Sampling Distribution Calculator

Take repeated samples of size $n$ and record $\bar{x}$ each time. Those means form their own distribution — the sampling distribution of the mean — and it is narrower than the population:

$\mu_{\bar{x}} = \mu, \qquad \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

That second quantity is the standard error (SE). It is not the population standard deviation, and substituting $\sigma$ for it is the single most common error on this topic.

When is $\bar{x}$ normal? Either condition suffices:

1. The population is normal — then $\bar{x}$ is exactly normal for any $n$, even $n = 2$.
2. The Central Limit Theorem — for any population with finite variance, $\bar{x}$ is approximately normal once $n$ is large. $n \ge 30$ is the usual rule of thumb; strongly skewed populations need more.

Also assumed: observations are independent. If sampling without replacement from a finite population, that holds well enough when $n \le 0.05N$; otherwise apply the finite population correction $\sqrt{(N-n)/(N-1)}$.

Standardise with the standard error, then read the cumulative distribution function $\Phi(z) = P(Z \le z)$:

$z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}$

Then $P(\bar{x} > a) = 1 - \Phi(z_a)$ and $P(a < \bar{x} < b) = \Phi(z_b) - \Phi(z_a)$.

Is $\sigma$ known? If you only have the sample standard deviation $s$, the statistic follows a $t$ distribution with $n-1$ degrees of freedom, not $z$. For $n \ge 30$ the two are close enough that many textbooks use $z$ anyway.

Normal approximation to the binomial. A count $X \sim \text{Bin}(n,p)$ is approximately $N(np,\; np(1-p))$ when $np \ge 10$ and $n(1-p) \ge 10$. Apply a continuity correction of $\pm 0.5$.

• Dividing by $\sigma$ instead of $\sigma/\sqrt{n}$. That answers a question about one individual, not about a mean of $n$.
• Dividing by $n$ instead of $\sqrt{n}$. The standard error shrinks with the square root: quadrupling $n$ only halves it.
• Assuming the CLT fixes everything. It says nothing about outliers, dependent observations, or a biased sampling method. A large $n$ of a bad sample is still a bad sample.
• Using $z$ when $\sigma$ is unknown and $n$ is small. Use $t_{n-1}$.
• Skipping the continuity correction in the binomial approximation — with $n$ near the $np \ge 10$ boundary it shifts the answer noticeably.
• Confusing $P(X > a)$ with $P(\bar{x} > a)$. The second is far smaller, because means vary less than individuals.