**Instructions:** This Normal Probability Calculator for Sampling Distributions will compute normal distribution probabilities for sample means \(\bar X \), using the form below. Please type the population mean (\(\mu\)), population standard deviation (\(\sigma\)), and sample size (\(n\)), and provide details about the event you want to compute the probability for (for the standard normal distribution, the mean is 0 and the standard deviation is 1):

#### More About this Normal Distribution Probability Calculator for Sampling Distributions Tool

When a sequence of normally distributed variables \(X_1, X_2, ...., X_n\) is averaged, we get the sample mean

\[\bar X = \frac{1}{n}\sum_{i=1}^n X_i\]Since any linear combination of normal variables is also normal, the sample mean \(\bar X\) is also normally distributed (assuming that each \(X_i\) is normally distributed). The distribution of \(\bar X\) is commonly referred as to the *sampling distribution of sample means*.

Assuming that \(X_i \sim N(\mu, \sigma^2)\), for all \(i = 1, 2, 3, ...n\), then \(\bar X\) is normally distributed with the same common mean \(\mu\), but with a variance of \(\displaystyle\frac{\sigma^2}{n}\). This tells us that \(\bar X\) is also centered at \(\mu \) but its dispersion is less than that for each individual \( X_i \). Indeed, the larger the sample size, the smaller the dispersion of \(\bar X\).

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