Calculator Form
Use summary statistics or raw sample values. The calculator assumes the population standard deviation is known.
Example Data Table
| Scenario | Sample Size | Sample Mean | Known Sigma | Confidence | Approximate Interval |
|---|---|---|---|---|---|
| Machine fill weights | 36 | 52.0 | 12.0 | 95% | [48.08, 55.92] |
| Daily service time | 64 | 104.5 | 20.0 | 90% | [100.39, 108.61] |
| Sensor calibration output | 100 | 23.8 | 5.0 | 99% | [22.51, 25.09] |
Formula Used
For a z confidence interval for a population mean, the calculator uses the known population standard deviation and the normal critical value.
Two-sided interval:
CI = x̄ ± zα/2 × (σ / √n)
When finite population correction is selected, the standard error becomes:
(σ / √n) × √((N - n) / (N - 1))
One-sided bounds:
Lower bound = x̄ − zα × SE
Upper bound = x̄ + zα × SE
- x̄ = sample mean
- σ = known population standard deviation
- n = sample size
- SE = standard error
- z = critical z score from the selected confidence level
- N = population size when finite population correction applies
How to Use This Calculator
- Choose whether you want to use summary statistics or raw sample data.
- Select the interval type: two-sided, lower bound, or upper bound.
- Pick a common confidence level or enter a custom percentage.
- Enter the known population standard deviation.
- For summary mode, enter the sample mean and sample size.
- For raw mode, paste values separated by commas, spaces, or new lines.
- Enable finite population correction only when sampling from a limited population without replacement.
- Click Calculate Interval to see the result above the form, view the graph, and export the report.
FAQs
1) What does this calculator estimate?
It estimates a confidence interval for a population mean using the z distribution. This is appropriate when the population standard deviation is known or when that assumption is justified by the problem setup.
2) When should I use a z interval instead of a t interval?
Use a z interval when the population standard deviation is known. If sigma is unknown and estimated from the sample, a t interval is usually the better choice.
3) What is the meaning of a 95% confidence level?
It means that if the same sampling method were repeated many times, about 95% of the constructed intervals would contain the true population mean.
4) Why does the interval become wider at higher confidence levels?
Higher confidence needs a larger z critical value. That increases the margin of error, so the interval widens to reflect greater certainty.
5) Can I paste raw observations instead of summary values?
Yes. In raw mode, the calculator reads your sample values, computes the sample mean and count, and then builds the z interval using the sigma you provide.
6) What is finite population correction?
It reduces the standard error when your sample is taken from a relatively small finite population without replacement. It matters most when the sample is a noticeable share of the population.
7) What do one-sided bounds mean?
A one-sided bound gives only a lower or upper confidence limit. It is useful when the question focuses on proving a minimum or maximum plausible mean.
8) Can I export the results?
Yes. After calculation, use the CSV button for spreadsheet-friendly data or the PDF button for a clean report you can save, print, or share.