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.
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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 confidence interval provides a range of plausible values for an unknown population parameter based on sample data. Unlike a single point estimate, a confidence interval acknowledges the inherent uncertainty in sampling and quantifies how precise the estimate is likely to be.
The width of a confidence interval depends on three factors: the confidence level (higher = wider), the variability in the data (more spread = wider), and the sample size (larger = narrower). The 95% confidence level is the most commonly used standard in scientific research.
Our confidence interval calculator supports both mean and proportion intervals. Simply enter your sample statistics and desired confidence level to get the interval bounds, margin of error, z-critical value, and a clear interpretation of the results.
Last Updated: Mar 9, 2026
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