justyourcalculatorConfidence Interval Calculator
Get the range that likely holds the true mean or proportion.
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Confidence Interval Calculator
Confidence intervals for a mean or proportion
Use t when σ is estimated from a small sample.
95% confidence interval
48.040036 to 51.959964
50 ± 1.959964
- Margin of error
- ±1.95996398
- Standard error (s / √n)
- 1
- z critical value
- 1.959964
- Lower bound
- 48.04003602
- Upper bound
- 51.95996398
How it was calculated
- z for 95% = 1.959964
- SE = s / √n = 10 / √100 = 1
- E = 1.959964 × 1 = 1.95996398
- CI = 50 ± 1.95996398 = [48.04003602, 51.95996398]
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How the confidence interval calculator works
A confidence interval is a range built from a sample that, with a stated confidence level, contains the true population value. For a mean it's x̄ ± z × σ / √n, where σ / √n is the standard error and z is the critical value for the confidence level (1.96 for 95%).
When the standard deviation comes from a small sample, the t-distribution with n - 1 degrees of freedom gives a slightly wider, more honest interval. For large samples the two match.
For a proportion, the simple (Wald) interval is p̂ ± z √(p̂(1 - p̂) / n). The Wilson score interval is also shown because it behaves better when n is small or p̂ is near 0% or 100%.
Using and checking your result
Published by JustYourCalculator. Check the units and assumptions above, and compare a known example before relying on the output. Calculations use browser arithmetic and may round displayed values.
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