Enter Your Values
Example Data Table
| Scenario | Type | Inputs | Confidence | Illustrative Output |
|---|---|---|---|---|
| Quality score study | Mean | x̄ = 42.7, s = 6.2, n = 25 | 95% | Two-sided mean interval around 42.7 |
| Customer conversion test | Proportion | 86 successes, 120 trials | 95% | Wilson interval for population proportion |
| Process spread review | Variance | s² = 18.4, n = 20 | 99% | Variance interval from chi-square limits |
Formula Used
1) Mean bound or interval
Standard error: SE = s / √n when sigma is unknown, or SE = σ / √n when sigma is known.
Two-sided interval: x̄ ± critical value × SE
Lower one-sided bound: x̄ − critical value × SE
Upper one-sided bound: x̄ + critical value × SE
2) Proportion bound or interval
Point estimate: p̂ = x / n
Wald method: p̂ ± z × √[p̂(1 − p̂) / n]
Wilson method uses the score-based center and adjusted half-width for more stable bounds, especially with smaller samples.
3) Variance bound or interval
For a normal population, the variance interval uses the chi-square distribution.
Two-sided interval: ((n − 1)s² / χ² upper, (n − 1)s² / χ² lower)
Lower one-sided bound: (n − 1)s² / χ²(1 − α)
Upper one-sided bound: (n − 1)s² / χ²(α)
This page uses standard critical-value approximations suitable for fast reporting and teaching workflows.
How to Use This Calculator
- Select the calculation type: mean, proportion, or variance.
- Choose whether you need a two-sided interval, a lower bound, or an upper bound.
- Enter the confidence level as a percentage.
- Fill in the relevant sample values for your chosen method.
- Click Calculate Bounds to place the result above the form.
- Review the estimate, bound values, critical value, and chart.
- Use the CSV or PDF buttons to export the visible result block.
Frequently Asked Questions
1) What is a confidence bound?
A confidence bound is a statistical limit that estimates where a population value likely falls. It can be lower, upper, or part of a two-sided interval.
2) When should I use a one-sided bound?
Use a one-sided bound when only one direction matters, such as proving a minimum quality level or ensuring a maximum error rate stays below a threshold.
3) Why does the calculator offer both z and t methods?
The z method is used when the population standard deviation is known. The t method is preferred when it is unknown and estimated from the sample.
4) Why is Wilson available for proportions?
Wilson bounds are often more stable than simple Wald bounds, especially when the sample size is modest or the estimated proportion is near 0 or 1.
5) What assumptions matter for variance bounds?
Variance bounds using chi-square methods assume the underlying population is normally distributed. Strong departures from normality can affect the accuracy of the interval.
6) Does a 95% confidence level mean 95% probability the parameter is inside?
Not exactly. It means that if the same sampling process were repeated many times, about 95% of similarly constructed intervals would capture the true parameter.
7) Can I use this for classroom and reporting work?
Yes. It is useful for assignments, dashboards, internal reviews, and quick statistical summaries. Highly regulated work should still be verified with validated software.
8) What does the graph represent?
The graph is a visual guide centered on the estimate. It marks lower and upper limits so you can quickly understand interval width and directional bounds.