Calculate probabilities for sample means using the Central Limit Theorem and normal distribution.
Last updated: March 2026
A sampling distribution is the probability distribution of a statistic (like the sample mean) calculated from repeated samples of the same population. If you drew many random samples from a population and calculated the mean of each, the distribution of those means would be the sampling distribution of the mean.
The Central Limit Theorem (CLT) is the foundation: regardless of the original population's shape, the sampling distribution of the mean is approximately normal (bell-shaped) when n is sufficiently large (typically n ≥ 30). This is remarkably powerful—it means you can use normal distribution methods for inference even if the raw data isn't normally distributed.
The sampling distribution has a mean equal to the population mean (μ) and standard deviation called the standard error, equal to σ/√n. This calculator finds the probability of observing a sample mean within a range given the population parameters and sample size.
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Because sample means vary less than individual observations. SE = σ/√n decreases with larger n. If you average 100 values, the average is more stable than any single value.
It measures how many standard errors your sample mean is from the population mean. z=1.5 means 1.5 standard errors above μ. Larger |z| indicates a more extreme sample mean.
The Central Limit Theorem states that for n ≥ 30, the sampling distribution is approximately normal regardless of population shape. For smaller n, you need the population itself to be normal.
Larger n makes SE smaller, so the distribution gets narrower and taller. Doubling n reduces SE by 1/√2. To halve SE, you need to quadruple n.
Yes, as long as n ≥ 30 (or n ≥ 15 for less extreme skewness). The CLT guarantees the sampling distribution is approximately normal, even if population data looks nothing like normal.
That's fine. Negative z means sample mean is below population mean. P(X̄ ≥ x̄) is just 1 minus the CDF value. The calculator handles this automatically.
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