Confidence Interval Calculator
Calculate confidence intervals for sample means and proportions. Estimate KPI ranges, survey margins of error, and conversion rate bounds at 90%, 95%, or 99% confidence.
Calculate confidence intervals for KPI estimates, survey results, and conversion rates. Choose between mean (continuous data) or proportion (yes/no, success/failure) modes.
Formula
CI for Mean: x̄ ± (Z × σ/√n) | CI for Proportion: p ± (Z × √[p(1−p)/n])
A confidence interval provides a range of values that likely contains the true population parameter (mean or proportion) with a specified level of confidence. The margin of error increases with higher confidence levels and decreases with larger sample sizes.
Worked Example: Survey Confidence Interval
NPS survey of 500 respondents. Mean NPS score: 42, Standard Deviation: 15. Find the 95% confidence interval.
Standard Error (SE) = 15 / √500 = 15 / 22.36 = 0.671
Z-score for 95% = 1.96
Margin of Error = 1.96 × 0.671 = 1.315
Lower Bound = 42 − 1.315 = 40.685
Upper Bound = 42 + 1.315 = 43.315
Interpretation: We are 95% confident the true NPS of your customer base lies between 40.7 and 43.3. A higher confidence level (99%) or larger margin of error can be achieved with a larger sample.
Frequently Asked Questions
Use this in your workflow
After calculating a confidence interval, use the Sample Size Calculator to determine how large your next survey should be for a target margin of error. Check the Standard Deviation Calculator if you need to compute SD from raw data. Browse all Free Business Calculators.
Worked example: NPS confidence interval
A useful starting point before entering your own figures above.
| Metric | Value |
|---|---|
| Survey type | NPS survey |
| Sample size | 500 respondents |
| Sample mean NPS | 42 |
| Standard deviation | 15 |
| Confidence level | 95% |
| Standard error | 0.671 |
| Margin of error | ±1.32 |
| 95% confidence interval | [40.68, 43.32] |
Interpretation: Based on 500 responses with a mean NPS of 42, we are 95% confident the true NPS of your entire customer base lies between 40.68 and 43.32. This margin of error reflects sampling variability — it would be tighter with a larger sample or narrower with lower confidence.
Worked example: Conversion rate confidence interval
Proportion confidence interval for A/B test results.
| Metric | Value |
|---|---|
| Experiment type | Website A/B test |
| Sample size | 2,000 visitors |
| Conversions | 180 (9%) |
| Sample proportion | 0.09 or 9% |
| Confidence level | 95% |
| Standard error | 0.0065 |
| Margin of error | ±1.27% (±0.0127) |
| 95% confidence interval | [7.73%, 10.27%] |
Interpretation: Your sample conversion rate was 9%, but the 95% confidence interval is [7.73%, 10.27%]. This is the plausible range for the true conversion rate. A larger sample (e.g., 5,000 visitors) would shrink the interval; a smaller sample would widen it.
Limitations & Assumptions
This calculator assumes your sample is random and representative, and uses the normal approximation (valid for large samples). For small samples (n < 30) with unknown population SD, use a t-distribution instead. Confidence intervals do not account for bias, data quality issues, or non-response. These results are for estimation and planning only — not statistical inference or hypothesis testing.
Common mistakes when calculating confidence intervals
Confusing margin of error with statistical significance
A confidence interval estimates the range of the true parameter. It is NOT a hypothesis test. A narrow 95% CI for a conversion rate of [8%, 10%] tells you where the true rate likely is — it does not tell you whether a treatment works or which variant is better.
Using the wrong formula for means vs. proportions
Mean CIs use the standard deviation: CI = x̄ ± (Z × σ/√n). Proportion CIs do not use a pre-computed SD; they compute it from the proportion itself: CI = p ± (Z × √[p(1−p)/n]). Using the wrong formula gives nonsensical results.
Misinterpreting the confidence level
A 95% confidence interval does NOT mean there is a 95% probability the true value is in the interval. It means that 95% of intervals calculated this way (from repeated samples) would contain the true value. For any given interval, the parameter is or is not in it — probability does not apply.
Using a small sample without adjusting for the t-distribution
For small samples (n < 30) with unknown population SD, use the t-distribution instead of the normal (Z) distribution. The t-distribution has heavier tails, producing wider intervals to account for extra uncertainty. Using Z-scores with small samples produces intervals that are too narrow.
Computing CI from non-random or biased samples
A confidence interval is only valid if the sample is random and representative. If your survey is biased (e.g., only online responses, self-selected participants), a narrow CI gives false precision — it says nothing about the bias. Always assess your sampling method.
When to use this calculator
- →Reporting survey results with a margin of error (e.g., "42 ± 1.3 with 95% confidence")
- →Estimating the range for KPIs like NPS, CSAT, or conversion rate
- →Determining whether your A/B test sample size is large enough
- →Building confidence ranges for business forecasts and dashboards
- →Planning the next survey — use the result with the Sample Size Calculator
Frequently asked questions
What is a confidence interval and when do I use it?
A confidence interval is a range of values that likely contains the true population parameter (mean or proportion). Use it to estimate KPI ranges, survey margins of error, conversion rate bounds, and A/B test results. Instead of reporting a single point estimate, report the range you are confident contains the truth.
What is the difference between mean and proportion confidence intervals?
Mean CI applies to continuous data (e.g., NPS scores, transaction values). It uses standard deviation: CI = x̄ ± (Z × σ/√n). Proportion CI applies to yes/no data (e.g., conversion, churn). It uses the proportion: CI = p ± (Z × √[p(1−p)/n]).
What does "95% confidence level" mean?
A 95% CI means: if you repeated your survey 100 times, approximately 95 of the intervals would contain the true population parameter. It does NOT mean there is a 95% chance the true value is in this particular interval.
Why does larger sample size shrink the confidence interval?
Larger samples reduce sampling variability. The standard error shrinks by √n, so doubling the sample size reduces the margin of error by ~29%, and quadrupling it cuts the margin in half. This is why larger surveys are more expensive but give tighter, more credible ranges.
When should I use 90%, 95%, or 99% confidence level?
Use 90% for exploratory work where a wider margin is acceptable. Use 95% as the standard for business reporting, surveys, and A/B tests. Use 99% only when you need very high confidence and can afford a much wider margin, such as medical decisions.
Can I use this for A/B tests?
Yes, use the proportion CI for yes/no outcomes (conversion, click, purchase). Enter the sample size and the observed proportion. The resulting CI shows the plausible range for the true conversion rate. Larger sample sizes give narrower intervals; compare treatment intervals to decide if one is clearly better.