Calculate confidence intervals for a population mean
Z-interval: x̄ ± z·σ/√n. It assumes σ is known or the sample is reasonably large (n ≥ 30). For a small sample with an estimated standard deviation, a t-interval is wider and more accurate.
How it works
Produces a range that would contain the true population parameter in a stated share of repeated samples, typically 95%. It replaces a single point estimate with an honest statement of how much sampling noise surrounds it.
CI = x̄ ± z × (s / √n)
x̄sample meanssample standard deviationnsample sizezcritical value: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%
- Standard error: 12.4 / √100 = 1.24
- Margin of error: 1.96 × 1.24 = 2.43
- Interval: 68.5 ± 2.43
- The confidence applies to the procedure, not to one interval. Saying there is a 95% probability the true mean lies inside this specific range is the standard misreading.
- Width shrinks with the square root of n, so halving the interval takes four times the sample.
- Use the t-distribution when n is under about 30 or the population standard deviation is unknown. For n = 10, the 95% multiplier is 2.262 rather than 1.96.
- Two overlapping confidence intervals do not prove the difference is insignificant. Test the difference directly rather than eyeballing the overlap.
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