Calculate 90%, 95%, and 99% confidence intervals for population means and proportions. Includes margin of error, standard error, critical values, and plain-English interpretation.
A confidence interval (CI) is a range of values calculated from sample data that is likely to contain the true population parameter with a specified level of confidence. For example, a 95% confidence interval for a mean states: "If we were to repeat this sampling process 100 times, approximately 95 of those intervals would contain the true population mean."
The key components are: a point estimate (the sample statistic), a margin of error (how far the interval extends in each direction), and a confidence level (typically 90%, 95%, or 99%).
| Confidence Level | Ξ± (alpha) | Ξ±/2 | Z* critical value | Use case |
|---|---|---|---|---|
| 90% | 0.10 | 0.05 | 1.645 | Less precise, smaller interval |
| 95% | 0.05 | 0.025 | 1.960 | Most common in research |
| 99% | 0.01 | 0.005 | 2.576 | High precision required |
The margin of error is half the width of the confidence interval: MOE = z* Γ SE, where SE is the standard error. The standard error of the mean is SE = s/βn. For proportions: SE = β(pΜ(1βpΜ)/n).
Reducing the margin of error requires either increasing the sample size or lowering the confidence level. Doubling the precision (halving the MOE) requires quadrupling the sample size because of the square root relationship.
When the population standard deviation is known, or when the sample size is large (n β₯ 30), the z-distribution is used. When the population SD is unknown and the sample is small (n < 30), the t-distribution is more appropriate because it has heavier tails, reflecting greater uncertainty about the population SD.
This calculator automatically applies the z-distribution for simplicity with a note that for small samples (n < 30), a t-distribution would give slightly wider intervals. The difference is minimal for n β₯ 30.
US political polls typically use a sample of approximately 1,000 likely voters. With n = 1,000 at 95% confidence, the margin of error for a proportion near 50% is approximately Β±3.1 percentage points. News outlets routinely report this alongside poll results, though it is often misunderstood by the public.
For the 2024 US presidential election, most reputable polls carried a Β±3β4% margin of error. A result showing Candidate A at 51% and Candidate B at 49% with a Β±3% MOE means the race is effectively statistically tied.
In the UK, opinion polls regulated by the British Polling Council (BPC) typically survey 1,000β2,000 people. The standard reporting convention requires disclosure of the margin of error and methodology. UK general election forecasters such as Electoral Calculus combine multiple polls using meta-analysis techniques to reduce uncertainty.
Clinical trials frequently report 95% confidence intervals alongside p-values. A drug study might report: "The treatment reduced blood pressure by a mean of 8.2 mmHg (95% CI: 6.1 to 10.3 mmHg)." This tells clinicians both the magnitude of the effect and the uncertainty around it.
Importantly, a confidence interval that does not cross zero (for a difference) or 1.0 (for a ratio/odds ratio) is considered statistically significant at the corresponding alpha level. This is equivalent to a p-value < 0.05 for a 95% CI.
In digital marketing and product development, A/B tests compare conversion rates between two versions of a webpage or feature. A confidence interval for the difference in proportions tells you whether the observed difference is statistically meaningful or likely due to random variation.
For example, if Version A converts at 4.2% and Version B at 5.1% with n = 2,000 in each group, the 95% CI for the difference might be [0.1%, 1.7%], suggesting a genuine improvement.
The width of a confidence interval decreases as sample size increases β but with diminishing returns. The margin of error is proportional to 1/βn. Increasing from n = 100 to n = 400 (four times larger) halves the margin of error. This is why there are diminishing returns to very large samples for practical purposes.
| Sample Size (n) | MOE for p=50%, 95% CI | CI Width |
|---|---|---|
| 50 | Β±13.9% | 27.8% |
| 100 | Β±9.8% | 19.6% |
| 400 | Β±4.9% | 9.8% |
| 1,000 | Β±3.1% | 6.2% |
| 2,000 | Β±2.2% | 4.4% |
| 10,000 | Β±1.0% | 2.0% |
It means that if you repeated the sampling process many times and constructed a 95% CI each time, approximately 95% of those intervals would contain the true population parameter. It does not mean there is a 95% probability the true value is in any specific interval.
A 99% CI is wider than a 95% CI because it requires greater certainty. The critical z-value increases from 1.96 to 2.576, making the interval about 31% wider. You gain more confidence but sacrifice precision (narrower interval).
Larger samples produce narrower CIs because the standard error (SE = s/βn) decreases. To halve the width of the interval, you must quadruple the sample size, as MOE is proportional to 1/βn.
Use the t-distribution when the population standard deviation is unknown and the sample size is small (n < 30). The t-distribution has heavier tails, producing wider intervals that account for the additional uncertainty. For n β₯ 30, z and t give very similar results.
The margin of error (MOE) is the maximum expected difference between the sample result and the true population value at a given confidence level. For a poll of n = 1,000 at 95% confidence with p near 50%, MOE β Β±1.96 Γ β(0.5Γ0.5/1000) β Β±3.1%.
This calculator provides CI estimates based on standard formulas and is appropriate for educational purposes. For published research, consult a biostatistician and use validated statistical software such as R, Stata, SPSS, or SAS to ensure correct assumptions (normality, independence) are met.
The standard error (SE) measures how much the sample mean (or proportion) is expected to vary from the true population value across repeated samples. SE = s/βn for means, and SE = β(pΜ(1βpΜ)/n) for proportions. A smaller SE means a more precise estimate.
If a proportion CI for survey approval is [0.42, 0.56], you can be 95% confident the true population approval rating falls between 42% and 56%. This is wider than typical news reporting, which often just states the midpoint (49%) without the full interval context.
Results are for educational purposes only. This calculator uses z-distribution approximations. For small samples (n < 30) or non-normal distributions, consult a qualified statistician.