Enter large-sample inputs
Worked examples
| Interval type | Inputs | Confidence | Point estimate | Approximate interval |
|---|---|---|---|---|
| One-sample mean | x̄ = 52.4, s = 12.1, n = 64 | 95% | 52.4 | 49.435 to 55.365 |
| One-sample proportion | x = 138, n = 200 | 95% | 0.690 | 0.626 to 0.754 |
| Difference of means | x̄1 = 74.8, s1 = 11.2, n1 = 80; x̄2 = 70.1, s2 = 10.4, n2 = 75 | 95% | 4.700 | 1.302 to 8.098 |
| Difference of proportions | x1 = 96, n1 = 150; x2 = 78, n2 = 145 | 95% | 0.102 | -0.010 to 0.214 |
Large-sample interval formulas
1) One-sample mean interval
Formula: x̄ ± z × (s / √n)
Use this when you estimate one population mean from a large sample. The center is the sample mean, and the margin depends on the z critical value and standard error.
2) One-sample proportion interval
Formula: p̂ ± z × √[p̂(1 − p̂) / n]
Use this when the data records success or failure. The estimate is the sample proportion, and the interval reflects normal approximation uncertainty.
3) Difference of means interval
Formula: (x̄1 − x̄2) ± z × √[(s1² / n1) + (s2² / n2)]
Use this for two independent large samples when you compare average outcomes. The interval estimates the true difference between population means.
4) Difference of proportions interval
Formula: (p̂1 − p̂2) ± z × √[(p̂1(1 − p̂1)/n1) + (p̂2(1 − p̂2)/n2)]
Use this for two independent groups with binary outcomes. The interval estimates how much the population proportions differ.
How to use this calculator
- Choose the interval type that matches your data.
- Select a standard confidence level or enter a custom z value.
- Enter the sample statistics for one group or two groups.
- Press Calculate interval to generate the bounds and interpretation.
- Review the margin of error, standard error, and assumptions warning.
- Use the CSV or PDF buttons to export the result summary.
- Read the graph to see the point estimate and interval span.
Frequently asked questions
1) What is a large-sample interval?
It is a confidence interval built with a normal approximation. As sample size grows, the sampling distribution becomes more stable, so z-based bounds work well for many means and proportions.
2) When should I use this instead of a small-sample method?
Use it when sample sizes are sufficiently large and the normal approximation is reasonable. For small samples, exact or t-based procedures are often safer and more accurate.
3) Why does the calculator ask for a z critical value?
The z critical value controls interval width. Higher confidence needs a larger z value, which increases the margin of error and makes the interval wider.
4) What does the margin of error tell me?
It shows how far the interval extends from the point estimate in each direction. Smaller standard errors or lower confidence levels produce narrower margins.
5) Why can a difference interval cross zero?
If zero lies inside the interval, the data is consistent with no population difference. That does not prove equality, but it weakens evidence for a real gap.
6) What checks matter for proportion intervals?
A common rule is to have at least 10 successes and 10 failures in each relevant sample. This helps the normal approximation behave more reliably.
7) Can I use sample standard deviation for a mean interval?
Yes. In large samples, using the sample standard deviation in the standard error is common because the large-sample normal approximation remains effective.
8) What do the export buttons save?
They save the result summary table. CSV works well for spreadsheets, while PDF is useful for reports, sharing, documentation, and audit-ready records.