Calculator Inputs
Example Data Table
| Scenario | Mode | Parameters | Question | Standard Error | Approximate Result |
|---|---|---|---|---|---|
| Exam Scores | Sample Mean | μ = 50, σ = 12, n = 36 | P(48 ≤ x̄ ≤ 52) | 2.0000 | 0.6827 |
| Daily Output | Sample Mean | μ = 100, σ = 15, n = 64 | P(x̄ ≥ 103) | 1.8750 | 0.0548 |
| Survey Support | Sample Proportion | p = 0.40, n = 200 | P(0.33 ≤ p̂ ≤ 0.47) | 0.0346 | 0.9564 |
| Quality Checks | Sample Mean | μ = 250, σ = 20, n = 49 | 95% central interval for x̄ | 2.8571 | 244.4000 to 255.6000 |
Formula Used
For sample means
x̄ is approximately normal with center μ and standard error:
SE = σ / √n
If sampling is without replacement from a finite population, then SE adjusted = (σ / √n) × √((N - n) / (N - 1)).
For sample proportions
p̂ is approximately normal with center p and standard error:
SE = √(p(1 - p) / n)
Finite population correction can also multiply the standard error when sampling without replacement.
Z score conversion
z = (observed value - center) / SE
Probabilities come from the standard normal curve after converting the bound or interval into z scores.
Central coverage interval
center ± zα/2 × SE
This gives the middle percentage of the sampling distribution for the selected confidence level.
How to Use This Calculator
- Choose whether you want a sample mean or sample proportion analysis.
- Enter the relevant population values and your sample size.
- Select the analysis type: lower tail, upper tail, between bounds, or central interval.
- Enter a threshold, two bounds, or a confidence level as needed.
- Enable finite population correction only when sampling without replacement from a limited population.
- Press the calculate button to see the result above the form.
- Review the standard error, z scores, probability, interval, and interpretation.
- Use the CSV or PDF buttons to export your result summary.
Frequently Asked Questions
1. What does this calculator estimate?
It estimates sampling distribution behavior using central limit theorem logic. You can compute standard errors, tail probabilities, interval probabilities, and central coverage intervals for sample means or sample proportions.
2. When is the central limit theorem reliable?
It is usually stronger with larger samples. For means, n of 30 or more is a common rule. For proportions, both np and n(1-p) should usually be at least 10.
3. Why does sample size matter so much?
Larger samples reduce standard error. That makes the sampling distribution tighter around the true center, so probabilities and intervals become more precise and less spread out.
4. What is standard error?
Standard error measures how much a sample statistic varies from one sample to another. It is the spread of the sampling distribution, not the spread of raw data values.
5. What is finite population correction?
It reduces standard error when you sample without replacement from a limited population. The correction matters more when the sample is a noticeable share of the full population.
6. What is the difference between x̄ and p̂?
x̄ is the sample mean for numerical measurements. p̂ is the sample proportion for yes or no outcomes. Both can be approximated with a normal curve under suitable conditions.
7. Why can proportion intervals look unrealistic sometimes?
Normal approximations can extend slightly below 0 or above 1 when proportions are near extremes or samples are small. That signals the approximation may be weak for that setup.
8. Does this guarantee a real sample result?
No. It gives an approximate long run pattern under stated assumptions. Any single sample may differ, but repeated sampling should follow the modeled behavior more closely.