95% Confidence Interval
70.71 – 79.29
MOE
±4.2941
z*
1.960
70.71
79.29
±4.2941
1.9600
2.1909
We are 95% confident the true population mean lies between 70.71 and 79.29.
CI = x̅ ± z* × (σ / √n)Calculates the confidence interval for a population mean when the population standard deviation is known (or approximated). The margin of error is z* times the standard error.
Where:
x̅= Sample mean (center of the interval)z*= Z-critical value for the chosen confidence levelσ= Population (or sample) standard deviationn= Sample sizeCI = p̂ ± z* × √(p̂(1 - p̂) / n)Calculates the confidence interval for a population proportion using the normal approximation. Valid when np̂ and n(1-p̂) are both at least 10.
Where:
p̂= Sample proportion (observed success rate)z*= Z-critical value for the chosen confidence leveln= Sample sizeSE = σ / √nThe standard error measures the typical distance between a sample mean and the true population mean. It decreases as sample size increases.
Where:
σ= Population standard deviationn= Sample sizeInputs
Result
With a sample mean of 75, SD of 12, and n=30, the 95% CI is 70.71 to 79.29. We are 95% confident the true population mean exam score falls in this range.
Inputs
Result
In a survey of 200 people where 60% said yes, the 95% CI for the true proportion is 53.21% to 66.79%. The margin of error is ±6.79 percentage points.
Inputs
Result
With a mean of 98.2°F, SD of 0.7, and 50 readings, the 99% CI is 97.95 to 98.45. The high confidence level widens the interval slightly compared to a 95% CI.
A confidence interval is a range of values that is likely to contain the true population parameter. For example, a 95% confidence interval means that if you repeated the sampling process 100 times, approximately 95 of those intervals would contain the true population value.
| Confidence Level | Z-Critical | Width |
|---|---|---|
| 90% | 1.645 | Narrowest |
| 95% | 1.960 | Moderate |
| 99% | 2.576 | Widest |
For a population mean with known standard deviation, use CI = x̅ ± z*(σ/√n). The sample mean x̅ is the center, z* is the critical value for your confidence level, σ is the standard deviation, and n is the sample size.
| Step | Formula | Example Value |
|---|---|---|
| Standard Error | σ/√n | 12/√30 = 2.19 |
| Margin of Error | z* × SE | 1.96 × 2.19 = 4.29 |
| Lower Bound | x̅ - MOE | 75 - 4.29 = 70.71 |
| Upper Bound | x̅ + MOE | 75 + 4.29 = 79.29 |
Higher confidence levels produce wider intervals. A 90% CI is the narrowest and least certain, a 95% CI is the standard in most research, and a 99% CI is the widest and most conservative. The trade-off is always between precision (narrow interval) and confidence (high certainty).
| Confidence | z* | MOE (if SE=1) |
|---|---|---|
| 90% | 1.645 | ±1.645 |
| 95% | 1.960 | ±1.960 |
| 99% | 2.576 | ±2.576 |
For proportions, the standard error formula changes to SE = √(p̂(1-p̂)/n), where p̂ is the sample proportion. The interval is then p̂ ± z*×SE. Proportions are bounded between 0 and 1, so the interval is clamped to this range.
| Parameter | Mean CI | Proportion CI |
|---|---|---|
| Center | x̅ (sample mean) | p̂ (sample proportion) |
| Standard Error | σ/√n | √(p̂(1-p̂)/n) |
| Typical use | Continuous data | Binary/yes-no data |
Increasing the sample size narrows the confidence interval because the standard error decreases proportionally to 1/√n. To cut the margin of error in half, you need to quadruple the sample size. This is why large studies produce more precise estimates.
| Sample Size | SE Factor | Relative MOE |
|---|---|---|
| n = 25 | 1/√25 = 0.200 | 100% |
| n = 100 | 1/√100 = 0.100 | 50% |
| n = 400 | 1/√400 = 0.050 | 25% |
| n = 1600 | 1/√1600 = 0.025 | 12.5% |
A 95% confidence interval does not mean there is a 95% chance the true value falls within the interval. The true parameter is fixed — it either is or is not in the interval. The correct interpretation: if you repeated the sampling process 100 times and computed a 95% CI each time, approximately 95 of those 100 intervals would contain the true population parameter. The remaining 5 would not.
For a concrete example: a survey of 200 people finds 60% support a policy. The 95% CI is 53.2% to 66.8%. This means the sampling method produces intervals that capture the true population proportion 95% of the time. The margin of error (±6.8 percentage points) quantifies the precision of the estimate. Doubling the sample size to 400 would reduce the margin of error to ±4.8% — a 29% improvement in precision for 2× the data collection effort.
The confidence level is a property of the method, not the interval. "We are 95% confident" means the procedure captures the true value 95% of the time, not that this specific interval has a 95% probability of being correct.
Confidence intervals for means and proportions use different standard error formulas. For a mean: SE = σ/√n, where σ is the standard deviation and n is the sample size. For a proportion: SE = √(p̂(1–p̂)/n), where p̂ is the sample proportion. Both intervals have the same structure: Point Estimate ± z* × SE, where z* is the critical value (1.645 for 90%, 1.960 for 95%, 2.576 for 99%).
A key difference: proportion standard error is maximized at p̂ = 0.5 (maximum uncertainty). A survey result of 50% ± 7% is wider than 80% ± 5.5% from the same sample size because 50/50 splits have the highest inherent uncertainty. The normal approximation for proportions requires np̂ ≥ 10 and n(1–p̂) ≥ 10 to be valid. For small samples or extreme proportions, use the Wilson interval (this calculator uses the normal approximation for simplicity).
| Parameter | Standard Error | Example (n=100) | z*=1.96 MOE |
|---|---|---|---|
| Mean (σ=15) | σ/√n | 15/√100 = 1.50 | ±2.94 |
| Proportion (p=0.5) | √(p(1-p)/n) | √(0.25/100) = 0.05 | ±9.80% |
| Proportion (p=0.8) | √(p(1-p)/n) | √(0.16/100) = 0.04 | ±7.84% |
The standard error is proportional to 1/√n, which means you must quadruple the sample size to cut the margin of error in half. Going from n=25 (SE factor 0.200) to n=100 (SE factor 0.100) halves the MOE. But going from n=100 to n=400 is needed to halve it again — diminishing returns that make very large samples expensive relative to the precision gained.
In practical terms: a national political poll with n=1,000 has a margin of error of approximately ±3.1% at 95% confidence. Doubling to n=2,000 reduces it to ±2.2%. Quadrupling to n=4,000 gets ±1.5%. For most business and research applications, n=300–500 provides sufficient precision (MOE of ±4–6%) at reasonable cost. Use the Statistics Calculator to compute the standard deviation from your raw data before feeding it into this confidence interval tool.
| Sample Size | SE Factor | Relative MOE | Typical Application |
|---|---|---|---|
| n = 30 | 0.183 | 100% | Small pilot study |
| n = 100 | 0.100 | 55% | Classroom research |
| n = 400 | 0.050 | 27% | Business survey |
| n = 1,000 | 0.032 | 17% | National poll |
| n = 10,000 | 0.010 | 5.5% | Large clinical trial |
The choice of confidence level balances certainty against precision. A 99% CI is approximately 32% wider than a 95% CI for the same data (z*=2.576 vs. 1.960). Use 90% for exploratory analysis where precision matters more than certainty. Use 95% as the default for published research (the universal standard). Use 99% when the cost of being wrong is high — pharmaceutical safety testing, engineering tolerances, or financial risk assessment.
In practice, the confidence level should be chosen before data collection, not after. Selecting the level that produces the "best" result is a form of p-hacking. If a 95% CI just barely excludes a critical value but a 99% CI includes it, the honest conclusion is that the evidence is suggestive but not definitive — not that the 95% level proves the point. Report the CI bounds and let the reader evaluate the practical significance.
Last Updated: Mar 26, 2026
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