Results
This section appears below the header and above the form after calculation.
Calculator
Plotly Graph
The chart shows the sampling distribution around the estimate and highlights the interval or bound.
Calculation History
Each new calculation is appended below. Export the history at any time.
| # | Timestamp | Mode | Style | Confidence | Estimate | Critical | Std. Error | Margin | Lower | Upper |
|---|---|---|---|---|---|---|---|---|---|---|
| No calculations yet. | ||||||||||
Example Data Table
These examples show typical use cases for means, proportions, and one-sided bounds.
| Scenario | Estimate | Spread / Input | n | Confidence | Interval / Bound |
|---|---|---|---|---|---|
| Mean interval with known σ | 52.40 | σ = 5.20 | 64 | 95% | 51.13 to 53.67 |
| Mean interval with unknown s | 87.30 | s = 12.50 | 25 | 99% | 80.45 to 94.15 |
| Proportion interval from counts | 0.740 | 148 / 200 | 200 | 90% | 0.689 to 0.791 |
| Lower one-sided mean bound | 16.80 | σ = 3.10 | 49 | 95% | 16.07 to ∞ |
Formula Used
1. Confidence Level and Alpha
Confidence Level C and significance level α are linked by:
α = 1 − C
For a two-sided interval, each tail uses α / 2. For a one-sided bound, one tail uses all of α.
2. Mean Interval with Known Population Deviation
SE = σ / √n
ME = z* × SE
CI = x̄ ± ME
3. Mean Interval with Unknown Population Deviation
SE = s / √n
ME = t* × SE
CI = x̄ ± ME
This page uses a strong analytical approximation for the t critical value, which is accurate for practical calculator work.
4. Proportion Interval
p̂ = x / n
SE = √[ p̂(1 − p̂) / n ]
ME = z* × SE
CI = p̂ ± ME
5. Finite Population Correction
When sampling without replacement from a limited population:
FPC = √[(N − n) / (N − 1)]
Adjusted SE = SE × FPC
How to Use This Calculator
- Choose the estimation mode for a known deviation, unknown deviation, or proportion.
- Select whether you want a two-sided interval or a one-sided bound.
- Enter the confidence level or alpha. The paired field updates automatically.
- Provide sample size and the required mean, deviation, successes, or sample proportion inputs.
- Turn on finite population correction only when sampling without replacement from a limited population.
- Press the calculate button to show the result above the form, update the graph, and store the row in history.
- Use the CSV and PDF buttons to export saved calculations for reporting or review.
FAQs
1. What does a confidence level show?
A confidence level states how often the method would capture the true population value over many similar samples. A 95% level means the procedure succeeds about 95% of repeated times.
2. When should I use z instead of t?
Use z when the population standard deviation is known or a normal approximation is appropriate. Use t when you estimate spread from the sample and the population deviation is unknown.
3. What is the significance level α?
Alpha is the remaining tail probability after choosing the confidence level. For a 95% confidence level, α equals 0.05. Two-sided intervals split that value across both tails.
4. Why does a larger sample size narrow the interval?
A bigger sample lowers the standard error because sampling variability shrinks as n grows. Smaller standard error reduces the margin of error and makes the interval tighter.
5. What is the difference between two-sided and one-sided bounds?
A two-sided interval estimates both lower and upper limits. A one-sided bound only protects one direction, so it gives a single limit and uses the full tail allocation on one side.
6. Can I use this calculator for proportions?
Yes. Choose the proportion mode and enter either successes with sample size or a direct sample proportion. The calculator then estimates the interval around that population proportion.
7. Why would I use finite population correction?
Use finite population correction when sampling without replacement from a limited population and your sample is a noticeable share of that population. It reduces standard error slightly.
8. Does 95% confidence mean a 95% probability for this interval?
Not exactly. In classical statistics, the true value is fixed and the interval changes across repeated samples. The 95% statement describes the long-run performance of the method.