# Normal Probability Calculator for Sampling Distributions

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, population standard deviation, 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):

Population Mean ($$\mu$$):
Population St. Dev. ($$\sigma$$):
Sample Size: ($$n$$)
Two-Tailed:
$$\le \bar X \le$$
Left-Tailed:
$$\bar X \le$$
Right-Tailed:
$$\bar X \ge$$

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 $$X_i ~ N(\mu, \sigma^2)$$, 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 $$\bar X$$.