What is a sampling distribution?

A sampling distribution is the probability distribution of a statistic (like the mean or proportion) calculated from all possible samples of a given size drawn from a population. It describes how the statistic varies from sample to sample.

What is standard error?

Standard error (SE) is the standard deviation of the sampling distribution. For the mean, SE = σ/√n. It measures how much sample means are expected to vary from the true population mean. Larger samples produce smaller standard errors.

How does sample size affect the sampling distribution?

As sample size increases, the sampling distribution becomes narrower (smaller standard error) and more normally distributed (Central Limit Theorem). Larger samples give more precise estimates of population parameters.

What is the Central Limit Theorem?

The Central Limit Theorem states that the sampling distribution of the mean approaches a normal distribution as sample size increases, regardless of the shape of the population distribution. This generally applies when n ≥ 30.

How do I interpret a z-score in this context?

A z-score tells you how many standard errors your sample mean is from the population mean. A z-score of 2.0 means your sample mean is 2 standard errors above the population mean, which has about a 4.5% probability of occurring by chance.

What is a confidence interval?

A confidence interval provides a range of plausible values for the population parameter. A 95% CI means that if you repeated your sampling many times, 95% of the intervals constructed would contain the true parameter value.

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Sampling Distribution Calculator

Calculate sampling distributions, standard error, and confidence intervals for statistical analysis.

Population Mean (μ) Population Std Dev (σ) Sample Size (n) Sample Mean (x̄)

Standard Error

Z-Score

P-value (two-tailed)

95% CI Lower

95% CI Upper