Calculator Inputs
Example Data Table
| Scenario | Estimate | Sample Size | Confidence | Lower Bound | Upper Bound |
|---|---|---|---|---|---|
| Mean, unknown deviation | 78.40 | 36 | 95% | 74.83 | 81.97 |
| Mean, known deviation | 152.00 | 64 | 99% | 148.13 | 155.87 |
| Proportion, Wilson | 0.6200 | 250 | 95% | 0.5585 | 0.6782 |
| Mean with finite population adjustment | 41.20 | 40 | 90% | 39.97 | 42.43 |
These rows illustrate typical outputs for different confidence interval assumptions.
Formula Used
Mean Interval with Known Population Deviation
Standard Error: SE = σ / √n
Margin of Error: ME = z × SE
Confidence Interval: x̄ ± ME
Mean Interval with Sample Deviation
Standard Error: SE = s / √n
Margin of Error: ME = t × SE
Confidence Interval: x̄ ± ME
Proportion Interval
Wald Standard Error: SE = √[ p̂(1 − p̂) / n ]
Wald Interval: p̂ ± z × SE
Wilson Interval: adjusted center and adjusted half width using z and n for improved small sample performance.
Finite Population Correction
FPC: √[(N − n) / (N − 1)]
This calculator multiplies the standard error by the finite population correction when that option applies.
How to Use This Calculator
- Select the interval type that matches your dataset.
- Enter the confidence level and sample size.
- Fill the mean and deviation fields for mean intervals.
- Enter successes for proportion intervals.
- Optionally add a population size and enable correction.
- Set decimal places and an optional unit label.
- Press the calculate button to view bounds, margin, width, and graph.
- Use the export buttons to save the result as CSV or PDF.
Frequently Asked Questions
1. What does a confidence interval range show?
It gives a lower and upper bound for a population value, based on your sample. The chosen confidence level describes how often intervals built this way would contain the true value across repeated sampling.
2. When should I use a z interval?
Use a z interval when the population deviation is known or when a normal approximation is justified. This is common for large samples or well-studied processes with stable variability estimates.
3. When should I use a t interval?
Use a t interval when estimating a population mean and the population deviation is unknown. It adjusts the critical value using sample size, which matters most for smaller datasets.
4. Why is the interval wider at higher confidence levels?
Higher confidence requires a larger critical value. That increases the margin of error, so the final interval becomes wider. More certainty about coverage needs a broader range.
5. What is the margin of error?
The margin of error is the distance from the point estimate to either interval bound. It depends on the critical value, variability, sample size, and whether finite population correction is used.
6. Why offer Wilson and Wald methods for proportions?
Wilson intervals usually behave better for smaller samples or proportions near zero or one. Wald intervals are simpler and familiar, but they can be less stable in edge cases.
7. What does finite population correction change?
It reduces the standard error when your sample is a meaningful share of a limited population. That can narrow the interval, especially when the sample size is large relative to population size.
8. Does a 95% interval mean a 95% chance the true value is inside?
Not exactly. In classical frequentist terms, the true value is fixed. The 95% statement refers to the long-run success rate of the interval-building method, not a probability on one finished interval.