Confidence Interval Calculator
Calculate the margin of error and confidence interval bounds for a sample mean, at any of the standard confidence levels.
Results from this calculator are estimates provided for general informational purposes only, based on formulas, rates, and standards commonly accepted as of 2026. Figures may differ slightly from other calculators or professional sources due to rounding methods, differing assumptions, or regional regulations, and rules may change over time. Always consult a qualified professional — such as a financial advisor, healthcare provider, or other relevant specialist — before making decisions based on these results.
Confidence Interval
19.7407 – 21.4593
Example
With a sample of 50, a mean of 20.6, and a standard deviation of 3.1, you can be 95% confident the true population mean falls between 19.741 and 21.459.
Margin of Error
± 0.8593 (4.17%)
Critical Z-Value
1.960
Standard Error
0.4384
Range Width
1.7186
Confidence region under the normal curve
What Is a Confidence Interval?
A confidence interval is a range of values, built around a sample estimate, within which the true population parameter likely falls. A 95% confidence interval doesn't mean there's a 95% chance the true mean is inside this specific interval — it means that if you repeated the same sampling process many times and built an interval each time, about 95% of those intervals would contain the true population mean.
The core formula is x̄ ± Z × (σ / √n), where x̄ is the sample mean,
Z is the critical value for your chosen confidence level, σ is the standard
deviation, and n is the sample size.
Common Critical Z-Values
| Confidence Level | Critical Z-Value |
|---|---|
| 80% | 1.282 |
| 90% | 1.645 |
| 95% | 1.960 |
| 98% | 2.326 |
| 99% | 2.576 |
Why Larger Samples Narrow the Interval
The margin of error shrinks with the square root of the sample size, so quadrupling your sample only halves the margin of error — a classic case of diminishing returns that explains why polls and studies often need very large samples to shrink an already-narrow interval further.
Confidence Level vs. Precision Tradeoff
Choosing a higher confidence level (say 99% instead of 90%) widens the interval, because you're demanding more certainty that the true value is captured. There's no free lunch: more confidence always costs precision unless you also increase the sample size.
When This Formula Applies
This z-based formula assumes the population standard deviation is known, or that the sample is large enough (typically n ≥ 30) for the sample standard deviation to be a reliable stand-in. For smaller samples with an unknown population standard deviation, a t-distribution-based interval is more appropriate.
Example — Your Current Inputs
With a sample of 50, a mean of 20.6, and a standard deviation of 3.1, you can be 95% confident the true population mean falls between 19.741 and 21.459.
Additional Example — Political Poll
A poll of 1,000 voters finds 52% support for a candidate. At 95% confidence with a standard deviation typical of proportion data, the margin of error works out to roughly ±3.1 points — the familiar "margin of error" figure reported alongside most published polls.
About These Parameters
- Sample Size, Mean & Standard Deviation
- These three numbers summarize your sample data — how many observations you collected, their average, and how spread out they are.
- Confidence Level
- How certain you want to be that the interval captures the true population value. 95% is the most common default across scientific research and polling.
Frequently Asked Questions
Does a 95% CI mean there's a 95% chance the true mean is in this range?
Not quite — the true mean either is or isn't in any specific interval. The 95% refers to the long-run reliability of the method: 95% of intervals built this way, across repeated sampling, would contain the true value.
Why did my interval get wider when I increased confidence?
A higher confidence level uses a larger critical z-value, which widens the margin of error — you're trading precision for certainty.
What if I don't know the population standard deviation?
For large samples, the sample standard deviation is a good enough substitute. For small samples, a t-distribution-based confidence interval — which uses slightly wider critical values — is more statistically appropriate.