Results: Confidence intervals rely on assumptions:
  • Independence: Observations must be independent (no clustering, matching, or repeated measures). Violations can severely underestimate SE and produce overly narrow intervals.
  • Representative Sampling: The sample should be randomly selected from the population of interest. Convenience samples or biased sampling invalidate interval estimates.
  • Normality (for means, small n): For small samples, the t-interval assumes the population is approximately normal. Check with histograms, Q-Q plots, or the Shapiro-Wilk test.
  • Binomial Conditions (for proportions): np ≥ 10 and n(1-p) ≥ 10 for Wald; Wilson relaxes this but still requires n to be reasonably large.
  If assumptions are violated, consider alternative methods like bootstrap confidence intervals, rank-based intervals, or Bayesian credible intervals.
• Use Sample Size Planning to Get Desired ME: If your confidence interval is too wide to be useful, you need a larger sample size. The relationship is: ME ∝ 1/√n. To cut the ME in half, you need 4 times the sample size. Use the Sample Size & Power calculator to plan studies prospectively: specify your target ME (or equivalently, target CI width) and desired confidence level, and it will tell you the required n. This is crucial for designing studies that will produce actionable results.

Point Estimate

The point estimate is your best single-value estimate of the population parameter based on the sample data. For a single mean, it's x̄. For a proportion, it's p̂ = x / n. For differences, it's x̄₁ - x̄₂ or p̂₁ - p̂₂. The point estimate serves as the center of the confidence interval. While it's our best guess, it's subject to sampling variability—if we took a different sample, we'd get a different point estimate. The confidence interval quantifies this uncertainty.

Standard Error (SE)

The standard error measures the variability of the point estimate across different samples. It's the standard deviation of the sampling distribution of the estimator. For a mean: SE = s / √n (or σ / √n if σ is known). For a proportion: SE = √(p̂(1-p̂) / n). Smaller SE means the estimate is more precise—repeated samples would give point estimates clustered more tightly around the true parameter. SE decreases as sample size increases, following a 1/√n relationship. The SE is the key input to calculating the margin of error.

Critical Value

The critical value is the z or t multiplier corresponding to your chosen confidence level. For a 95% z-interval, it's 1.96. For a 95% t-interval with df = 20, it's approximately 2.086. The critical value determines how many standard errors you extend on each side of the point estimate to form the interval. Higher confidence levels require larger critical values (e.g., 99% uses 2.576 for z, wider than 95% at 1.96), resulting in wider intervals. The t critical value depends on degrees of freedom and converges to the z value as df increases.

Margin of Error (ME)

The margin of error is the half-width of the confidence interval, calculated as ME = Critical Value × SE. It represents the maximum expected difference between the point estimate and the true parameter at the chosen confidence level. For example, if x̄ = 50 and ME = 3, you report "50 ± 3" or [47, 53]. The ME combines the precision of your estimate (SE) and the confidence you want (critical value). To reduce ME, you can either increase sample size (reducing SE) or accept lower confidence (reducing the critical value). Reporting the ME alongside the point estimate is a clear way to communicate uncertainty.

Confidence Interval [Lower, Upper]

The confidence interval is the range [Point Estimate - ME, Point Estimate + ME], providing a plausible set of values for the true population parameter. A 95% CI means that if we repeated the sampling and interval construction process many times, 95% of the resulting intervals would contain the true parameter. It does NOT mean the parameter has a 95% probability of being in this specific interval—once calculated, the parameter is either in it or not. Use the CI to assess both statistical significance (does it exclude a null value like 0 or 1?) and practical significance (is the range narrow enough to inform decisions?).

Error-Bar Chart

The error-bar chart provides a visual representation of the confidence interval. The central point represents the point estimate, and the error bars extend to the lower and upper confidence limits. This makes it easy to visually assess:

• The precision of the estimate (shorter bars = more precise)
• Whether the interval excludes a reference value (e.g., does a difference interval exclude 0?)
• Comparisons between multiple groups (if comparing error bars from different samples)

When comparing two error bars, overlapping bars do NOT necessarily mean no significant difference—formal statistical tests or explicit difference intervals are needed for proper inference.

Interpretation for Differences

When constructing a CI for a difference (mean₁ - mean₂ or prop₁ - prop₂), special interpretation applies:

• If the CI excludes 0: There is a statistically significant difference at the (1 - confidence level) significance level. For example, a 95% CI excluding 0 corresponds to p < 0.05 in a two-tailed test.
• If the CI includes 0: The difference is not statistically significant at that level. However, the width of the interval still provides useful information about the precision of the estimate.
• Practical vs Statistical Significance: Even if 0 is excluded, check whether the interval range is practically meaningful. A CI of [0.1, 0.5] might be statistically significant but too small to matter in practice.