Confidence Interval Calculator
Calculate the margin of error and confidence interval bounds for a sample mean, at any of the standard confidence levels.
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.