Confidence Interval Calculator
A confidence interval (CI) gives a range of plausible values for a population parameter based on a sample. A 95% CI means: if we repeated this study many times, 95% of the resulting intervals would contain the true population value.
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Tip: Increasing sample size narrows the CI (more precision). Doubling n reduces the margin of error by a factor of √2 ≈ 1.41.
- 1Calculate the sample mean (x̄) and standard deviation (s)
- 2Determine the z* critical value for your confidence level (95% → z* = 1.96)
- 3Calculate standard error: SE = s / √n
- 4Margin of error: ME = z* × SE
- 5CI = (x̄ − ME, x̄ + ME)
n=100, mean=50, SD=10, 95% CI=(48.04, 51.96)ME = 1.96 × 10/√100 = 1.96
n=30, mean=75, SD=15, 99% CI=(67.94, 82.06)ME = 2.576 × 15/√30 = 7.06
| Confidence Level | Z* Value | α (significance) |
|---|---|---|
| 80% | 1.282 | 0.20 |
| 90% | 1.645 | 0.10 |
| 95% | 1.960 | 0.05 |
| 99% | 2.576 | 0.01 |
| 99.9% | 3.291 | 0.001 |
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Fun Fact
A 95% confidence interval does NOT mean there is a 95% probability the true value is in that interval — it either is or it isn't. The 95% refers to the long-run frequency of the method.
References
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