Sampling Distribution Calculator

What is Sampling Distribution Calculator?

A Sampling Distribution Calculator analyzes the variability of statistics across repeated samples. Instead of focusing on individual data points, it models the distribution of a statistic—commonly the sample mean \(\bar X\) or sample proportion \(\hat p\)—under repeated random sampling from a population with mean \(\mu\) and standard deviation \(\sigma\). This perspective enables probability questions (e.g., how likely is \(\bar X\) to exceed a threshold?), hypothesis tests, and confidence intervals. Key results include exact formulas under normality and large-sample approximations through the Central Limit Theorem (CLT).

About the Sampling Distribution Calculator

Enter summary statistics or raw data and this tool returns standard errors, z/t scores, tail probabilities, and confidence intervals. It supports one- and two-sample means, proportions, and difference statistics. Assumption checks (independence, adequate \(n\), nonzero \(\sigma\) or valid \(s\)) are surfaced, with notes on when to prefer t instead of z and when normal approximations for proportions are reasonable. Every step—standardization, critical values, and interval construction—appears alongside neatly formatted formulas that resize to fit any screen.

How to Use this Sampling Distribution Calculator

1. Choose a statistic: \(\bar X\), \(\hat p\), difference of means, or difference of proportions.
2. Provide inputs (\(\mu,\sigma,n\) or \(\bar x,s,n\); for proportions, \(p\) or \(\hat p, n\)).
3. Select a task: probability, hypothesis test, or confidence interval; pick tail(s) and confidence level if needed.
4. Click calculate to view standardization and result; adjust sample size to explore precision via SE.
5. Download or copy the step-by-step formulas and conclusions for reports or assignments.

Examples

Example 1: Mean probability

\(\mu=50,\; \sigma=10,\; n=36\). Find \(\Pr(\bar X>53)\).

Example 2: Proportion probability

\(p=0.60,\; n=200\). Find \(\Pr(\hat p<0.55)\).

Example 3: Two-sample CI for means

\(\bar x_1=72, s_1=8, n_1=40;\; \bar x_2=68, s_2=10, n_2=50\). 95% CI for \(\mu_1-\mu_2\).

Example 4: t-based CI (small n)

\(\bar x=5.2, s=1.5, n=9\). 95% CI for \(\mu\).

FAQs

When do I use z versus t?

Use z when \(\sigma\) is known or \(n\) is very large; otherwise use t with \(s\) and appropriate degrees of freedom.

What conditions justify the normal approximation for \(\hat p\)?

Typically when \(np\ge10\) and \(n(1-p)\ge10\) (or with \(\hat p\) after observing data) and sampling is independent.

How does increasing sample size affect results?

It reduces standard error \((\propto1/\sqrt n)\), narrowing confidence intervals and making extreme sample statistics less likely.

Can R² or other fit metrics replace sampling distributions?

No. Fit metrics describe in-sample fit; sampling distributions support inference about populations and uncertainty quantification.

What if my data are skewed or heavy‑tailed?

For means, the CLT often helps at larger \(n\); for small \(n\), consider robust methods or transformations.

How do I get a two‑sided p‑value?

Double the smaller tail area: \(p=2\min\{\Pr(Z\le z),\,\Pr(Z\ge z)\}\) (or use the t distribution accordingly).

Does the calculator handle paired data?

Yes—compute differences within pairs, then treat the mean difference with one‑sample methods on those differences.

What about difference of proportions?

The SE is \(\sqrt{p_1(1-p_1)/n_1+p_2(1-p_2)/n_2}\); use pooled \(p\) in certain hypothesis tests.